GRE Quant: Variation & Proportionality

Variation and proportionality explain how one quantity changes with another in a defined relationship. In direct variation, values move together, and in inverse variation, one value increases as the other decreases. Proportionality also extends to higher powers, such as one quantity being proportional to the square of another. These relationships help you connect quantities, understand scaling, and follow how changes pass through a setup. On the GRE, variation and proportionality appear in a wide mix of questions and word problems that test how accurately you read and apply these relationships.

The following video, part of our GRE prep course online, explains variation and proportionality in a clear and easy to follow way. It shows how to work with direct and inverse relationships, set up proportional expressions correctly, and handle cases involving higher powers. The lesson then works through GRE-style problems so you see the ideas applied clearly and can carry the method into your GRE prep, quizzes, GRE sectional mocks, and GRE mocks.

Overview of Variation

Variation involves relationships between variables using a constant.

The Variation Equation

When a variable A varies directly with the square of B and inversely with C, the relationship is expressed as: A is proportional to B² / C

To turn this into an equation, a constant k is introduced: A = k * (B² / C)

Step 1: Find the Constant (k)

Given the initial values A = 100, B = 20, and C = 5:

• 100 = k * (20² / 5)
• 100 = k * (400 / 5)
• 100 = k * 80
• k = 5/4 (or 1.25)

Step 2: Solve for the Unknown

To find C when A = 60 and B = 8, use the constant k = 5/4:

• 60 = (5/4) * (8² / C)
• 60 = (5/4) * (64 / C)
• 60 = 80 / C
• 60 * C = 80
• C = 80/60 = 4/3

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GRE-style Practice Question 1

Correct Answer: D

Increase by 1500%

 

For a detailed explanation, please refer to the video presented earlier on this page.

 

Following is a concise, step-wise written explanation…

 

Question

The quantities P and Q are related by the equation P = k / Q², where k is a constant. A decrease of 75% in the value of Q leads to what effect in the value of P?

1. Increase by 50%
2. Increase by 300%
3. Increase by 400%
4. Increase by 1500%
5. Increase by 1600%

 

Step-by-step Solution

1. Understand the initial state

Let the initial value of Q be 100. Initial P = k / 100² = k / 10000.

2. Calculate the new Q

A 75% decrease means the new Q is 25% of the original. New Q = 25.

3. Calculate the new P

New P = k / 25² = k / 625.

4. Compare the values

To find the relationship, divide the new P by the initial P: (k / 625) / (k / 10000) = 10000 / 625 = 16. The new P is 16 times the original P.

5. Find the percentage increase

An increase to 16 times the original value is a 1500% increase. Calculation: (16 – 1) * 100 = 1500%.

 

Correct Answer: D

Increase by 1500%

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GRE-style Practice Question 2

Correct Answer: $3,000

 

For a detailed explanation, please refer to the video presented earlier on this page.

 

Following is a concise, step-wise written explanation…

 

Question

Jack’s salary includes a fixed component and a variable component. The variable component involves t for every additional car that he sells after the first 20 cars. In May, Jack sold 30 cars and drew a salary of $4,000. In June, he sold 45 cars and drew a salary of $5,500. What is the fixed component of Jack’s salary?

 

Step-By-Step Solution

1. Identify cars sold over the 20 car limit for May: 30 minus 20 equals 10 cars.
2. Identify cars sold over the 20 car limit for June: 45 minus 20 equals 25 cars.
3. Determine the difference in salary between the two months: $5,500 minus $4,000 equals $1,500.
4. Determine the difference in the number of commissionable cars sold: 25 minus 10 equals 15 cars.
5. Calculate the commission value (t) for one car: $1,500 divided by 15 equals $100 per car.
6. Calculate the total variable component for the month of May: 10 cars times $100 equals $1,000.
7. Subtract this variable component from the total May salary to isolate the fixed component: $4,000 minus $1,000 equals $3,000.
8. Verify using June data: 25 cars times $100 equals $2,500 in variable pay. $5,500 total salary minus $2,500 variable pay equals $3,000 fixed component.

 

Correct Answer: $3,000

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GRE-style Practice Question 3

Correct Answer: C
The two quantities are equal.

 

For a detailed explanation, please refer to the video presented earlier on this page.

 

Following is a concise, step-wise written explanation…

 

Question

The total rent of a guest house includes a fixed cost and an additional cost per guest, which is the same for each guest. The total rent is $2,000 for 50 guests and $3,200 for 100 guests.

Quantity A

The total rent for 200 guests

Quantity B

$5,600

1. Quantity A is greater.
2. Quantity B is greater.
3. The two quantities are equal.
4. The relationship cannot be determined from the information given.

 

Step 1: Find the cost per guest

The number of guests increased by 50 (from 50 to 100). The total rent increased by $1,200 (from $2,000 to $3,200). Since the fixed cost does not change, the $1,200 increase is due to the 50 extra guests. Cost per guest = $1,200 / 50 = $24.

Step 2: Find the fixed cost

Total rent for 50 guests is $2,000. The cost for these 50 guests is 50 * $24 = $1,200. Fixed cost = Total rent – Variable cost. Fixed cost = $2,000 – $1,200 = $800.

Step 3: Calculate Quantity A

Total rent for 200 guests = Fixed cost + (200 * Cost per guest). Total rent = $800 + (200 * $24). Total rent = $800 + $4,800 = $5,600.

Step 4: Compare

Quantity A is $5,600. Quantity B is $5,600.

 

Correct Answer: C
The two quantities are equal.

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GRE-style Practice Question 4

Correct Answer: 100 million

 

For a detailed explanation, please refer to the video presented earlier on this page.

 

Following is a concise, step-wise written explanation…

 

Question

A bacterial population grows by the same factor on passage of each hour. The bacterial population 8 hours ago was 1 thousand and the population two hours ago was 1 million. What will be the population of bacteria two hours from now?

1. 0.01 million
2. 0.1 million
3. 10 million
4. 100 million
5. 1 billion

 

Step by Step Solution

Identify the growth period

The time between 8 hours ago and 2 hours ago is 6 hours. During these 6 hours, the population grew from 1 thousand to 1 million.

Calculate the growth factor

To get from 1 thousand to 1 million, the population increased by a factor of 1000. Since this happened over 6 hours, we let r be the hourly growth factor. r⁶ = 1000

Determine the future time gap

The time between 2 hours ago and 2 hours from now is 4 hours. We need to find the population after these 4 hours of growth.

Apply the growth factor

The growth over 4 hours is r⁴. We know r⁶ = 10³. This means r² = 10. Therefore, r⁴ = (r²)² = 10² = 100.

Calculate the final population

Multiply the population from 2 hours ago by the 4 hour growth factor. 1 million * 100 = 100 million.

 

Correct Answer: 100 million

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