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All questions of Data Sufficiency: 600 Level for GMAT Exam

Compound P contains just three chemicals — X, Y, and Z. The three chemicals are required to be in the following ratio: 5 grams of X, 15 grams of Y, and z grams of Z. If 25 grams of X and a proportional amount of Y are added to the compound, what is the weight of Z that must also be added to maintain the required overall proportion?
(1) The weight of Compound P is always three times the weight of Z.
(2) The weight of Z is always twice the weight of X.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'D'. Can you explain this answer?

Pioneer Academy answered

Statement (1) says that the weight of Compound P is always three times the weight of Z.

If the weight of Compound P is always three times the weight of Z, we can set up an equation based on the given ratio:

5g of X : 15g of Y : zg of Z = 25g of X : yg of Y : 3zg of Z

Since the weight of Compound P is always three times the weight of Z, we have:

25g of X + yg of Y + 3zg of Z = 3zg of Z

This equation simplifies to:

25g of X + yg of Y = 2zg of Z

We can see that the equation only involves X, Y, and Z, which are the chemicals mentioned in the problem. Therefore, statement (1) alone is sufficient to answer the question.

Now, let's consider statement (2), which states that the weight of Z is always twice the weight of X.

If the weight of Z is always twice the weight of X, we can set up another equation based on the given ratio:

5g of X : 15g of Y : zg of Z = 25g of X : yg of Y : 2xg of X

Since the weight of Z is always twice the weight of X, we have:

25g of X + yg of Y + 2xg of X = zg of Z

This equation involves X, Y, and Z, which are the chemicals mentioned in the problem. Therefore, statement (2) alone is also sufficient to answer the question.

Both statement (1) and statement (2) individually provide enough information to determine the weight of Z that must be added to maintain the required overall proportion. Therefore, the correct answer is (D) EACH statement ALONE is sufficient to answer the question asked.

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How many positive prime numbers are less than the integer n?
(1) 14 < n < 20
(2) 13 < n < 17
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'B'. Can you explain this answer?

Rahul Kapoor answered

Statement (1): 14 < n < 20
The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. There are a total of 8 prime numbers in this range.

Statement (2): 13 < n < 17
The prime numbers less than 17 are 2, 3, 5, 7, 11, and 13. There are a total of 6 prime numbers in this range.

Since statement (2) provides a narrower range for n (13 < n < 17) compared to statement (1) (14 < n < 20), statement (2) alone is sufficient to determine the number of positive prime numbers less than n. Therefore, the answer is option B: Statement (2) alone is sufficient, but statement (1) alone is not sufficient to answer the question asked.

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A certain mixture of paint requires blue, yellow, and red paints in ratios of 2:3:1, respectively, and no other ingredients. If there are ample quantities of the blue and red paints available, is there enough of the yellow paint to make the desired amount of the mixture?
(1) Exactly 20 quarts of the mixture are needed.
(2) Exactly 10 quarts of the yellow paint are available.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Kalyan Nair answered

Statement 1:
Exactly 20 quarts of the mixture are needed.

Statement 2:
Exactly 10 quarts of the yellow paint are available.

Approach:
We need to determine whether there is enough yellow paint to make the desired amount of the mixture. To do this, we can compare the available yellow paint with the required amount based on the given ratio.

Analysis:
Let's assume that the total amount of paint required for the mixture is represented by the variable 'x'. Based on the given ratio, we can determine the amounts of blue, yellow, and red paints required as follows:
- Blue paint: (2/6) * x = x/3
- Yellow paint: (3/6) * x = x/2
- Red paint: (1/6) * x = x/6

We need to compare the amount of yellow paint available with x/2.

From Statement 1:
Exactly 20 quarts of the mixture are needed.

This statement alone does not provide any information about the availability of yellow paint. Therefore, it is not sufficient to answer the question.

From Statement 2:
Exactly 10 quarts of the yellow paint are available.

This statement gives us the exact amount of yellow paint available, which is 10 quarts. However, without knowing the total amount of paint required for the mixture, we cannot determine whether there is enough yellow paint. Therefore, this statement alone is not sufficient to answer the question.

Combined Analysis:
Combining both statements, we know that exactly 20 quarts of the mixture are needed and exactly 10 quarts of yellow paint are available.

Since the total amount of paint required is 20 quarts, we can determine the required amount of yellow paint using the given ratio:
Yellow paint = (3/6) * 20 = 10 quarts

Since the available yellow paint is exactly 10 quarts, which is equal to the required amount, we can conclude that there is enough yellow paint to make the desired amount of the mixture.

Therefore, both statements together are sufficient to answer the question.

Answer:
Option (C)

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A mixture is composed of ingredients A, B, C, and D. How much more (in grams) of ingredient A than ingredient D is in the mixture?
(1) The ingredients A, B, C, and D are in the ratio 10:5:4:2, respectively.
(2) The amount (in grams) of ingredient B is 4 more than that of ingredient C.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Talent Skill Learning answered

Statement (1): The ingredients A, B, C, and D are in the ratio 10:5:4:2, respectively.
This statement provides the ratio of the ingredients but does not give us any specific quantities. Without knowing the actual quantities, we cannot determine the difference between the amounts of ingredient A and ingredient D. Therefore, statement (1) alone is not sufficient.

Statement (2): The amount (in grams) of ingredient B is 4 more than that of ingredient C.
This statement only provides information about ingredients B and C, but it does not mention anything about ingredients A or D. Without knowing the quantities of ingredients A and D, we cannot determine the difference between them. Hence, statement (2) alone is not sufficient.

Considering both statements together:
While statement (1) gives us the ratio of the ingredients, and statement (2) gives us information about ingredients B and C, we still don't have any specific quantities for ingredients A and D. Therefore, when we combine both statements, we still cannot determine the exact difference in grams between ingredient A and ingredient D. Thus, both statements together are not sufficient.

Therefore, the answer is option C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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Jar 1 contains a solution which is 30% nitric acid and 70% water, while Jar 2 contains a solution which is 60% nitric acid and 40% water. How many ml of Jar 1 should be mixed with 100 ml of Jar 2 solution, in order to form an intended solution S?
(1) Solution S is intended to be 40% nitric acid and 60% water.
(2) If similar quantities of solution S and the solution in Jar 1 are taken, then solution S will contain 4/3 times the nitric acid than that present in Jar 1.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'D'. Can you explain this answer?

Rahul Kapoor answered

Given
Jar 1 - 30% Nitric acid & 70% water
Jar 2 - 60% Nitric acid & 40% water.
Quantity of Jar 2 = 100ml
Quantity of Jar 1 = X ml
No info in prompt about intended solution S.
Lets look at St.1 Alone
Intended Solution S = 40% Nitric acid & 60% water.
Quantity of Solution S = (100+X) ml
Hence Quantity of Nitric acid = Quantity of Nitric acid in Jar 1 + Quantity of Nitric acid in Jar 2
40%*(100+X) = 30% of X + 60% of 100
Equation in single unknown, solving which we can deduce Quantity of Jar 1 hence, statement 1 is Sufficient.
Statement 2 Alone
When similar quantities of Solution S & Jar 1 are taken, the Nitric acid quantity in Solution 4/3 times of Nitric acid in Jar 1.
which means, if 100 ml of Solution S & Jar 1 are taken,
Quantity of Nitric acid in Jar 1 is 30% of 100 ml = 30ml
Therefore, Quantity of Nitric acid in Solution S = 4/3*(30) = 40 ml
Hence Solution S contains 40% Nitric acid & 60% water.
Which is similar to info provided in St.1, solving which we can deduce Quantity of Jar 1. Statement 2 is Sufficient.
Statement 1 & Statement alone are Sufficient.

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Three liquids, A, B and C, are formed by mixing petrol and spirit in varying ratios. What is the percent of petrol in liquid C?
(1) The ratio of petrol and spirit in A and B are 2 : 3 and 3 : 4, respectively.
(2) If 20 liters of A, 21 liters of B and 27 liters of C are mixed, the resulting ratio of petrol and spirit is 29 : 39.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Pioneer Academy answered

To solve this problem, let's analyze each statement:

Statement (1) provides the ratios of petrol and spirit in liquids A and B.

The ratio of petrol to spirit in liquid A is 2:3, and in liquid B, it is 3:4.

However, this statement alone does not provide any information about the composition or ratio of petrol and spirit in liquid C. Therefore, statement (1) alone is not sufficient to answer the question asked.

Now let's consider statement (2). It states that when 20 liters of A, 21 liters of B, and 27 liters of C are mixed, the resulting ratio of petrol to spirit is 29:39.

This statement provides information about the overall mixture of A, B, and C, but it does not specifically give the composition of liquid C. Therefore, statement (2) alone is not sufficient to answer the question asked.

Since neither statement alone is sufficient, let's analyze them together:

Combining both statements, we know the ratios of petrol and spirit in liquids A and B, as well as the overall mixture ratio when A, B, and C are mixed.

However, we still don't have any direct information about the composition or ratio of petrol and spirit in liquid C. Therefore, even when considering both statements together, we cannot determine the percent of petrol in liquid C.

Hence, the correct answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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<x> is defined to be the smallest integer greater than or equal to x. What is the value of <x>?
1) x>0
2) x<1
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Talent Skill Learning answered

Statement (1) <x> can take multiple values: 1(0 < x ≤ 1), 2(1 < x ≤ 2) and so on. Not sufficient.
Statement (2) <x> can take multiple values: 1(0 < x ≤ 1), 0(−1 < x ≤ 0) and so on. Not sufficient.
Combined (1) and (2) taken together will result in 0 < x < 1. We can find the unique value of <x>, which is 11. Sufficient.

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What is the value of u?
(1) |u| + |v| = 0
(2) u² + v² = 0
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'D'. Can you explain this answer?

Talent Skill Learning answered

Statement (1): |u| + |v| = 0
The sum of the absolute values of u and v is equal to 0. Since absolute values are always non-negative, for their sum to be 0, both u and v must be 0. Therefore, u = 0. Statement (1) alone is sufficient to determine the value of u.

Statement (2): u² + v² = 0
The sum of the squares of u and v is equal to 0. Since squares are always non-negative, for their sum to be 0, both u²and v² must be 0. This means both u and v must be 0. Therefore, u = 0. Statement (2) alone is also sufficient to determine the value of u.

Since both statement (1) and statement (2) individually provide sufficient information to determine the value of u, the answer is D: EACH statement ALONE is sufficient to answer the question asked.

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If a and b are both two digit integers, is a + b a multiple of 11?
(1) The tens digit of a is equal to the units digit of b, and the tens digit of b is equal to the units digit of a
(2) Both a and b are odd
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'A'. Can you explain this answer?

Rahul Kapoor answered

Statement 1: The tens digit of a is equal to the units digit of b, and the tens digit of b is equal to the units digit of a
This means the numbers will be of the form xy and yx.

On adding, we get (10x + y) + (10y + x) = 11x + 11y
This clearly is divisible by 11
SUFFICIENT

Statement 2: Both a and b are odd
The different values of a and b can be:
a = 23 and b = 35
Here a + b = 58. Not a multiple of 11

a = 11 and b = 33
a + b = 44. Multiple of 11
INSUFFICIENT

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If positive integer x is a multiple of 6 and positive integer y is a multiple of 14, is xy a multiple of 105?
(1) x is a multiple of 9.
(2) y is a multiple of 25.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'B'. Can you explain this answer?

Pioneer Academy answered

Let's analyze the given statements:

Statement (1): x is a multiple of 9. This statement tells us that x is a multiple of 3, which is one of the prime factors of 105. Therefore, if statement (1) is true, xy will be a multiple of 105 regardless of the value of y.

Statement (2): y is a multiple of 25. This statement tells us that y is a multiple of 5, another prime factor of 105. However, it does not provide any information about whether x is a multiple of 3 or 7, so we cannot determine if xy is a multiple of 105 based on statement (2) alone.

Combining both statements, we know that x is a multiple of 3 and y is a multiple of 5. However, we still don't know if x is a multiple of 7. Without knowing if x is divisible by 7, we cannot conclusively determine if xy is a multiple of 105.

Therefore, the answer is (B): Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

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A certain aquarium holds three types of fish: angelfish, swordtails and guppies. What is the ratio of the number of guppies to the number of angelfish?
(1) There are 200 fish in the aquarium
(2) 45 percent of the fish are swordtails, and there are twice as many swordtails as there are angelfish.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'B'. Can you explain this answer?

Rahul Kapoor answered

Statement (1) states that there are 200 fish in the aquarium. However, this information alone does not provide any specific details about the distribution or ratio of the different types of fish in the aquarium. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2) states that 45 percent of the fish are swordtails, and there are twice as many swordtails as there are angelfish. From this information, we can determine the ratio of swordtails to angelfish, but it doesn't provide any direct information about the number of guppies or their ratio to angelfish. Therefore, statement (2) alone is not sufficient to answer the question.

Considering both statements together, we know from statement (1) that there are 200 fish in the aquarium. From statement (2), we know that 45 percent of the fish are swordtails and there are twice as many swordtails as angelfish.

Since statement (2) provides the information about the ratio of swordtails to angelfish, we can determine the number of angelfish in the aquarium. If the number of angelfish is represented by A, then the number of swordtails is 2A. The total number of angelfish and swordtails is A + 2A = 3A.

We also know from statement (1) that there are 200 fish in total. Therefore, 3A = 200.

Solving this equation, we find A = 66.67, which is not a whole number. This indicates that the ratio of guppies to angelfish cannot be determined with certainty, as it depends on the exact number of fish in the aquarium.

Therefore, when both statements are considered together, they are still not sufficient to answer the question.

Based on our analysis, the answer is B: Statement (2) alone is sufficient, but statement (1) alone is not sufficient to answer the question asked.

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If x, y, and z are all positive integers, how many trailing zeros are contained in the product of (5^x)(2^y)(3^z)?
(1) x = 12
(2) x < y
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Rahul Kapoor answered

Statement (1): x = 12.
If x = 12, then the product becomes (5 * 12)(2y)(3z) = 60(2y)(3z). The prime factorization of 60 is 2^2 * 3 * 5. Since we have a 2 and a 5 in the product, we can create trailing zeros by multiplying any number of pairs of 2s and 5s. However, statement (1) alone does not provide any information about y or z, so we cannot determine the number of trailing zeros.

Statement (2): x < y.
Statement (2) provides a relationship between x and y but does not give any specific values. Therefore, statement (2) alone does not provide enough information to determine the number of trailing zeros.

Combining both statements, we know that x = 12 and x < y. However, we still do not have any information about z. Without knowing the values of y and z, we cannot determine the number of trailing zeros.

Therefore, both statements together are sufficient to determine that x = 12 and x < y, but they are not sufficient to answer the question about the number of trailing zeros. The answer is option C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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Is j prime?
(1) j + 1 is not prime
(2) j is odd.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'E'. Can you explain this answer?

Rahul Kapoor answered

Statement 1: j + 1 is not prime
There are several values of j that satisfy statement 1. Here are two:
Case a: j = 3, in which case j+1 = 4 and 4 is not prime. In this case j IS prime
Case b: j = 9, in which case j+1 = 10 and 10 is not prime. In this case j is NOT prime
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: j is odd
There are several values of j that satisfy statement 1. Here are two:
Case a: j = 3 (and 3 is odd). In this case j IS prime
Case b: j = 9 (and 9 is odd). In this case j is NOT prime
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statements 1 and 2 combined
There are several values of j that satisfy BOTH statements. Here are two:
Case a: j = 3, in which case j is odd AND j+1 is not prime. In this case j IS prime
Case b: j = 9, in which case j is odd AND j+1 is not prime. In this case j is NOT prime
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

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What is the average weight of all the students in the class?
(1) The ratio of the average weight of all the boys to girls the average weight of all the girls in the class is 3:2
(2) The number of boys and girls in the class are 40 and 60 respectively
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'E'. Can you explain this answer?

Rahul Kapoor answered

Statement (1): The ratio of the average weight of all the boys to the average weight of all the girls is 3:2. This statement provides information about the relative weights of boys and girls, but it doesn't provide any information about the total number of boys or girls or their individual weights. Therefore, it is not sufficient to determine the average weight of all the students.

Statement (2): The number of boys and girls in the class are 40 and 60, respectively. This statement provides information about the total number of boys and girls in the class but doesn't provide any information about their weights. Knowing the number of students alone is not enough to calculate the average weight of all the students.

Since neither statement alone provides sufficient information, we cannot determine the average weight of all the students in the class by using either statement alone. Therefore, the answer is option E: Statements (1) and (2) together are not sufficient to answer the question asked, and additional data are needed.

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If r is not equal to 0, is r²/|r| < 1?
(1) r > -1
(2) r < 1
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Athira Choudhury answered

No, the expression r^2/|r| is not defined when r is not equal to 0. This is because the denominator |r| represents the absolute value of r, which is always non-negative. Therefore, when r is not equal to 0, the expression r^2/|r| will have a non-zero denominator, making it undefined.

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A certain alloy contains Lead, copper and tin. How many pounds of tin are in 56 pounds of the alloy?
(1) By weight the alloy is 3/7 lead and 5/14 copper.
(2) By weight the alloy is 6 parts lead and 5 parts copper.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'A'. Can you explain this answer?

Rahul Kapoor answered

Statement (1) says that by weight, the alloy is 3/7 lead and 5/14 copper.

From this information, we can determine the weight of tin in the alloy. Since the alloy is composed of lead, copper, and tin, the remaining weight after accounting for lead and copper would be the weight of tin.

Let's assign variables:
L = weight of lead
C = weight of copper
T = weight of tin

According to statement (1), the alloy is 3/7 lead and 5/14 copper. Therefore, we have the following equations:

L + C + T = total weight of the alloy (56 pounds in this case)
L = (3/7)(total weight of the alloy)
C = (5/14)(total weight of the alloy)

Using these equations, we can solve for the weight of tin, T.

From the given information, we can deduce that statement (1) alone is sufficient to determine the weight of tin in the alloy.

Therefore, the correct answer is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

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If x, y, and z are positive integers, is x+y divisible by 5?
1) x+z is divisible by 5
2) y+z is divisible by 5
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'E'. Can you explain this answer?

Rahul Kapoor answered

Statement (1): x+z is divisible by 5.
This statement alone does not provide any information about whether x+y is divisible by 5. We don't have any direct relationship between x+z and x+y.

Statement (2): y+z is divisible by 5.
Similarly, this statement alone does not give us any information about the divisibility of x+y by 5. There is no direct relationship between y+z and x+y.

Since neither statement alone is sufficient to answer the question, and the statements do not provide enough information when considered together, the answer is (E): Statements (1) and (2) together are not sufficient to answer the question asked, and additional data are needed.

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Today, Samson’s Train traveled at a constant rate along a straight North-South route that included the famous 10-mile long Chasm Bridge. Was Samson’s Train on the Chasm Bridge at 2:00 p.m.?

(1) At 9:00 a.m., Samson’s Train was 200 miles north of the Chasm Bridge, and at 4:00 p.m, the train was 140 miles south of Chasm Bridge.
(2) Samson’s Train started its trip at 6:00 a.m., and traveled at a rate of less than 70 miles per hour.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'E'. Can you explain this answer?

Pioneer Academy answered

Statement (1) provides information about the position of Samson's Train at two different times: 9:00 a.m. and 4:00 p.m. We know that at 9:00 a.m., the train was 200 miles north of the Chasm Bridge, and at 4:00 p.m., it was 140 miles south of the bridge. However, we don't have any information about the speed of the train or its direction. Therefore, statement (1) alone is not sufficient to determine if the train was on the Chasm Bridge at 2:00 p.m.

Statement (2) gives the starting time of the trip and mentions that the train traveled at a rate of less than 70 miles per hour. However, there is no information about the position of the train at any specific time or its direction of travel. Therefore, statement (2) alone is not sufficient to determine if the train was on the Chasm Bridge at 2:00 p.m.

When we combine the information from both statements, we still don't have any additional details about the position of the train at 2:00 p.m. Therefore, statements (1) and (2) together are not sufficient to answer the question.

In conclusion, the answer is (E): Statements (1) and (2) together are not sufficient to answer the question asked, and additional data are needed.

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If |x| < 4, what is the value of x?
(1) x is an integer divisible by 3
(2) x is an integer divisible by 2
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Sharmila Singh answered

If |x| is greater than or equal to 0, then x could be any real number.

If |x| is less than 0, then it is not possible since the absolute value of a number cannot be negative.

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Each of the 600 elements of Set X is a distinct integer. How many of the integers in Set X are positive odd integers?
(1) Set X contains 150 even integers.
(2) 70% of the odd integers in Set X are positive.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Talent Skill Learning answered

Statement (1): Set X contains 150 even integers.
From this statement, we know that out of the 600 elements in Set X, 150 are even integers. However, this statement doesn't provide any information about the odd integers or their distribution. We cannot determine the exact number of positive odd integers in Set X based on this statement alone. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): 70% of the odd integers in Set X are positive.
This statement tells us that 70% of the odd integers in Set X are positive. However, we don't have any information about the total number of odd integers in Set X or the total number of integers in Set X. Without knowing these quantities, we cannot determine the number of positive odd integers. Thus, statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we still cannot determine the exact number of positive odd integers in Set X. Statement (1) provides information about even integers, while statement (2) gives information about the percentage of positive odd integers but not their actual count. Without more specific data, we cannot answer the question.

Therefore, both statements together are sufficient to answer the question, but neither statement alone is sufficient. The answer is C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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If k is an integer, is k³ > k?
(1) k is not negative
(2) k is prime
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'B'. Can you explain this answer?

Talent Skill Learning answered

To determine whether k³ > k, let's analyze each statement:

Statement (1): k is not negative
If k is not negative, it means k can be zero or a positive number. We can test this statement with examples:

For k = 0, we have 0³ = 0, which is not greater than 0. So, k³ is not greater than k in this case.
For k = 1, we have 1³ = 1, which is equal to 1. So, k³ is not greater than k in this case.
For k = 2, we have 2³ = 8, which is greater than 2. So, k³ is greater than k in this case.

From the examples above, we can see that statement (1) alone is not sufficient to determine whether k³ > k.

Statement (2): k is prime
If k is prime, it means k is a positive integer greater than 1 that has no divisors other than 1 and itself. We can test this statement with examples:

For k = 2, we have 2³ = 8, which is greater than 2. So, k³ is greater than k in this case.
For k = 3, we have 3³ = 27, which is greater than 3. So, k³ is greater than k in this case.
For k = 4, we have 4³ = 64, which is greater than 4. So, k³ is greater than k in this case.

From the examples above, we can see that statement (2) alone is sufficient to determine that k³ > k.

Therefore, the answer is option B: Statement (2) alone is sufficient, but statement (1) alone is not sufficient to answer the question asked.

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The operation* represents either addition, subtraction, or multiplication of integers, what is the value of 10?
(1) 02 = 2
(2) 2*0 = 2
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'D'. Can you explain this answer?

Rahul Kapoor answered

(1) 0*2 = 2

From this statement, we know that the result of 02 is 2. However, this statement does not provide any direct information about the value of 10. It only tells us the result of a different operation. Therefore, statement (1) alone is not sufficient to determine the value of 1*0.

(2) 2*0 = 2

From this statement, we know that the result of 20 is 2. Similar to statement (1), this statement does not directly provide information about the value of 10. It only tells us the result of a different operation. Therefore, statement (2) alone is not sufficient to determine the value of 1*0.

By analyzing each statement individually, we can see that neither statement alone is sufficient to determine the value of 10. We need additional information to determine the operation represented by the asterisk (). Therefore, the answer is (D) EACH statement ALONE is sufficient to answer the question asked.

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Out of 25 Gryffindor students, 16 play chess, and 11 play quidditch. How many of Gryffindor students play both chess and quidditch ?
(1) 3 Gryffindor students play neither chess, nor quidditch.
(2) For every 3 Gryffindor students who play neither chess, nor quidditch, there are 5 Gryffindor students who play both chess and quidditch.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'D'. Can you explain this answer?

Malavika Choudhury answered

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked

Statement 1: 3 Gryffindor students play neither chess nor quidditch.

From this statement, we know that there are 3 students who do not play chess or quidditch. Let's assume that x students play both chess and quidditch.

Number of students who play chess = 16
Number of students who play quidditch = 11
Number of students who play neither chess nor quidditch = 3

We can represent the information in a Venn diagram:

```
Chess
/ \
/ \
/ \
/ \
/ \
Both
/ \
/ \
/ \
/ \
/ \
/ \
Quidditch
```

From the diagram, we can see that the number of students who play both chess and quidditch is given by the formula:

x = Number of students who play chess + Number of students who play quidditch - Total number of students

Substituting the given values, we have:

x = 16 + 11 - 25
x = 27 - 25
x = 2

Therefore, 2 Gryffindor students play both chess and quidditch.

Statement 1 alone is sufficient to answer the question.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked

Statement 2: For every 3 Gryffindor students who play neither chess nor quidditch, there are 5 Gryffindor students who play both chess and quidditch.

Let's assume that the number of Gryffindor students who play neither chess nor quidditch is 3n, where n is a positive integer. According to the statement, the number of Gryffindor students who play both chess and quidditch is 5n.

Using the given values, we can set up the following equation:

3n + 5n = 25

Simplifying, we get:

8n = 25

Since n is a positive integer, there is no solution for this equation. Therefore, statement 2 alone is not sufficient to answer the question.

Statement (2) ALONE is not sufficient to answer the question asked.

Statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient

From statement 1, we know that 3 students play neither chess nor quidditch.

From statement 2, we know that for every 3 students who play neither chess nor quidditch, there are 5 students who play both chess and quidditch.

Combining the information from both statements, we can conclude that there are 3 students who play neither chess nor quidditch, and for every 3 students, there are 5 students who play both chess and quidditch.

Therefore, the number of Gryffindor students who play both chess and quidditch is:

3 * 5 = 15

Statements (1) and

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Bottles A and B contain alcohol-water solutions in ratios of 1:3 and 4:1 respectively. X liters of the solution from bottle A are to be mixed with Y liters of the solution from bottle B. What will be the percentage of alcohol in the resultant solution?
(1) X = Y
(2) Total 10 liters of solutions from bottles A and B are mixed.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'A'. Can you explain this answer?

Rahul Kapoor answered

Let's analyze the statements:

Statement (1) alone: X = Y

If X = Y, it means that an equal volume of solution is taken from bottles A and B. Since bottle A has a 1:3 alcohol-water ratio and bottle B has a 4:1 ratio, the resulting mixture will have equal amounts of alcohol and water. Therefore, the percentage of alcohol in the resultant solution will be 50%.

Statement (1) alone is sufficient to answer the question.

Statement (2) alone: Total 10 liters of solutions from bottles A and B are mixed.

From statement (2) alone, we know the total volume of the mixture is 10 liters. However, we don't have any information about the specific volumes of solution X and Y taken from bottles A and B. Without knowing the individual quantities mixed, we cannot determine the percentage of alcohol in the resultant solution.

Statement (2) alone is not sufficient to answer the question.

Combining both statements:

From statement (1), we know that X = Y. However, we still don't have any information about the specific values of X and Y.

From statement (2), we know the total volume of the mixture is 10 liters.

Even when considering both statements together, we still don't have enough information to determine the values of X and Y or the respective quantities of alcohol and water in the resultant solution. Therefore, statements (1) and (2) together are not sufficient to answer the question.

The correct answer is (A): Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

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How many triangles can be formed using 8 points in a given plane?
(1) A triangle is formed by joining 3 distinct points in the plane
(2) Out of 8 given points, three are collinear
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'B'. Can you explain this answer?

Abhishek Choudhury answered

Statement (1):
A triangle is formed by joining 3 distinct points in the plane.

This statement alone is not sufficient to determine the number of triangles that can be formed using 8 points. It only tells us that a triangle can be formed by joining 3 distinct points, but it does not provide any information about the total number of points or their arrangement in the plane.

Statement (2):
Out of 8 given points, three are collinear.

This statement alone is not sufficient to determine the number of triangles that can be formed using 8 points. It only tells us that three points are collinear, but it does not provide any information about the remaining 5 points or their arrangement in the plane.

Combined statements (1) and (2):
By combining the two statements, we can determine the number of triangles that can be formed using 8 points in a given plane.

If three points are collinear, they cannot form a triangle. Therefore, in order to form a triangle, we need to choose 3 points out of the remaining 5 points that are not collinear.

The total number of triangles that can be formed can be calculated using the combination formula:

C(n,r) = n! / (r!(n-r)!)

Where n is the total number of points and r is the number of points to be chosen to form a triangle.

In this case, n = 5 (as 3 points are collinear) and r = 3.

C(5,3) = 5! / (3!(5-3)!) = 5! / (3!2!) = (5x4x3x2x1) / (3x2x1x2x1) = 10

Therefore, the number of triangles that can be formed using 8 points in a given plane is 10.

Conclusion:
From the analysis above, we can conclude that statement (2) alone is sufficient to answer the question asked, but statement (1) alone is not sufficient. Therefore, the correct answer is option B.

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In a certain mixture of juice and water, if only water was added to the mixture, how much water was added to the mixture?
(1) After adding water, water made up fifteen percent of the mixture.
(2) Before adding water, water made up ten percent of the mixture.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'E'. Can you explain this answer?

Notes Wala answered

Statement (1) alone: After adding water, water made up fifteen percent of the mixture.

Statement (1) tells us the proportion of water in the mixture after adding water, which is fifteen percent. However, it doesn't provide any information about the initial composition of the mixture or the amount of water originally present. Without knowing the initial composition, we cannot determine the amount of water added to the mixture. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2) alone: Before adding water, water made up ten percent of the mixture.

Statement (2) provides information about the proportion of water in the mixture before adding water, which is ten percent. However, it doesn't give us any information about the final composition or the amount of water added. Without knowing the change in composition or the specific quantities involved, we cannot determine the amount of water added to the mixture. Therefore, statement (2) alone is not sufficient to answer the question.

Combining both statements:

By combining the statements, we know that before adding water, water made up ten percent of the mixture (statement 2), and after adding water, water made up fifteen percent of the mixture (statement 1). However, this information alone does not provide enough information to determine the amount of water added to the mixture. We need additional data such as the total volume or quantity of the mixture to make that determination.

Since statements (1) and (2) together are not sufficient to answer the question and additional data are needed, the correct answer is (E): Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

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How much water (in grams) should be added to a 35%-solution of acid to obtain a 10%-solution?
(1) There are 50 grams of the 35%-solution.
(2) In the 35%-solution the ratio of acid to water is 7:13.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'A'. Can you explain this answer?

Rahul Kapoor answered

Statement (1): There are 50 grams of the 35%-solution. This statement provides information about the initial amount of the 35%-solution. We know the concentration (35%) and the quantity (50 grams). From this information, we can calculate the amount of acid in the solution (35% of 50 grams) and determine the amount of water in the solution (100% - 35% of 50 grams). With this information, we can calculate how much water should be added to obtain a 10%-solution.

Statement (2): In the 35%-solution, the ratio of acid to water is 7:13. This statement provides information about the ratio of acid to water in the 35%-solution. However, it does not give us any information about the quantity or concentration of the solution. Without knowing the amount or concentration of the solution, we cannot determine the amount of water needed to obtain a 10%-solution.

When we consider both statements together, we know the initial amount of the 35%-solution (50 grams) from Statement (1) and the ratio of acid to water in the solution from Statement (2). With this information, we can calculate the amount of acid and water in the initial solution. However, we still don't have enough information to determine how much water should be added to obtain a 10%-solution. We need to know the desired quantity or final concentration of the solution.

Therefore, Statement (1) alone is sufficient to answer the question, but Statement (2) alone is not sufficient. The correct answer is option A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

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What is the number of multiples of 4 in 5 consecutive integers?
1) The median of them is 4
2) The average (arithmetic mean) of them is a multiple of 4
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'D'. Can you explain this answer?

Rahul Kapoor answered

To determine the number of multiples of 4 in 5 consecutive integers, let's analyze each statement:

Statement (1) tells us that the median of the five consecutive integers is 4. Since there are five integers, the middle number must be 4. Let's assume the five consecutive integers are x, x+1, x+2, x+3, and x+4. Since 4 is the median, we have (x+2) = 4, which implies x = 2. Therefore, the five consecutive integers are 2, 3, 4, 5, and 6. Among these, there is only one multiple of 4, which is 4 itself. Hence, statement (1) alone is sufficient to answer the question.

Statement (2) tells us that the average (arithmetic mean) of the five consecutive integers is a multiple of 4. Let's assume the five consecutive integers are a, b, c, d, and e. We can express the average as (a+b+c+d+e)/5. If this average is a multiple of 4, it means that the sum a+b+c+d+e must be divisible by 4. However, statement (2) does not provide any specific values for a, b, c, d, and e, or their relationships, so we cannot determine the number of multiples of 4. Hence, statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we know that the median is 4 (statement 1) and the average is a multiple of 4 (statement 2). From statement 1, we can determine that the five consecutive integers are 2, 3, 4, 5, and 6. From statement 2, we know that the sum of these integers is a multiple of 4. The sum is 2+3+4+5+6 = 20, which is divisible by 4. Therefore, the number of multiples of 4 in these five consecutive integers is 1.

Thus, each statement alone is sufficient to answer the question. The answer is D: EACH statement ALONE is sufficient to answer the question asked.

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How many integers are there between C and D?
(1) Neither c nor d is an integer.
(2) c – d = 3
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

EduRev GMAT answered

- Statement (1): Neither $ c $ nor $ d $ is an integer. This tells us that $ c $ and $ d $ are not whole numbers but doesn't provide any information on their difference. Insufficient alone.

- Statement (2): $ c - d = 3 $. This provides the difference but not the specific values of $ c $ and $ d $. We can't determine integers between them. Insufficient alone.

- Combined: Knowing $ c - d = 3 $ and neither is an integer, $ c $ and $ d $ must be such that there's exactly one integer in between. For example, if $ d = 1.5 $ and $ c = 4.5 $, the integer 3 lies in between. Hence, together they are sufficient.

The answer is C.

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If x and y are positive integers, is x even?
(1) x^y + y^xis even.
(2) y^x + 4x is odd.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Rahul Kapoor answered

Statement (1): x^y + y^x is even.
The sum of xy + yx can be simplified to 2xy, which is even. This statement implies that 2xy is even, but it doesn't provide any information about the value of x or y individually. Therefore, statement (1) alone is not sufficient to determine if x is even.

Statement (2): y^x + 4x is odd.
If y^x + 4x is odd, it means that the sum of yx and 4x is odd. This implies that both terms must have the same parity (either both even or both odd) since the sum of two terms with different parity would be even. Since 4x is always even (for any positive integer x), it means that yx must also be even. However, we still don't know the specific values of x and y, so we cannot determine if x is even or odd. Statement (2) alone is not sufficient to determine if x is even.

Combining both statements, we know that 2xy is even (from statement 1) and yx is even (implied by statement 2). This means that both xy and yx must be even. If both xy and yx are even, it implies that x and y are both even. Therefore, by considering both statements together, we can conclude that x is even. Hence, both statements together are sufficient to answer the question.

The answer is option C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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Tom and Samuel are riding motorcycles down the highway at constant speeds. If Tom is now 2 miles ahead of Samuel, how many minutes before Tom is 3 miles ahead of Samuel?
(1) Tom is traveling at 70 miles per hour and Samuel is traveling at 60 miles per hour.
(2) Tom left 5 minutes before Samuel.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'A'. Can you explain this answer?

Geetika Sarkar answered

Understanding the Problem
Tom is currently 2 miles ahead of Samuel, and we need to determine how long it will take for Tom to be 3 miles ahead of Samuel.
Analyzing Statement (1)
- Tom's Speed: 70 miles per hour
- Samuel's Speed: 60 miles per hour
Relative Speed Calculation
- The relative speed of Tom with respect to Samuel = Tom's speed - Samuel's speed = 70 mph - 60 mph = 10 mph.
Time to Increase Lead
- Tom is currently 2 miles ahead and needs to be 3 miles ahead. Therefore, he needs to extend his lead by 1 mile.
- Time = Distance / Speed = 1 mile / 10 mph = 0.1 hours = 6 minutes.
Thus, Statement (1) alone is sufficient.
Analyzing Statement (2)
- Tom left 5 minutes before Samuel, which gives him a head start. However, we lack information about their speeds.
- Without knowing how fast each is traveling, we cannot determine if the time until Tom is 3 miles ahead can be calculated.
Thus, Statement (2) alone is not sufficient.
Conclusion
Since Statement (1) is sufficient on its own to answer the question and Statement (2) is not, the correct answer is option A:
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

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Fanny and Alexander are y miles apart and are traveling in a straight line toward each other at a constant rate of 25 mph and x mph respectively. What is the value of x?
(1) 1.5 hours before they meet they were 135 miles apart
(2) y = 360 miles
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'A'. Can you explain this answer?

Rahul Kapoor answered

Statement (1): 1.5 hours before they meet, they were 135 miles apart.
From this statement, we can calculate the rate at which Fanny and Alexander are closing the distance between them. In 1.5 hours, Fanny travels 1.5 * 25 = 37.5 miles, and Alexander travels 1.5 * x miles. The total distance they cover in 1.5 hours is 37.5 + 1.5x miles. Since they were 135 miles apart before this time, we can write the equation:

37.5 + 1.5x = 135

Simplifying the equation:

1.5x = 135 - 37.5
1.5x = 97.5
x = 97.5 / 1.5
x = 65

Therefore, statement (1) alone is sufficient to determine the value of x.

Statement (2): y = 360 miles.
This statement provides the initial distance between Fanny and Alexander, but it doesn't give us any information about their rates or the time it takes for them to meet. Without the rate at which they are traveling, we cannot determine the value of x. Therefore, statement (2) alone is not sufficient to determine the value of x.

Since statement (1) alone is sufficient to answer the question, but statement (2) alone is not, the answer is A: Statement (1) alone is sufficient, but statement (2) alone is not sufficient to answer the question asked.

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If x is a positive integer, is x⁵+ 1 an odd number?
(1) x is the smallest integer that is divisible by all integers from 21 to 24, inclusive.
(2) 5^x is an odd number
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'A'. Can you explain this answer?

Ananya Iyer answered

Statement (1): x is the smallest integer that is divisible by all integers from 21 to 24, inclusive.

To determine if x^5 - 1 is odd, we need to know the value of x.

If x is an even number, then x^5 will also be even, and subtracting 1 will result in an odd number. However, if x is an odd number, then x^5 will also be odd, and subtracting 1 will result in an even number.

From statement (1), we know that x is divisible by all integers from 21 to 24. Let's examine these numbers:

21 = 3 * 7
22 = 2 * 11
23 = prime number
24 = 2^3 * 3

The smallest integer that is divisible by all of these numbers is the least common multiple (LCM) of these numbers, which is 2^3 * 3 * 7 * 11 = 2^3 * 3 * 7 * 11.

Since x is the smallest integer divisible by these numbers, we can conclude that x must be equal to 2^3 * 3 * 7 * 11.

Statement (2): 5x is an odd number

If 5x is an odd number, then x must be an odd number.

From statement (1), we found that x is equal to 2^3 * 3 * 7 * 11, which is an even number. Therefore, statement (2) is not consistent with statement (1).

Combining Statements (1) and (2):

Since statement (2) contradicts statement (1), the statements cannot be combined to provide a consistent answer. Therefore, the correct answer is option A - Statement (1) alone is sufficient, but statement (2) alone is not sufficient to answer the question asked.

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Total amount collected from an officer for a charity cause was $250. If each employee contributed either $20 or $25, how many employees contributed $20?
I. The number of employees in the office is 11.
II. The number of employees who gave $25 was 1 more than the number of employees who gave $20.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'D'. Can you explain this answer?

Pioneer Academy answered

1. Let's assume x employees contributed $20 and y contributed $25 then as per the stem 20x + 25y = 250
2. We can simplify this equation to 4x + 5y = 50 by multiplying both sides with 1/5
3. The question asks us to determine value of x

Statement-I: The number of employees in the office is 11 (Sufficient)
This states that x + y = 11. Now we have two equations and two unknowns hence we can solve the equations to get the values of x

Statement-II: The number of employees who gave $25 was 1 more than the number of employees who gave $20 (Sufficient)
This states that y = x + 1. Again we have two equations and two unknowns hence we can solve the equations to get the values of x

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What is the units digit of positive integer y?
(1) The units digit of y² is 1
(2) The units digit of y does not equal 1
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Rahul Kapoor answered

Statement 1
(1) The units digit of y² is 1
The unit digit of y can be either 1 or the unit digits can be 9.
The statement is not sufficient to answer the question.
Hence, we can eliminate A and D.

Statement 2
(2) The units digit of y does not equal 1
Not sufficient, as the unit digit can be any integer between 2 and 9, both inclusive.
We can eliminate B.

Combined
From statements 1 and 2, we can conclude that the unit digit = 9.
Sufficient.

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A college admissions committee will grant a certain number of $10,000 scholarships, $5,000 scholarships, and $1,000 scholarships. If no student can receive more than one scholarship, how many different ways can the committee dole out the scholarships among the pool of 10 applicants?
(1) In total, six scholarships will be granted.
(2) An equal number of scholarships will be granted at each scholarship level
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Nandita Yadav answered

Statement (1) alone:
- With statement (1) alone, we know that a total of six scholarships will be granted.
- However, we do not know how many scholarships of each amount will be granted. Therefore, statement (1) alone is not sufficient to determine the number of different ways the scholarships can be distributed among the applicants.

Statement (2) alone:
- With statement (2) alone, we know that an equal number of scholarships will be granted at each scholarship level.
- This means that there will be an equal number of $10,000, $5,000, and $1,000 scholarships awarded.
- However, we do not know the total number of scholarships that will be granted. Therefore, statement (2) alone is not sufficient to determine the number of different ways the scholarships can be distributed among the applicants.

Both statements together:
- Combining both statements, we know that a total of six scholarships will be granted and an equal number of scholarships will be granted at each level.
- With this information, we can determine that there will be 2 scholarships at each level (since 2 x 3 = 6).
- Now, we can calculate the number of ways the scholarships can be distributed among the applicants by considering the combinations of applicants for each scholarship level.
- Therefore, both statements together are sufficient to answer the question.
- The correct answer is option 'C'.

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If r and s are each greater than 0, is (r + n)/(s + n) > r/s?
(1) n > 0
(2) r < s
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'A'. Can you explain this answer?

Rahul Kapoor answered

Statement (1): n > 0
Statement (2): r < s

Let's evaluate each statement:

Statement (1) alone: Since n > 0, adding a positive value to both the numerator and denominator of a fraction will increase its value. Therefore, (r + n)/(s + n) > r/s is true. Statement (1) alone is sufficient to answer the question.

Statement (2) alone: The relationship between r and s does not provide any information about the value of n or how it affects the inequality (r + n)/(s + n) > r/s. Statement (2) alone is not sufficient to answer the question.

Since statement (1) alone is sufficient, but statement (2) alone is not, the answer is A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

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Is integer x positive?
(1) x > x³
(2) x < x²
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'A'. Can you explain this answer?

Rahul Kapoor answered

Statement (1) states that x is greater than x³. For this statement to hold true, x³ must be negative. Since x is an integer, the only way for x³ to be negative is if x is negative. If x is negative, it cannot be positive. Therefore, statement (1) alone is sufficient to determine that x is not positive.

Statement (2) states that x is less than x². This inequality does not provide enough information to determine if x is positive or negative. For example, if x = -2, then x² = 4 and x is negative. However, if x = 1/2, then x² = 1/4 and x is positive. Statement (2) alone is not sufficient to answer the question.

By considering statement (1) alone, we can conclude that x is not positive. Therefore, statement (1) alone is sufficient to answer the question.

Hence, the correct answer is A: Statement (1) alone is sufficient, but statement (2) alone is not sufficient to answer the question asked.

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Is x > y
(1) x + y > 0
(2) y² > x²
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'E'. Can you explain this answer?

Rahul Kapoor answered

To determine if x > y, let's analyze the given statements:

Statement (1): x + y > 0
This statement alone tells us that the sum of x and y is greater than zero, but it doesn't provide any specific information about the relationship between x and y individually. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): y²> x²
This statement tells us that the square of y is greater than the square of x. However, this doesn't necessarily mean that y is greater than x because the squares of both numbers could be equal while the numbers themselves have different signs. For example, if x = -2 and y = 2, the condition in statement (2) would be satisfied, but x would still be greater than y. Therefore, statement (2) alone is not sufficient to answer the question.

Considering both statements together, we still don't have enough information to determine if x > y. Statement (1) provides information about the sum of x and y, while statement (2) provides information about the squares of x and y. However, the relationship between x and y is not explicitly established.

Therefore, the answer is (E): Statements (1) and (2) together are not sufficient to answer the question asked, and additional data are needed.

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A bag contains a mixture of beans and pulses. To achieve 20 percent beans in the mixture, what percent of the mixture should be taken out and replaced with pulses?
(1) The mixture originally has 40 percent beans and 60 percent pulses.
(2) Total quantity of the mixture is 20 lb.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Rahul Kapoor answered

Statement (1) alone: The mixture originally has 40% beans and 60% pulses.

Statement (1) provides the initial composition of the mixture, stating that it contains 40% beans and 60% pulses. However, it does not give any information about the quantity or size of the mixture. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2) alone: The total quantity of the mixture is 20 lb.

Statement (2) gives us the total quantity of the mixture, which is 20 lb. However, it doesn't provide any information about the composition or percentage of beans and pulses in the mixture. Without knowing the initial composition, we cannot determine the required percentage of the mixture to be replaced with pulses. Therefore, statement (2) alone is not sufficient to answer the question.

Combining both statements:

Together, we know that the mixture originally has 40% beans and 60% pulses, and the total quantity of the mixture is 20 lb. However, we still don't have enough information to determine the exact percentage of the mixture that needs to be replaced with pulses to achieve 20% beans. We need to know the current quantity of beans and pulses in the mixture to make this determination.

Since neither statement alone is sufficient to answer the question, but together they provide some useful information, the correct answer is (C): BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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The functions pq denotes p + q and the function max (p, q) denotes the maximum value of p and q. For example, 12 = 1+2 = 3 and max (1, 2) = 2. What is the value of 2*(Max (p, q))?
(1) Max (5, p) = 6
(2) Max (3, q) = 3
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Notes Wala answered

(1) Max(5, p) = 6

From this statement, we know that the maximum value between 5 and p is 6. However, this information alone does not provide enough information to determine the value of 2*(Max(p, q)). We don't know the relationship between q and p or the specific values of q and p. Therefore, statement (1) alone is not sufficient to answer the question.

(2) Max(3, q) = 3

From this statement, we know that the maximum value between 3 and q is 3. Similar to statement (1), this information alone does not provide enough information to determine the value of 2*(Max(p, q)). We don't know the relationship between p and q or the specific values of p and q. Therefore, statement (2) alone is not sufficient to answer the question.

By analyzing each statement individually, we can see that neither statement alone is sufficient to determine the value of 2*(Max(p, q)). However, by combining both statements, we can conclude that the maximum value between p and q is 6, and the maximum value between q and 3 is 3. Therefore, the maximum value of p and q is 6. Substituting this into 2*(Max(p, q)), we get 2*6 = 12. Therefore, both statements together are sufficient to answer the question.

Hence, the answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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What is the value of │x + 7│?
(1) │x + 3│= 14
(2) (x + 2)² = 169
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Rahul Kapoor answered

To determine the value of |x + 7|, let's evaluate each statement:

Statement (1): |x + 3| = 14

Considering the absolute value, we can rewrite the equation as two separate cases:

x + 3 = 14:
Solving this equation gives us x = 11. Therefore, |x + 7| = |11 + 7| = |18| = 18.

-(x + 3) = 14:
Simplifying, we have -x - 3 = 14.
Rearranging, we get -x = 17, and dividing by -1, we have x = -17. Therefore, |x + 7| = |-17 + 7| = |-10| = 10.

Statement (1) alone gives us two possible values for |x + 7|: 18 or 10.

Statement (2): (x + 2)² = 169

Taking the square root of both sides, we have x + 2 = ±13.

Case 1: x + 2 = 13
Solving this equation gives us x = 11. Therefore, |x + 7| = |11 + 7| = |18| = 18.

Case 2: x + 2 = -13
Solving this equation gives us x = -15. Therefore, |x + 7| = |-15 + 7| = |-8| = 8.

Statement (2) alone gives us two possible values for |x + 7|: 18 or 8.

Combining both statements, we see that the only common value for |x + 7| is 18.

Therefore, both statements together are sufficient to answer the question, but neither statement alone is sufficient. The answer is (C).

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What is the remainder when 6^x is divided by 10?
1. x is a positive integer
2. x is a positive even integer
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'D'. Can you explain this answer?

Talent Skill Learning answered

Statement 1: x is a positive interger. Meaning X > 0 and is an Integer.
Now, (6)^P (where P is a positive Integer) will always give the last digit as 6 and hence remainder will always remain 6. → SUFFICIENT

Statement 2: This is just a filtered part of Statement 1. Meaning in Statement 1, I already said we will get a single solution for X > 0 and X as an integer. Now, this includes both even and Odd. This, Statement 2 is also sufficient.
Hence, D.

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There are exactly 6 teams in league x. What was the total number of games played by the 6 teams last season?
(1) Each team in league x played each of the other teams at least once.
(2) No team in league x played more than 7 games.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'E'. Can you explain this answer?

Rutuja Banerjee answered

Statement (1): Each team in league x played each of the other teams at least once.
This statement implies that each team played 5 games because there are 6 teams in total and each team plays against the other 5 teams. However, we don't know if there were any additional games played beyond these 5.

Statement (2): No team in league x played more than 7 games.
This statement sets an upper limit on the number of games played by any team. However, it doesn't provide any information about the minimum number of games played by each team or the total number of games played by all teams combined.

Combined statements:
From statement (1), we know that each team played at least 5 games. However, this information alone is not sufficient to determine the total number of games played by all teams because we don't know if there were any additional games beyond the minimum requirement.

From statement (2), we know that no team played more than 7 games. This information alone is not sufficient to determine the total number of games played by all teams because we don't know the minimum number of games played by each team.

Therefore, the combined statements are also not sufficient to answer the question because we need to know both the minimum and maximum number of games played by each team in order to determine the total number of games played by all teams.

In conclusion, neither statement alone nor the combined statements provide enough information to determine the total number of games played by the 6 teams in league X. Additional data is needed to answer the question. Thus, the correct answer is option E.

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Aaron travels from town X to town Y and then back from town Y to town X, taking different routes in each direction. If his speed when travelling from town X to town Y is 36 miles per hour, and his speed when travelling in the opposite direction is 48 miles per hour, what is his average speed for the entire journey?
1) The length of the return trip is 24% of the entire distance traveled.
2) The length of the return trip is 100 miles.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'E'. Can you explain this answer?

Rahul Kapoor answered

Statement (1): The length of the return trip is 24% of the entire distance traveled.
This statement alone is not sufficient to determine the average speed. While it tells us the proportion of the return trip distance to the total distance, it doesn't provide the actual distance or any information about the speed for the initial trip.

Statement (2): The length of the return trip is 100 miles.
This statement alone is not sufficient either. It gives us the distance of the return trip, but we still don't have information about the distance of the initial trip or the speed for the initial trip.

Therefore, neither statement alone is sufficient to answer the question. Considering both statements together also doesn't provide enough information to calculate the average speed. We need additional data about the distance of the initial trip to determine Aaron's average speed. The correct answer is (E): Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

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What is the average (arithmetic mean) number of runs per game scored by The New York Yankees last season ?
(1) Last season, The New York Yankees played 160 games
(2) Last season, The New York Yankees scored four runs per game in exactly 1/5 of their games, five runs per game in exactly 3/4 of their games, and nine runs per game in exactly 1/20th of their games.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'B'. Can you explain this answer?

Pioneer Academy answered

Statement (1) says that last season, The New York Yankees played 160 games.

Knowing the number of games played is helpful, but it does not provide any information about the number of runs scored in each game. Without this information, we cannot determine the average number of runs per game. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2) provides information about the number of runs scored per game in different fractions of their games.

According to statement (2), the Yankees scored four runs per game in exactly 1/5 of their games, five runs per game in exactly 3/4 of their games, and nine runs per game in exactly 1/20th of their games.

This statement gives us the distribution of runs per game in terms of fractions, but it does not provide the actual number of games in each category. Without knowing the number of games falling into each category, we cannot calculate the average number of runs per game. Therefore, statement (2) alone is not sufficient to answer the question.

Combining both statements, we know that the Yankees played 160 games (from statement 1) and we have information about the distribution of runs per game in terms of fractions (from statement 2). However, we still do not have the specific number of games falling into each category. Without this information, we cannot determine the average number of runs per game.

Hence, the correct answer is (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

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Any decimal that has only a finite number of nonzero digits is a terminating decimal. For example, 12, 0.13, and 4.068 are three terminating decimals. If j and k are positive integers and the ratio j/k is expressed as a decimal, is j/k a terminating decimal?
(1) k = 3
(2) j is an odd multiple of 3.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'C'. Can you explain this answer?

Rahul Kapoor answered

To determine if the ratio j/k is a terminating decimal, we need to consider both statements.

Statement (1) tells us that k = 3. If k is equal to 3, it means the denominator has a prime factor of 3. Since any power of 3 in the denominator will result in a repeating decimal, we can conclude that the ratio j/k is not a terminating decimal. Therefore, statement (1) alone is sufficient to answer the question.

Statement (2) tells us that j is an odd multiple of 3. This information alone does not provide any details about the denominator k or the factors of k. It does not give us enough information to determine if the ratio j/k is a terminating decimal. Therefore, statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we know that k = 3 (from statement 1) and j is an odd multiple of 3 (from statement 2). Since the denominator is 3, which has a prime factor of 3, the ratio j/k will not be a terminating decimal. Therefore, both statements together are sufficient to answer the question.

Hence, the answer is C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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A cyclist rides to the top of a hill and then coasts back down by the same route. The entire trip takes 4 hours. What is his average (arithmetic mean) speed when climbing the hill?
(1) The cyclist's average speed over the entire trip was 6 miles per hour.
(2) The distance to the top of the hill was 12 miles, and the cyclist's average speed on the downward trip was 18 miles per hour.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'B'. Can you explain this answer?

Rahul Kapoor answered

Statement 1 gives the information about the total distance. But there is no clue of the speed climbing the hill. Not sufficient

Statement 2 - Distance for downtrip will also be 12. We can find the time taken for covering downtrip which is 12/18 or 2/3 hr.
Time taken for uphill = 4-2/3
= 10/3
Distance = 12

Avg speed can be found out. Sufficient

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If [x] denotes the least integer greater than or equal to x, what is the value of [x+1] = ?
(1) [x] = 1
(2) [2x] = 1
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'D'. Can you explain this answer?

Shivam Ghoshal answered

To find the value of [x 1], we need to determine the least integer greater than or equal to x+1. Let's analyze the given statements:

Statement (1): [x] = 1
This statement tells us that the least integer greater than or equal to x is 1. However, it does not provide any information about the value of x+1. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): [2x] = 1
This statement tells us that the least integer greater than or equal to 2x is 1. Again, it does not provide any direct information about the value of x+1. Therefore, statement (2) alone is not sufficient to answer the question.

Combining both statements:
By combining the information from both statements, we can deduce that the least integer greater than or equal to x is 1 and the least integer greater than or equal to 2x is 1. Since the least integer greater than or equal to 2x is 1, it implies that 2x lies between 1 and 2. Therefore, x must lie between 0.5 and 1. By adding 1 to x, we can conclude that x+1 lies between 1.5 and 2. Therefore, the least integer greater than or equal to x+1 is 2.

Hence, by combining both statements, we can determine that [x 1] = 2. Therefore, both statements together are sufficient to answer the question.

Therefore, the correct answer is option (C): BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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What is the value of |x|?
(1) |x² + 16| - 5 = 27
(2) x^2 = 8x - 16
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
d)
EACH statement ALONE is sufficient to answer the question asked
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Correct answer is option 'D'. Can you explain this answer?

Rahul Kapoor answered

Statement (1): |x² + 16| - 5 = 27

Adding 5 to both sides, we have |x² + 16| = 32.

Considering the absolute value, we can rewrite the equation as two separate cases:

x² + 16 = 32:
Solving this equation gives us x² = 16, which means x = ±4. Therefore, |x| = 4.

-(x² + 16) = 32:
Simplifying, we have -x² - 16 = 32.
Rearranging, we get -x² = 48, and dividing by -1, we have x² = -48. Since the square of any real number is non-negative, there is no real solution for x in this case.

Statement (1) alone gives us two possible values for |x|: 4 or no solution.

Statement (2): x² = 8x - 16

Rearranging, we get x² - 8x + 16 = 0. This equation can be factored as (x - 4)(x - 4) = 0, which means x = 4.

Statement (2) alone gives us the value x = 4, which means |x| = 4.

Considering both statements together, we see that they both give the same value for |x|, which is 4.

Therefore, each statement alone is sufficient to answer the question, and the answer is (D).

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