Test: Quantitative Reasoning (Level 700) - 1 - GMAT MCQ

Let's assume the number of mobile phones purchased is "n."

Cost price per mobile phone = $120 + 5% of $120 = $120 + (5/100) * $120 = $120 + $6 = $126

Selling price per mobile phone = $180 - 5% of $180 = $180 - (5/100) * $180 = $180 - $9 = $171

The profit per mobile phone = Selling price - Cost price = $171 - $126 = $45

Total profit for "n" mobile phones = $45 * n

Profit percentage = (Total profit / Total cost price) * 100 = [(45 * n) / (126 * n)] * 100 = (45 / 126) * 100 = 0.357 * 100 = 35.7%

Approximating to the nearest whole number, the profit percentage is 36%.

Therefore, the correct answer is (B) 36%.

To arrange the letters of the word "COMPUTER" such that vowels occupy the even positions, we can follow the steps you mentioned:

1. Identify the vowels in the word "COMPUTER": O, U, and E.
2. There are 3 vowels, so there are 3 possible choices for the even positions.
3. The remaining 5 consonants (C, M, P, T, R) will occupy the odd positions.
4. There are 5 consonants, so there are 5 possible choices for the odd positions.
5. We need to multiply the number of choices for even positions by the number of choices for odd positions to get the total number of arrangements.

Number of choices for even positions = 4P3 = 4! / (4 - 3)! = 4 × 3 × 2 = 24

Number of choices for odd positions = 5!

Total number of arrangements = Number of choices for even positions × Number of choices for odd positions = 24 × 120 = 2880

Therefore, the correct number of ways to arrange the letters in the word "COMPUTER" if the vowels occupy the even positions is 2880.

The correct answer is (E) 2880.

Given: Area of the square cross-section = 4 square feet Volume of the iron bar = 8.4 cubic feet Speed of the conveyor = 360 feet per hour

To find the time, we can use the formula: Time = Distance / Speed

First, let's find the length of the iron bar. Since the cross-section is a square with an area of 4 square feet, the side length of the square can be found by taking the square root of the area. Side length of the square cross-section = √4 = 2 feet

Since the volume of the iron bar is given by: Volume = Area * Length 8.4 cubic feet = 4 square feet * Length Length = 8.4 cubic feet / 4 square feet = 2.1 feet

Now, let's calculate the distance the iron bar needs to travel: Distance = Length = 2.1 feet

Finally, we can calculate the time it takes for the iron bar to move through the conveyor: Time = Distance / Speed = 2.1 feet / 360 feet per hour

Converting the time from hours to seconds: Time = (2.1 feet / 360 feet per hour) * (1 hour / 3600 seconds) Time = 0.00583333 hours * 3600 seconds/hour Time ≈ 21 seconds

Therefore, it takes approximately 21 seconds for a volume of 8.4 cubic feet of the iron bar to move through the conveyor.

The correct answer is (A) 21.

7x = 10ⁿ–1....10ⁿ = 7x + 1
So 7x has to be of type 9, 99, 999, and so on.
Let us check which is the first multiple of 7 in the series..
We can save time by knowing some multiples of 7..
9=7+2...NO
99=70+29...NO, 70+28 is.
999 = 777+210+12...NO
9999=7777+2121+101=7777+2121+70+31...NO
99999=77777+21210+1019=77777+21210+700+280+39....NO
999999=777777+212121+10101=777777+212121+7000+2800+280+21...YES
That is 6 times 9s...Now 999999=10⁶-1
The next multiple in the series will be twice the number of 9s, so 2*6 times 9s or 10^12
Thus 99th smallest value of n will be 99∗6 = 594

To calculate the probability that the sum of the three numbers on the balls selected will be odd, we need to consider the possible combinations of odd and even numbers.

There are two cases to consider:

• If all three numbers selected are odd, the sum will be odd.
• If two numbers selected are odd and one is even, the sum will also be odd.

Case 1: All three numbers are odd The probability of selecting an odd number from a range of 1 to 100 is 1/2 (since half of the numbers are odd). Since we are selecting three numbers with replacement, the probability of selecting three odd numbers is (1/2) * (1/2) * (1/2) = 1/8.

Case 2: Two numbers are odd and one number is even The probability of selecting an odd number is 1/2, and the probability of selecting an even number is also 1/2. Therefore, the probability of selecting two odd numbers and one even number is (1/2) * (1/2) * (1/2) * 3 = 3/8, where the factor of 3 accounts for the three different positions in which the even number can be selected.

Adding up the probabilities from both cases: 1/8 + 3/8 = 4/8 = 1/2

Therefore, the correct answer is option C: 1/2.

Let's rephrase the given information and calculations:

Assuming we have three cup sizes: Small, Medium, and Large, Let the price of a Small cup be 2x. Let the price of a Medium cup be 5x. Let the price of a Large cup be Ax.

According to the given information, The product of the prices (2x * 5x * Ax) equals 800.

After a price increase, The new product of the prices ((2x + 6) * (5x + 6) * Ax) equals 3200.

By dividing both of these equations, we find: (10x² + 42x + 36) / (40x²) = 1.

Simplifying the equation gives us: 30x² - 42x - 36 = 0.

Further simplification leads to: 5x² - 7x - 6 = 0.

Factoring the equation, we get: (5x + 3)(x - 2) = 0.

Therefore, x = 2.

Now, we can determine the prices for each cup size: Small cup price = 4. Medium cup price = 10. Large cup price = 20 (as the product is 800).

To find the sum of the original unit prices, we add: Small price + Medium price + Large price = 4 + 10 + 20 = 34.

The correct answer is: A

Use the combined time formula where combined time  , where time for Misha = w, combined time = q and time for Petro = x.
Therefore, 

Multiply the equation by x+w to find qx+qw=xw.

Subtract by qx and factor x to find that qw=x(w–q).
Divide by w–q to find that 

Given:

A vessel is filled with liquid, 3 parts of which are water and 5 parts syrup.

Explanation:

3 parts water and 5 parts syrup.

So water = 3/8 and syrup = 5/8

To make them equal,

4/8 of water and syrup should be there

Let x be the amount of liquid we replace by water,

Water = 3x/8 and Syrup = 5x/8

Now,

Water before replacement + 5x/8 = syrup before replacement – 5x/8

⇒ syrup – water(before replacement) = 10x/8 = 5x/4

⇒ 5/8 part – 3/8 part = 5x/4

⇒ 1/4 part = 5x/4

∴ 1/5 of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup.

It is clear that all the answer options given are composite number. So, the divisor is not prime.

Approach to find the highest power of a composite number that divides n!
Step 1: Prime factorize the divisor, ‘x’ in this case.
Step 2: Compute the highest power of each of the prime factors that divides n!
Step 3: The highest power of x that divides n! is determined by the power of that prime factor which available in the least number.

Key Inference: The value of 'y' will be the highest for such an x whose highest prime factor is the smallest.

Prime Factorize numbers given in the 5 answer options
Option A: 111 = 3 × 37. The highest prime factor of 111 is 37.
Option B: 462 = 2 × 3 × 7 × 11. The highest prime factor of 462 is 11.
Option C: 74 = 2 × 37. The highest prime factor of 74 is 37.
Option D: 33 = 3 × 11. The highest prime factor of 33 is 11.
Option E: 210 = 2 × 3 × 5 × 7. The highest prime factor of 210 is 7.

210 is the number which has the smallest value of the highest prime factor among the 5 given options.
So, the value of y will be highest for 210.

Choice E is the correct answer.

Step 1: List Down Possibilities
The student is required to solve 6 out of 10 questions.
Questions are divided into 2 sections of 5 questions each.
Not more than 4 questions can be selected from any section.

Step 2: List Down Possibilities

Step 3: Count Number of Outcomes for each Possibility and Add
Possibility 1: Section 1: 4 Questions | Section 2: 2 Questions
This can be done in 5C₄ × 5C₂ = 5 × 10 = 50 ways

Possibility 2: Section 2: 3 Questions | Section 2: 3 Questions
This can be done in 5C₃ × 5C₃ = 10 × 10 = 100 ways

Possibility 3: Section 1: 2 Questions | Section 2: 4 Questions
This can be done in 5C₂ × 5C₄ = 10 × 5 = 50 ways

Total number of ways = 50 + 100 + 50 = 200 ways

Choice D is the correct answer.

To determine the approximate percent change in the number of unemployed construction workers over the given period, we need to calculate the difference in the unemployment rates and the change in the number of construction workers.

The unemployment rate dropped from 16% to 9%, indicating a decrease of 16 - 9 = 7 percentage points.

Next, we know that the number of construction workers increased by 20% from September 1, 1992, to September 1, 1996.

To calculate the percent change in the number of unemployed construction workers, we can assume that the change in the number of unemployed workers is directly proportional to the change in the number of construction workers. Therefore, the percent change in the number of unemployed workers will be the same as the percent change in the number of construction workers.

Since the number of construction workers increased by 20%, we can approximate the percent change in the number of unemployed construction workers as -20% (negative sign indicates a decrease).

Hence, the approximate percent change in the number of unemployed construction workers over this period is a 30% decrease, as given by option (B).

Step 1: What are the factors of 36?
1, 2, 3, 4, 6, 9, 12, 18, and 36.
Of these, 1, 2, 3, 4, 6, and 9 are single digit factors and can therefore, be digits of the 3 digit numbers.

Step 2: List Down Possibilities and Count
Possibility 1: Let 9 be one of the 3 digits.
The product of the remaining 2 digits will, therefore, be 4.

Possibility 2: Let 6 be one of the 3 digits.
The product of the remaining 2 digits will, therefore, be 6.


Possibility 3: 4 is one of the three digits
The product of the remaining 2 digits is 9.

Possibility 4: Let 3 be one of the three digits
The product of the remaining 2 digits is 12.


Possibility 5: Let 2 be one of the three digits
The product of the remaining 2 digits is 18.

Possibility 6: Let 1 be one of the three digits
The product of the remaining 2 digits is 36.

Number of such 3-digit positive integers is calculated by adding all the outcomes.
The outcomes from possibilities 4, 5, and 6 should not be counted because they have already been counted in the earlier possibilities.
the total Number of such 3-digit positive integers are 6 + 3 + 3 + 6 + 3 = 21 Numbers

Choice A is the correct answer.

To find the combined tax rate for John and Ingrid, we need to calculate the total amount of tax they pay and divide it by their total income.

John's income is $56,000, and he pays 30% tax. Therefore, his tax amount is 0.3 * $56,000 = $16,800.

Ingrid's income is $72,000, and she pays 40% tax. Therefore, her tax amount is 0.4 * $72,000 = $28,800.

The total tax paid by John and Ingrid is $16,800 + $28,800 = $45,600.

The combined income of John and Ingrid is $56,000 + $72,000 = $128,000.

To calculate the combined tax rate, we divide the total tax paid by the combined income and multiply by 100 to get the percentage:

Combined tax rate = ($45,600 / $128,000) * 100 ≈ 35.625 ≈ 35.6%

Therefore, the combined tax rate for John and Ingrid is approximately 35.6%, as given by option (D).

Any positive integer will have '1' and the number itself as factors. That makes it a minimum of 2 factors (except '1' which has only one factor). If the positive integer has only one more factor, then in addition to 1 and the number, the square root of the number should be the only other factor.

There are two key points in the above finding. The number has to be a perfect square. And the only factor other than 1 and the number itself should be its square root.

Therefore, if a positive integer has only 3 factors, then it should be a perfect square and it should be the square of a prime number.

How many numbers from {1, 2, 3, 4, .... 100} have exactly 3 factors?
Let us look at an example. 4 has the following factors: 1, 2, and 4 (exactly 3 factors). It is the square of '2' which a prime number.
Squares of numbers that are not prime numbers will have more than 3 factors. For instance, 36 is a perfect square. But it has 9 factors.

Number of squares of prime numbers from 1 to 100 that have exactly 3 factors are 4, 9, 25, and 49. i.e., 4 numbers

Step 1: Compute the total number of possibilities
Number of ways of selecting two distinct integers from the set of first 100 positive integers = 100C₂ ways.

 

Step 2: Compute the number of favourable outcomes
The product of two distinct numbers 'a' and 'b' will be 4 when one of the numbers is 1 and the other is 4. There is only one set that will result in this product.
The same holds good for the other 3 numbers as well. Product of two distinct numbers 'a' and 'b' will be 9 when one of the numbers is 1 and the other is 9 and so on.
Therefore, there are 4 outcomes in which the product of the two numbers will result in a number that has exactly 3 factors.

Step 3: Compute the required Probability

To solve this problem, we need to express the decimal 5.125 as a fraction. Since the remainder is always less than the divisor, the quotient in this case would be the whole number part, which is 5, and the remainder would be the fractional part, which is 0.125. Therefore, we have:

a/b = 5 + 0.125

Now, let's analyze each statement separately:

Statement (A): r = 1

Since the decimal part of the quotient is 0.125, the remainder cannot be 1. Therefore, statement (A) is not necessarily true.

Statement (B): q > r

In this case, q = 5 and r = 0.125. Since 5 is greater than 0.125, statement (B) is true.

Statement (C): b = 8

The value of the divisor, b, is not given in the problem. Therefore, we cannot determine its exact value based on the information provided. Statement (C) cannot be concluded.

Statement (D): 8r = b

The remainder, r, is 0.125. Multiplying this by 8 gives us 1, which matches the value of the divisor b (since 8 x 0.125 = 1). Therefore, statement (D) is true.

Statement (E): 8b = r

This statement suggests that the fractional part of the quotient (0.125) is equal to 8 times the divisor b. However, this is not true since 8 times any positive integer will result in a value greater than 1. Therefore, statement (E) is not true.

Based on the analysis above, the correct answer is D: 8r = b, which must be true.

The common difference is not same, and they do not fall in pattern of squares cubes etc. Rather I tried with 2²−2,3²−3, but it ended there because next should have been 4²−4 or 12.
So, it was 4th or 5th trial that I stumbled upon the correct (or should I say one of the correct ) pattern.
Took out the difference between each consecutive terms.
(6−2),(22−6),(56−22),(114−56)……
4,16,34,58…… No pattern that can justify each number.
So, went with one more difference.
(16−4),(34−16),(58−34)…… 
12,18,24……… Ok, so I have a pattern here. Each difference of difference increases by 6.
Next, should be 24 + 6 or 30 and D - 58 = 30 or D = 88, where D is the difference between 114 and next term.
Thus, answer is 114 + 88 or 202.

Numbers which are divisible by 2:

Numbers which are divisible by 3:

Numbers which are divisible by 6:

Hence, 26 + 17 - 8 = 35, Option (A)

Given:
A nation’s population triples every 100 years.
Thus the percentage at which the population grows = 200%
Let us assume that every t represents one block of 100 years.
Thus -
72,900,000 = 100,000 ∗ (1 + 200/100)t
Dividing the equation by 100,000 on both the sides and reducing the fractions we get -
729 = (1 + 2)^t
27² = 3^t
32 ∗ 3 = 3^t
Therefore t = 6
We had assumed 't' to be a block of 100 years, so 6 represents 600 years.
Year in which the population becomes 729 ∗ 10⁵
= 1200 + 600 = 1800

We can set the number of rooms to be 60, which is a convenient choice as it is the least common multiple (LCM) of 3, 4, and 5.

By multiplying 3/4 by 60, we find that 45 rooms were rented, leaving 15 rooms unrented.

Similarly, by multiplying 3/5 by 60, we determine that 36 rooms are air conditioned, while 24 rooms are not.

Therefore, out of the 36 air conditioned rooms, 24 were rented, leaving 12 unrented.

Consequently, the percentage of air conditioned rooms that were not rented is calculated as 12/15, which simplifies to 4/5 or 80%.