GRE Quant: Ratio-Proportion

A ratio shows how two quantities compare, and a proportion shows a part out of the whole based on that ratio. For example, if the ratio of boys to girls in a class is 2 to 3, then the proportion of boys in the class is 2 out of 5. These simple concepts play a vast role in arithmetic and help you understand share, distribution, and relative size in a clear way. From this base, the GRE builds a wide range of interesting and nuanced questions that test how well you track relationships between quantities across different settings and formats.

The following video, part of our GRE course, explains ratio and proportion in a simple and structured manner. It shows how to set up ratios correctly, convert them into proportions, handle scaling, and manage changing totals. The video then applies the method on GRE-style problems so you experience the application first-hand. Use this learning steadily across your GRE prep, drills, and GRE practice tests.

Ratio and Proportion: What is the Difference?

Ratio

A ratio describes the quantity of one thing in relation to another thing.

• Example: If there are 2 red balls and 1 yellow ball, the ratio is expressed as 2 : 1.
• Pronunciation: It is spoken as a to b.

Proportion

A proportion describes the quantity of one thing in relation to the whole thing (the total).

• Example: In a group of 3 balls where 2 are red, the proportion of red balls is 2 in 3.
• Pronunciation: It is spoken as a in b or a in total.

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A Conceptual Practice Set…

Correct Answers

• A/C = 3/32
• A/D = 3/80
• A : B : C = 3 : 12 : 32
• A : B : C : D = 3 : 12 : 32 : 80

 

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

 

Following is a concise, step-wise written explanation…

 

Question

A/B = 1/4; B/C = 3/8; C/D = 2/5

1. A/C = ?
2. A/D = ?
3. A : B : C = ?
4. A : B : C : D = ?

 

Step by Step Solution

Find A/C

Multiply the first two ratios:

(A/B) * (B/C)

= (1/4) * (3/8)

A/C = 3/32

Find A/D

Multiply all three ratios:

(A/B) * (B/C) * (C/D)

= (1/4) * (3/8) * (2/5) A/D

= (1 * 3 * 2) / (4 * 8 * 5)

= 6/160

A/D = 3/80

Find A : B : C

To combine A/B = 1/4 and B/C = 3/8, the value of B must be the same in both.

Multiply A/B by 3: A/B = 3/12

Multiply B/C by 4: B/C = 12/32

Now B is 12 in both.

A : B : C = 3 : 12 : 32

Find A : B : C : D

From the previous question, A : B : C = 3 : 12 : 32.

The given ratio C/D is 2/5.

To match the value of C, multiply C/D by 16: C/D = (2 * 16) / (5 * 16) = 32/80

Now C is 32 in both sets.

A : B : C : D = 3 : 12 : 32 : 80

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A Conceptual Example

Correct Answers

• Multiplication by a positive integer p
• Division by a positive integer p

 

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

 

Following is a concise, step-wise written explanation…

 

Question

Which of the following operations, when carried out on both the numerator and the denominator of a fraction will always result in an equivalent fraction?

1. Multiplication by a positive integer p
2. Division by a positive integer p
3. Adding a positive integer p
4. Subtracting a positive integer p
5. Squaring (the numerator as well as the denominator)
6. Taking square root (of the numerator as well as the denominator)

 

Step-by-Step Explanation

Concept of Equivalent Fractions

An equivalent fraction represents the same value as the original fraction. This is only maintained when you scale the parts of the fraction by the same factor through multiplication or division.

Testing Each Operation

• Multiplication and Division: If you have a fraction 1/2 and multiply both parts by 2, you get 2/4. If you divide both parts of 2/4 by 2, you get 1/2. The value 0.5 does not change. These operations always work.
• Addition and Subtraction: If you take 1/2 and add 1 to both parts, you get 2/3. Since 0.5 does not equal 0.66, the fraction is not equivalent. The same happens with subtraction.
• Powers and Roots: If you take 1/2 and square both parts, you get 1/4. Since 0.5 does not equal 0.25, it is not equivalent. Taking a square root also changes the ratio.

Conclusion

Only multiplying or dividing the top and bottom by the same non-zero number keeps the fraction’s value the same.

 

Correct Answers

• Multiplication by a positive integer p
• Division by a positive integer p

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An Explanatory Example…

Correct Answer: -1320.

 

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

 

Following is a concise, step-wise written explanation…

 

Question

a:b = 1:3. When a is increased by 20 and b is decreased by 40, the ratio becomes 1:4. What is the value of 2a + 3b?

 

Step-by-step Explanation

Step 1: Define the variables

Since the ratio of a to b is 1:3, we can write: a = x b = 3x

Step 2: Set up the equation

According to the problem, when a increases by 20 and b decreases by 40, the new ratio is 1:4. (x + 20) / (3x – 40) = 1 / 4

Step 3: Solve for x

Cross-multiply to find x: 4 * (x + 20) = 1 * (3x – 40) 4x + 80 = 3x – 40 4x – 3x = -40 – 80 x = -120

Step 4: Find the values of a and b

a = x = -120 b = 3x = 3 * (-120) = -360

Step 5: Calculate the final expression

Substitute the values into 2a + 3b: 2 * (-120) + 3 * (-360) -240 + (-1080) -1320

 

Correct Answer: -1320

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GRE-style Example 1

Correct Answer: A
Quantity A is greater.

 

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

 

Following is a concise, step-wise written explanation…

 

Question

p : q = 1 : 3

Quantity A

(p + 1) / 10

Quantity B

(q + 1) / 30

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-by-step Explanation

Step 1: Simplify the ratio

The ratio p : q = 1 : 3 means that q = 3 * p.

Step 2: Substitute into Quantity B

Replace q with 3 * p in the expression for Quantity B: Quantity B = (3 * p + 1) / 30

Step 3: Make the denominators the same

To compare easily, multiply the top and bottom of Quantity A by 3: Quantity A = 3 * (p + 1) / (3 * 10) Quantity A = (3 * p + 3) / 30

Step 4: Compare the numerators

Now we compare (3 * p + 3) from Quantity A with (3 * p + 1) from Quantity B. Since 3 is greater than 1, (3 * p + 3) will always be greater than (3 * p + 1) regardless of the value of p.

 

Correct Answer: A
Quantity A is greater.

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GRE-style Example 2

Correct Answer: D
The relationship cannot be determined from the information given.

 

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

 

Following is a concise, step-wise written explanation…

 

Question

p and q are positive integers.

Quantity A

p / q

Quantity B

(p – 1) / (q – 1)

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 by step explanation

1. To compare fractions, use the picking numbers method.
2. Case 1: Let p = 2 and q = 3. Quantity A = 2 / 3 (which is 0.66). Quantity B = (2 – 1) / (3 – 1) = 1 / 2 (which is 0.50). In this case, Quantity A is greater.
3. Case 2: Let p = 3 and q = 2. Quantity A = 3 / 2 (which is 1.5). Quantity B = (3 – 1) / (2 – 1) = 2 / 1 (which is 2.0). In this case, Quantity B is greater.
4. Because Quantity A was greater in one scenario and Quantity B was greater in another, the relationship is not consistent.

 

Correct Answer: D
The relationship cannot be determined from the information given.

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GRE-style Example 3

Correct Answer: 300.

 

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

 

Following is a concise, step-wise written explanation…

 

Question

The ratio of boys to girls in two classes, A and B, is 1:2 and 3:1. When the two classes are combined, the ratio becomes 1:1. If there are 60 boys in A, what is the total number of students in the two classes combined?

 

Step by step explanation

Step 1: Analyze Class A

In Class A, the ratio of boys to girls is 1:2. We are told there are 60 boys.

• If 1 unit = 60 boys, then 2 units = 60 * 2 = 120 girls.
• Total students in Class A = 60 boys + 120 girls = 180 students.

Step 2: Set up Class B

In Class B, the ratio of boys to girls is 3:1. Let the number of boys be 3x and the number of girls be x.

Step 3: Combine the classes

When combined, the total number of boys must equal the total number of girls because the ratio is 1:1.

• Total boys = 60 + 3x
• Total girls = 120 + x

Set them equal to each other: 60 + 3x = 120 + x

Step 4: Solve for x

• Subtract x from both sides: 60 + 2x = 120
• Subtract 60 from both sides: 2x = 60
• Divide by 2: x = 30

Step 5: Calculate total students

• Total boys = 60 + 3(30) = 60 + 90 = 150
• Total girls = 120 + 30 = 150
• Total combined students = 150 + 150 = 300

 

Correct Answer: 300

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