## GRE Ratio and Proportion

**24. GR Ratio Proportion**

#### GRE Topic: Ration and Proportion – Word Problem.

##### Ratios and Proportions – Word Problem: An Overview

Ratios and proportions are fundamental concepts tested in the GRE quantitative reasoning section. A ratio compares two quantities, often expressed as a fraction or using a colon. A proportion is an equation that states that two ratios are equal. Understanding how to work with ratios and proportions is crucial for solving word problems on the GRE.

### 1. Understanding Ratios:

A ratio is a way to compare two quantities. It can be written in different forms, such as a fraction (a/b), with a colon (a:b), or using the word “to” (a to b). Ratios can represent various relationships, such as the ratio of boys to girls in a class or the ratio of ingredients in a recipe.

**Example:** In a class of 30 students, there are 18 boys and 12 girls. The ratio of boys to girls can be represented as 18/12, 18:12, or 18 to 12. This ratio can be simplified by dividing both numbers by their greatest common divisor (6), resulting in a simplified ratio of 3/2, 3:2, or 3 to 2.

### 2. Understanding Proportions:

A proportion is an equation that states that two ratios are equal. Proportions are used to solve problems that involve relationships between different quantities or rates.

**Example:** If 3 pounds of apples cost $6, how much would 5 pounds of apples cost? This problem can be set up as a proportion, with the ratio of pounds to cost being equal for both quantities:

(3 pounds) / ($6) = (5 pounds) / (x dollars)

By cross-multiplying and solving for x, we can determine that 5 pounds of apples would cost $10.

**3. Solving Ratio and Proportion Word Problems:**

To solve ratio and proportion word problems, follow these steps:

- Read the problem carefully and identify the known quantities and the unknown value(s) you need to find. b. Set up a ratio or proportion based on the relationships between the known and unknown quantities. c. Simplify the ratio, if necessary, and solve the proportion to find the unknown value(s). d. Check your answer to ensure it is reasonable and makes sense in the context of the problem.

**Example:** A car can travel 300 miles on 15 gallons of gas. How many miles can the car travel on 25 gallons of gas?

a. Known quantities: 300 miles (distance), 15 gallons (gas used); Unknown value: miles traveled on 25 gallons of gas b. Set up a proportion: (300 miles) / (15 gallons) = (x miles) / (25 gallons) c. Solve the proportion: x = (300 miles * 25 gallons) / 15 gallons d. Check your answer: x = 500 miles. This answer is reasonable and makes sense in the context of the problem.

By understanding ratios and proportions and following the above steps, you will be well-prepared to solve ratio and proportion word problems on the GRE.

**4. Common Mistakes and Misconceptions in Ratio and Proportion Word Problems:**

a. Misinterpreting the ratio order: When working with ratios, it is essential to maintain the order of the quantities being compared. For example, reversing the order will lead to an incorrect solution if a ratio represents the number of boys to girls.

b. Failing to simplify ratios: Before solving a proportion, ensure that the ratios are in their simplest form. This will make calculations more manageable and help avoid errors.

c. Confusing ratios with fractions: Although ratios can be expressed as fractions, they differ. Ratios represent comparing two quantities, while fractions represent a single quantity, usually a part of a whole.

d. Incorrectly setting up proportions: When setting up balances, it is crucial to ensure that the units being compared are consistent across both ratios. Mixing units can lead to incorrect solutions.

5. Strategies for Avoiding Common Mistakes and Misconceptions in Ratio and Proportion Word Problems:

a. Read the problem carefully: Make sure to read the problem thoroughly and understand the context. Identify the known quantities, unknown value(s), and their relationship.

b. Write down the known information: Jot down the given information and relationships in the problem. This will help you avoid confusion and make identifying errors or misconceptions easier.

c. Double-check your calculations: After solving the problem, review your calculations to ensure you have accurately simplified ratios, maintained the correct order, and set up the proportions correctly.

d. Practice, practice, practice: As with any topic, practice is essential to develop a strong understanding and avoid common mistakes and misconceptions. Work through a variety of ratio and proportion word problems to solidify your understanding and improve your problem-solving skills.

In conclusion, ratio and proportion word problems are a critical topics in the GRE quantitative reasoning section. By understanding common mistakes and misconceptions, practicing various problem types, and using the strategies outlined in this extended concept, you will be well-prepared to tackle ratio and proportion word problems on the GRE and improve your overall performance.

### 4. Additional Ratio and Proportion Concepts:

a. Part-to-Part and Part-to-Whole Ratios: Ratios can represent a comparison of one part to another part (part-to-part) or one part to the whole (part-to-whole). It is crucial to distinguish between these two types of ratios when solving word problems.

**Example:** In a class of 24 students, there are 14 boys and 10 girls. The part-to-part ratio of boys to girls is 14:10 (or 7:5 when simplified). The part-to-whole ratio of boys to the entire class is 14:24 (or 7:12 when simplified).

b. Rate Problems: Rate problems are a type of proportion problem that involves comparing two quantities with different units. Common rate problems include speed (distance/time), work (work done/time), and concentration (quantity/volume).

**Example:** If a car travels 60 miles in 1.5 hours, what is its average speed? Set up a proportion to compare the distance and time: (60 miles) / (1.5 hours) = (x miles) / (1 hour). Solving for x, we find the car’s average speed is 40 miles per hour.

c. Proportional Relationships: When two quantities are directly proportional, they increase or decrease together at a constant rate. When two quantities are inversely proportional, one quantity increases as the other decreases.

**Example:** The weight of an object is directly proportional to its volume. If an object with a volume of 5 cubic feet weighs 25 pounds, an object with a volume of 10 cubic feet will weigh 50 pounds.

**5. Challenging Ratio and Proportion Word Problems:**

You may encounter more complex ratio and proportion word problems as you progress through more difficult GRE questions. These problems may involve multiple ratios or proportions, combining different concepts, or applying ratios and proportions to real-world scenarios.

**Example:** A mixture contains liquids A and B in a ratio of 3:2. If 10 liters of liquid A are added to the mixture, the ratio of liquids A and B becomes 5:2. What was the initial volume of the mixture?

### 6. Tips for Tackling Challenging Ratio and Proportion Word Problems:

a. Break down the problem into smaller parts: Analyze the problem and identify the different components. Solve each part separately before combining them to find the overall solution.

b. Use a systematic approach: Carefully set up the ratios and proportions, ensuring that the units are consistent, and simplify if necessary. Double-check your work to avoid errors.

c. Be persistent: Challenging word problems may require multiple attempts or different approaches. Don’t be discouraged if you don’t solve the problem immediately. Keep practicing and refining your problem-solving skills.

In summary, mastering ratio and proportion word problems on the GRE requires a strong understanding of the core concepts, familiarity with different problem types, and practice solving a range of difficulty levels. By applying the strategies outlined in this extended concept and practicing various types of ratio and proportion word problems, you will be well-equipped to tackle even the most challenging questions on the GRE.

### Ratio and Proportion Applications in Other GRE Quantitative Reasoning Topics:

a. Geometry: Ratio and proportion concepts can be applied to geometry problems, such as scaling figures, calculating similar triangles, or determining the ratio of areas or volumes.

**Example:** Two similar triangles have corresponding sides in the ratio of 3:5. If the area of the smaller triangle is 27 square units, what is the area of the larger triangle?

b. Algebra: Ratios and proportions can also be used to solve algebraic equations involving multiple variables or to represent the relationship between different quantities.

**Example:** If x:y = 4:5 and y:z = 3:2, find the ratio x:y:z.

c. Data Analysis: Ratio and proportion concepts can help analyze data in tables, charts, or graphs by comparing quantities or determining relationships between different data points.

**Example:** The ratio of men to women at a conference is 2:3. If there are 240 attendees, how many women are at the conference?

### Advanced Ratio and Proportion Problem-Solving Techniques:

a. Working with multiple ratios: When dealing with multiple ratios in a single problem, it may be necessary to manipulate the ratios to make them compatible before combining or comparing them.

**Example:** If the ratio of apples to oranges is 3:4 and the ratio of oranges to bananas is 2:5, find the ratio of apples to oranges to bananas.

b. Combining ratio and proportion with other mathematical concepts: Some problems may require the integration of ratio and proportion concepts with other mathematical concepts, such as probability, combinations, or permutations.

**Example:** In a box, the ratio of red to blue marbles is 3:7. If there are 20 marbles in the box, what is the probability of randomly drawing a red marble?

### Tips for Mastering Advanced Ratio and Proportion Problem-Solving Techniques:

a. Understand the underlying concepts: Make sure you have a strong foundation in ratio, proportion, and other mathematical images that may appear in more advanced problems.

b. Practice a wide range of problems: To develop your problem-solving skills and become more comfortable with advanced techniques, practice solving various issues that combine ratio and proportion concepts with other mathematical concepts.

## GRE Practice questions with an answer key.

**Question types: Multiple Choice (Five answer choices; one correct answer) **

#### Difficulty level: medium

**Question 1:**

In a bag of colored marbles, the ratio of red marbles to blue marbles is 3:5. If there are 24 blue marbles, how many red marbles are in the bag?

A) 9 B) 12 C) 14 D) 15 E) 18

**Question 2:**

Two rectangles have the same height. The ratio of the base of the first rectangle to the base of the second rectangle is 4:7. If the area of the first rectangle is 60 square units, what is the area of the second rectangle?

A) 105 square units B) 120 square units C) 140 square units D) 210 square units E) 245 square units

**Question 3:**

If the ratio of the width to the length of a rectangle is 5:8, and the perimeter of the rectangle is 78 units, what is the length of the rectangle?

A) 15 units B) 24 units C) 30 units D) 32 units E) 48 units

**Question 4:**

A store sells three types of pens: A, B, and C. The ratio of the number of pens A to the number of pens B is 2:3, and the ratio of the number of pens B to the number of pens C is 4:5. If the store has a total of 165 pens, how many pens B are there? A) 40 B) 45 C) 60 D) 75 E) 80

**Question 5:** A car travels 360 miles in 6 hours. If the car maintains the same average speed, how far will it travel in 8 hours? A) 480 miles B) 520 miles C) 560 miles D) 600 miles E) 640 miles

**Answer Key:**

- D
- C
- B
- C
- A

### Question types: Multiple Choice (Five answer choices; one correct answer)

### Difficulty level: Hard

**Question 1:**

The ratio of the sides of two similar triangles is 3:5. If the area of the smaller triangle is 54 square units, what is the area of the larger triangle? A) 150 square units B) 180 square units C) 225 square units D) 270 square units E) 300 square units

**Question 2:**

A container has a mixture of liquids X and Y in the ratio of 5:7. When 15 liters of liquid Y is added to the mixture, the ratio of liquid X to liquid Y becomes 5:8. What was the initial volume of the mixture? A) 45 liters B) 60 liters C) 75 liters D) 90 liters E) 105 liters

**Question 3:**

A train travels from Town A to Town B at an average speed of 60 miles per hour and then from Town B to Town C at an average speed of 90 miles per hour. The total distance between Town A and Town C is 360 miles. If the total travel time from Town A to Town C is 5 hours, what is the distance between Town A and Town B? A) 100 miles B) 120 miles C) 150 miles D) 180 miles E) 200 miles

**Question 4: **

In a certain factory, the ratio of the number of machines producing Product A to the number of machines producing Product B is 3:2. If each machine producing Product A can make 50 units per day and each machine producing Product B can make 75 units per day, what is the total number of units produced by the factory in a day, given that there are 10 machines producing Product A?

A) 825 units B) 1,000 units C) 1,250 units D) 1,500 units E) 1,750 units

**Question 5:**

If x:y = 2:3 and y:z = 4:5, find the value of x:z.

A) 6:15 B) 8:15 C) 10:15 D) 12:15 E) 16:15

**Answer Key:**

- E
- B
- D
- C
- B

### Question types: multiple choice (Five answer choices; one correct answer)

### Difficulty level: challenging

**Question 1:**

A company sells products P and Q in the ratio 4:5. The profit margin for product P is 30%, while the profit margin for product Q is 40%. If the total revenue from selling product P is $12,000, what is the total profit for both products?

A) $6,000 B) $7,000 C) $7,800 D) $9,000 E) $10,000

**Question 2:**

The ratio of the sides of two similar triangles is 2:5. If the perimeter of the smaller triangle is 36 units, what is the area of the larger triangle, given that the area of the smaller triangle is 32 square units?

A) 80 square units B) 200 square units C) 250 square units D) 320 square units E) 400 square units

**Question 3:**

In a sports league, the ratio of the number of teams from City A to the number of teams from City B is 3:4. If each team from City A has 25 players and each team from City B has 20 players, what is the ratio of the total number of players from City A to the total number of players from City B when there are 12 teams from City

A? A) 3:4 B) 5:6 C) 6:5 D) 15:16 E) 20:21

**Question 4:**

The ratio of the speeds of two cars, A and B, is 3:4. Both cars start at the same time from the same point and travel in the same direction along a straight road. Car B reaches its destination 45 minutes earlier than Car A. If the distance between the starting point and the destination is 180 miles, what is the speed of Car A?

A) 40 mph B) 48 mph C) 60 mph D) 72 mph E) 80 mph

**Question 5:**

In a mixture of two liquids, the ratio of liquid L to liquid M is 7:5. If 14 liters of liquid M is added to the mixture, the new ratio becomes 7:6. What is the initial volume of the mixture?

A) 84 liters B) 98 liters C) 105 liters D) 112 liters E) 120 liters

**Answer Key:**

- C
- E
- D
- B
- A

### Question types: Quantity Comparison (four answer choices; one correct answer)

### Difficulty level: hard

**Question 1: **

**Quantity A:** The area of a triangle with sides of lengths 6, 8, and 10 units. **Quantity B:** The area of a trapezoid with bases of lengths 8 and 12 units and a height of 5 units

A) Quantity A is greater.

B) Quantity B is greater.

C) The two quantities are equal.

D) The relationship cannot be determined from the information given.

**Question 2: **

Quantity A: The speed of a car that travels 300 miles in 5 hours. Quantity B: The speed of a car that travels 500 miles in 8 hours

A) Quantity A is greater.

B) Quantity B is greater.

C) The two quantities are equal.

D) The relationship cannot be determined from the information given.

**Question 3: **

A container has a mixture of liquids X and Y in the ratio of 4:5. When 10 liters of liquid X is added to the mixture, the ratio of liquid X to liquid Y becomes 6:5.

**Quantity A:** The initial volume of liquid X

**Quantity B:** The initial volume of liquid Y

A) Quantity A is greater.

B) Quantity B is greater.

C) The two quantities are equal.

D) The relationship cannot be determined from the information given.

**Question 4: **

The ratio of the sides of two similar triangles is 2:3. If the area of the smaller triangle is 18 square units, what is the area of the larger triangle? Quantity A: The area of the larger triangle.

Quantity B: 27 square units

A) Quantity A is greater.

B) Quantity B is greater.

C) The two quantities are equal.

D) The relationship cannot be determined from the information given.

**Question 5: **

In a class of 60 students, the ratio of boys to girls is 3:2. Quantity A: The number of boys in the class Quantity B: 36

A) Quantity A is greater.

B) Quantity B is greater.

C) The two quantities are equal.

D) The relationship cannot be determined from the information given.

**Answer Key:**

- B
- A
- B
- A
- C

### Question types: Quantity Comparison (four answer choices; one correct answer)

### Difficulty level: challenging

**Question 1:**

A car travels from Town A to Town B at an average speed of 40 miles per hour and then from Town B to Town C at an average speed of 80 miles per hour. The total distance between Town A and Town C is 300 miles. Quantity A: The distance between Town A and Town B

Quantity B: 120 miles

A) Quantity A is greater.

B) Quantity B is greater.

C) The two quantities are equal.

D) The relationship cannot be determined from the information given.

**Question 2: **

A shop sells two types of pens: A and B. The ratio of the number of pens A to the number of pens B is 2:3. If each pen A costs $4 and each pen B costs $6, and the total revenue from selling all the pens is $720.

Quantity A: The number of pens A

Quantity B: 150

A) Quantity A is greater.

B) Quantity B is greater.

C) The two quantities are equal.

D) The relationship cannot be determined from the information given.

**Question 3:**

In a right triangle, one leg is 7 units long, and the other leg is 24 units long. Quantity A: The area of the triangle. Quantity B: 84 square units

A) Quantity A is greater.

B) Quantity B is greater.

C) The two quantities are equal.

D) The relationship cannot be determined from the information given.

**Question 4: **

A company sells products P and Q in the ratio 3:2. The profit margin for product P is 20%, while the profit margin for product Q is 50%. If the total revenue from selling product P is $15,000, what is the total profit for both products? Quantity A: The total profit for both products Quantity B: $9,000

A) Quantity A is greater.

B) Quantity B is greater.

C) The two quantities are equal.

D) The relationship cannot be determined from the information given.

**Question 5: **

If x:y = 5:8 and y:z = 2:3, find the value of x:z.

Quantity A: The ratio x:z

Quantity B: 5:12

A) Quantity A is greater.

B) Quantity B is greater.

C) The two quantities are equal.

D) The relationship cannot be determined from the information given.

**Answer Key:**

- D
- B
- C
- A
- C

### Multiple Select (three or more answer choices; one or more correct answers)

### Difficulty level: hard

**Question 1:**

In a certain factory, the ratio of the number of machines producing Product A to the number of machines producing Product B is 2:3, and the ratio of the number of machines producing Product B to the number of machines producing Product C is 4:5. If the factory has a total of 110 machines, which of the following could be the number of machines producing Product

A? A) 20 B) 25 C) 40 D) 45

**Question 2:**

A car travels from Town A to Town B at an average speed of 60 miles per hour and then from Town B to Town C at an average speed of 90 miles per hour. The total distance between Town A and Town C is 420 miles. Which of the following could be the distance between Town A and Town B?

A) 100 miles, B) 120 miles, C) 160 miles, D) 180 miles

**Question 3:**

In a class of 90 students, the ratio of boys to girls is 3:2. Which of the following statements are true?

A) There are 36 boys in the class. B) There are 54 girls in the class. C) There are 54 boys in the class. D) There are 36 girls in the class.

**Question 4:**

A container has a mixture of liquids X and Y in the ratio 3:4. When 12 liters of liquid Y is added to the mixture, the ratio of liquid X to liquid Y becomes 3:5. Which of the following could be the initial volume of the mixture?

A) 42 liters B) 56 liters C) 63 liters D) 84 liters

**Question 5:**

If x:y = 3:4 and y:z = 5:6, which of the following could be the value of x:z?

A) 9:16 B) 15:24 C) 15:30 D) 15:32

**Answer Key:**

- A, C
- A, B, D
- C, D
- A, B
- B, D

**Question types: Number entry **

**Difficulty level: hard**

**Question 1: **

The ratio of the sides of two similar triangles is 3:7. If the area of the smaller triangle is 81 square units, what is the area of the larger triangle? Enter your answer as a whole number.

**Answer: 539**

**Question 2: **

A company sells products P and Q in the ratio 5:3. The profit margin for product P is 25%, while the profit margin for product Q is 45%. If the total revenue from selling product P is $20,000, what is the total profit for both products? Enter your answer as a whole number.

**Answer: 11,700**

**Question 3:**

A car travels from Town A to Town B at an average speed of 45 miles per hour and then from Town B to Town C at an average speed of 90 miles per hour. The total distance between Town A and Town C is 360 miles. What is the distance between Town A and Town B? Enter your answer as a whole number.

**Answer: 120**

**Question 4:**

In a certain factory, the ratio of the number of machines producing Product A to the number of machines producing Product B is 3:5. If each machine producing Product A can make 80 units per day and each machine producing Product B can make 60 units per day, what is the total number of units produced by the factory in a day, given that there are 15 machines producing Product A? Enter your answer as a whole number.

**Answer: 2,100**

**Question 5: **

If x:y = 4:5 and y:z = 3:4, find the value of x:z. Enter the first number of the ratio (x) only.

**Answer: 12**