Test: Quantitative Reasoning (Level 600) - 2 - GMAT MCQ

To find the number of integers that satisfy the inequality x + 4 > |x + 10|, we can analyze the different cases based on the absolute value function.

Case 1: x + 10 ≥ 0 When x + 10 is non-negative, the absolute value |x + 10| can be written without the absolute value signs. Thus, the inequality becomes: x + 4 > x + 10

By subtracting x from both sides, we get: 4 > 10

This inequality is false, so there are no solutions in this case.

Case 2: x + 10 < 0 When x + 10 is negative, the absolute value |x + 10| becomes the negation of the expression inside the absolute value signs. Thus, the inequality becomes: x + 4 > -(x + 10)

Simplifying the inequality, we have: x + 4 > -x - 10

By adding x to both sides, we get: 2x + 4 > -10

Next, by subtracting 4 from both sides, we have: 2x > -14

Finally, dividing both sides by 2 gives: x > -7

This inequality means that x must be greater than -7. Since we are looking for integers, we can see that there are no integers greater than -7 that satisfy this condition.

Therefore, combining both cases, we find that there are no integers that satisfy the inequality x + 4 > |x + 10|. Hence, the answer is E: 0.

Formula:

Average = Sum of all the Values / Total Number of Values

Sum of All the values = Average * Total Number of Values

Given information:

The average of 13 numbers is 68.

Sum of 13 numbers = Average * 13 = 68 * 13 = 884.

The average of the first 7 numbers is 63.

Sum of first 7 numbers = Average * 7 = 63 * 7 = 441.

Sum of last 6 numbers:

Sum of last 6 numbers = Sum of 13 numbers - Sum of first 7 numbers = 884 - 441 = 443.

The average of the last 7 numbers is 70.

Sum of last 7 numbers = Average * 7 = 70 * 7 = 490.

7th number:

Since there are 13 numbers, the 7th number from the beginning is the same as the 7th number from the end.

Therefore, the 7th number = Sum of last 7 numbers - Sum of last 6 numbers = 490 - 443 = 47.

We have a circular table with six fixed seats. Juan must sit on the seat closest to the window, and Jamal must sit next to Juan.

Let's consider Juan and Jamal as a single entity since they must sit together. So, we have Juan and Jamal and four other friends.

We can fix Juan on the window seat. Then, Jamal can sit next to Juan in two possible positions.

Now, we have four remaining friends to be seated in the remaining four seats.

The number of ways to arrange the four friends in the remaining four seats is given by 4!, which is equal to 4 × 3 × 2 × 1 = 24.

Therefore, the total number of ways Juan and his five friends can sit is 2 (for Juan and Jamal) multiplied by 4! (for the remaining friends).

2 × 4! = 2 × 24 = 48.

Hence, the correct answer is C, 48.

When Tammy bikes the course at 30 miles per hour, the time taken can be calculated using the formula:

Time = Distance / Speed

So, the time taken to complete the race course at 30 miles per hour is x / 30 hours.

When Tammy returns home along the same route at 10 miles per hour, the time taken is x / 10 hours.

According to the given information, the total time taken for both trips is 2 hours. Therefore, we can set up the equation:

x / 30 + x / 10 = 2

To solve this equation, we can first find a common denominator by multiplying both sides by 30:

30 * (x / 30) + 30 * (x / 10) = 2 * 30

Canceling out the denominators, we get:

x + 3x = 60

Combining like terms, we have:

4x = 60

Dividing both sides by 4, we find:

x = 15

Therefore, the length of the race course is 15 miles.

So, the correct answer is C) 15.

To find the value of (<-2.1>)(<2.1>)(<2.9>), we need to evaluate the greatest integer less than or equal to each of the given numbers.

The greatest integer less than or equal to -2.1 is -3. The greatest integer less than or equal to 2.1 is 2. The greatest integer less than or equal to 2.9 is 2.

Therefore, the expression becomes: (-3)(2)(2).

Multiplying these values, we get: -3 * 2 * 2 = -12.

Hence, the correct answer is D) -12.

To determine which statements must be true given that ab > b, let's analyze each option:

I. a/b > 0: This statement is not necessarily true. It is possible for a/b to be negative if both a and b have different signs. Therefore, statement I is not always true.

II. (a - b) > 0: This statement is also not necessarily true. For example, if a = 3 and b = 2, then (a - b) = 3 - 2 = 1, which is not greater than 0. Therefore, statement II is not always true.

III. (a + b) > 1: This statement is not implied by the given condition ab > b. There is no direct relationship between (a + b) and ab > b. Therefore, statement III is not necessarily true.

Based on the analysis above, the only statement that must be true is statement I. So, the correct answer is A) I only.

I. a/b > c: This statement is not necessarily true. It is possible for a/b to be less than c if a is negative and b is positive. For example, if a = -6, b = 3, and c = 4, then a/b = -6/3 = -2, which is less than c. Therefore, statement I is not always true.

II. a/c > b: This statement is not necessarily true either. For example, if a = 4, b = 3, and c = 2, then a/c = 4/2 = 2, which is not greater than b. Therefore, statement II is not always true.

III. a/bc > 1: This statement is not necessarily true as well. For example, if a = 2, b = 3, and c = 1, then a/bc = 2/(3*1) = 2/3, which is less than 1. Therefore, statement III is not always true.

Based on the analysis above, none of the statements (I, II, or III) are necessarily true given the condition a > bc. Therefore, the correct answer is E) None of these.

To find the least value of n for which the product of the first n positive integers is divisible by 405, we need to determine the prime factorization of 405.

Prime factorization of 405: 405 = 3⁴ * 5¹

To ensure that the product of the first n positive integers is divisible by 405, we need to have at least four factors of 3 and at least one factor of 5.

Let's analyze the given answer choices:

(A) 6: The product of the first 6 positive integers is 1 * 2 * 3 * 4 * 5 * 6 = 720, which is not divisible by 405. Therefore, option A is not the correct answer.

(B) 8: The product of the first 8 positive integers is 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 = 40,320, which is divisible by 405 since it contains at least four factors of 3 and one factor of 5. However, we need to find the least value of n, so we will continue to check the remaining options.

(C) 9: The product of the first 9 positive integers is 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 = 326,880, which is divisible by 405. Since this is the smallest value that satisfies the divisibility condition, option C is the correct answer.

(D) 10: The product of the first 10 positive integers is 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 = 3,628,800, which is divisible by 405. However, option C already provides a smaller value for n, so option D is not the correct answer.

(E) 12: The product of the first 12 positive integers is 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 = 479,001,600, which is divisible by 405. However, option C already provides a smaller value for n, so option E is not the correct answer.

Therefore, the correct answer is (C) 9, as it represents the least value of n for which the product of the first n positive integers is divisible by 405.

Let's calculate the number of cartons of milk that will be sold during Ben's shift.

During the first hour, 75% of the milk is sold. This means 75/100 * 780 = 585 cartons of milk are sold.

After the first hour, there are 780 - 585 = 195 cartons of milk remaining.

During the second hour, 20% of the remaining milk is sold. This means 20/100 * 195 = 39 cartons of milk are sold.

After the second hour, there are 195 - 39 = 156 cartons of milk remaining.

Now, let's calculate how many cartons of milk Ben will need to restock the milk cooler.

The milk cooler originally holds 780 cartons of milk, and there are currently 156 cartons remaining. Therefore, Ben will need to restock the cooler with 780 - 156 = 624 cartons of milk.

Since there are 1000 cartons of milk in the back room, Ben will not need 1000 - 624 = 376 cartons of milk.

Therefore, the correct answer is B) 376.

The remainder when a positive integer y is divided by 7 is 2, resulting in possible values of y such as 2, 9, 16, 23, 30, 37, 44, 51, 58, and so on.

Similarly, when y is divided by 11, the remainder is 3, leading to possible values of y such as 3, 14, 25, 36, 47, 58, 69, 80, 91, 102, and so on.

The first common value between the two sets is 58.

To find the sum of its digits, we add 5 and 8, resulting in 13.

Therefore, the rephrased answer is: Option E.

To find the total number of riders, we can express it as either 5p + 2 or 6q, where p and q are positive integers.

By analyzing the given conditions, we observe that the first valid number of riders is 12, and the subsequent valid numbers follow a pattern of 12 + multiples of 30 (the least common multiple of 5 and 6).

Thus, the total number of riders can be represented as 30k + 12.

Considering the range of riders between 29 and 150, the valid options are 42, 72, 102, and 132.

Among the provided answer choices, the sum of the greatest possible number of people in line and the least possible number of people in line is 42 + 132, which equals 174. Therefore, option C (174) is the correct answer.

To solve this problem, let's denote the first term of the arithmetic progression as 'a' and the common difference as 'd'.

The 7th term of the arithmetic progression can be represented as: a + 6d

According to the given information, the square of the 7th term is equal to the product of the 3rd and 17th terms: (a + 6d)² = (a + 2d)(a + 16d)

Expanding the left side of the equation: a² + 12ad + 36d² = a² + 18ad + 32d²

Simplifying the equation: 6ad + 4d² = 0

Factoring out '2d': 2d(3a + 2d) = 0

Since the common difference 'd' cannot be zero (as it is positive), we can conclude that: 3a + 2d = 0

Rearranging the equation to solve for 'a': 3a = -2d a = -2d/3

The ratio of the first term to the common difference is given by a/d: a/d = (-2d/3) / d a/d = -2/3

Therefore, the ratio of the first term to the common difference is 2:3, which corresponds to option A.

Let's assume the number of nickels in the purse is 'n'. Since we know there are a total of 30 coins, the number of quarters can be calculated as '30 - n'.

The value of each nickel is $0.05, and the value of each quarter is $0.25. We can set up an equation based on the total value of the coins:

0.05n + 0.25(30 - n) = 4.70

Simplifying the equation:

0.05n + 7.5 - 0.25n = 4.70

Combining like terms:

-0.20n = 4.70 - 7.5

-0.20n = -2.80

Dividing both sides by -0.20:

n = -2.80 / -0.20

n = 14

Therefore, the purse contains 14 nickels, which corresponds to option B.

According to the given information, in three years, Janice will be three times as old as her daughter. So, we can form the equation:

J + 3 = 3(D + 3)

Simplifying this equation, we have:

J + 3 = 3D + 9

J = 3D + 9 - 3

J = 3D + 6

Now, let's consider the second piece of information. Six years ago, Janice's age was her daughter's age squared. We can write this as:

J - 6 = (D - 6)²

Expanding the equation:

J - 6 = D² - 12D + 36

J = D² - 12D + 42

Now, we have a system of equations:

J = 3D + 6 J = D² - 12D + 42

We can substitute the value of J from the first equation into the second equation:

3D + 6 = D² - 12D + 42

Rearranging this equation:

D^2 - 15D + 36 = 0

Factoring the quadratic equation:

(D - 3)(D - 12) = 0

This equation has two possible solutions:

D - 3 = 0 --> D = 3 D - 12 = 0 --> D = 12

Since the daughter's age cannot be negative, we take D = 12.

Now, substituting D = 12 into the first equation:

J = 3(12) + 6 J = 36 + 6 J = 42

Therefore, Janice is currently 42 years old.

The correct answer is (D) 42.

Let's calculate the production rates of Machine A and Machine B per hour:

Machine A's production rate = 1,050 components / 5 hours = 210 components per hour Machine B's production rate = 1,050 components / (15/2) hours = 1,050 components / 7.5 hours = 140 components per hour

When Machine A and Machine B work simultaneously for T hours, the total number of components produced is equal to the sum of their individual productions:

Total production = (Machine A's production rate + Machine B's production rate) × T

Substituting the values we calculated earlier:

Total production = (210 + 140) × T = 350 × T

We know that the total production is equal to 1,050 components, so we can solve for T:

350 × T = 1,050 T = 1,050 / 350 T = 3

Therefore, when the machines work simultaneously for 3 hours, Machine B would have produced:

Machine B's production = Machine B's production rate × T = 140 components per hour × 3 hours = 420 components

Hence, Machine B has produced 420 electrical components at the end of T hours. Therefore, the correct answer is (C) 420.

The total work to be done, which is equivalent to the least common multiple (LCM) of 6 and 8, is 24 units. Jerry's rate is 4 units per minute, while Adam's rate is 3 units per minute. When working together, they can complete 7 units per minute.

In 2 minutes of working together, they complete 14 units. Therefore, there are 10 units remaining to be done, which is calculated by subtracting 14 units from the total of 24 units. Adam alone will be able to complete these remaining 10 units in 10/3 minutes.

Therefore, the answer is C.

Now any of the alternate 4 places can be filled by 4 male or female in 4!
Similarily,
other 4 alternate places can be filled in 4!
Hence required probability= 2*4!*4!=1152

Internet - 11% of 300 = 33
Radio - 15% of 300 = 45

First mixture: milk and water in the ratio of 3x:4x.
Second mixture: milk and water in the ratio 5y:2y.
We want their mixture to have equal quantities of milk and water: 3x + 5y = 4x + 2y --> x/y = 3.

Let me 5 color be C1, C2, C3, C4, C5
In the worst case scenario, Ben selects all 6 pencils of each of the 4 colors. Total ways = 6X4 = 24
Then Ben must select at least 2 pencil from the 5th color to ensure that he selects at least 2 pencils of each color.
Therefore, minimum number of ways = 24+2 = 26. Hence, option C is the correct choice.