Ratio and Proportion: Civil Service Exam Guide

Ratio and proportion are among the most frequently tested concepts in the Civil Service Exam Numerical Ability section. These topics measure a test taker’s skill in comparing quantities, identifying relationships, and solving real-world problems involving scaling, distribution, and relative values. Mastering ratio and proportion ensures you can handle questions on mixtures, speed, work problems, and basic quantitative reasoning.

This guide provides an in-depth explanation of ratios and proportions, step-by-step examples, and practical strategies for achieving a high score.

What Is a Ratio?

A ratio compares two or more quantities of the same kind. It expresses how much of one quantity exists relative to another. Ratios may be written in several formats:

• Using a colon: 3 : 5
• As a fraction: 3/5
• In words: 3 to 5

For example, if there are 15 boys and 10 girls in a classroom, the ratio of boys to girls is:

15 : 10

which simplifies by dividing both terms by 5 into:

3 : 2

Thus, the ratio of boys to girls is 3 : 2.

Simplifying Ratios

A ratio is simplified the same way fractions are simplified—by dividing each term by their greatest common divisor (GCD).

Example:
Simplify the ratio 48 : 60.

1. Find the GCD of 48 and 60, which is 12.
2. Divide both terms by 12:
   • 48 ÷ 12 = 4
   • 60 ÷ 12 = 5

So the simplified ratio is 4 : 5.

Equivalent Ratios

Equivalent ratios express the same relationship but in scaled form. You obtain equivalent ratios by multiplying or dividing both terms by the same number.

Example:

Given 2 : 3, equivalent ratios include:

• Multiply both terms by 2 → 4 : 6
• Multiply both terms by 5 → 10 : 15
• Divide both terms by 2 → 1 : 1.5 (not always required but still equivalent)

Equivalent ratios maintain the same proportional relationship.

What Is Proportion?

A proportion is an equation stating that two ratios are equal.

Example:
2/3 = 4/6 is a proportion.

Proportions are often used to find missing values in problems involving scaling, distribution, distance, or comparison.

Solving Proportions Using Cross Multiplication

The most common technique for solving proportions is cross multiplication.

Given:

a/b = c/d

Cross multiply:

a × d = b × c

Example:

Solve for x:
4/5 = x/20

Cross multiply:

• 4 × 20 = 5 × x
• 80 = 5x
• x = 16

Thus, x = 16.

Types of Ratio and Proportion Questions in the Civil Service Exam

You will encounter several variations of ratio and proportion problems:

• Basic ratio simplification
• Finding missing values using proportions
• Dividing quantities according to a ratio
• Mixture and solution problems
• Speed, distance, and time using ratios
• Work problems involving proportion
• Percent ratio relationships

Understanding these forms helps you quickly identify the approach needed.

Dividing a Quantity in a Given Ratio

One common exam task is dividing a total amount based on a specific ratio.

Example:

Divide 700 pesos in the ratio 3 : 4.

1. Add the parts: 3 + 4 = 7.
2. Each “part” equals 700 ÷ 7 = 100.
3. Multiply each ratio term by 100:
   • Person A gets 3 × 100 = 300
   • Person B gets 4 × 100 = 400

Thus, the amount is divided into 300 and 400 pesos.

Ratio Word Problems

These appear often and require interpreting real-life situations.

Example:

The ratio of red balls to blue balls is 5 : 7. If there are 35 blue balls, how many red balls are there?

Use a proportion:

5/7 = x/35

Cross multiply:

• 7x = 175
• x = 25

There are 25 red balls.

Working With Part-Whole Ratios

Sometimes ratios represent portions of a whole.

Example:

A class ratio of boys to girls is 2 : 3. If there are 30 students in total, how many are boys?

1. Add ratio parts: 2 + 3 = 5.
2. Each part is 30 ÷ 5 = 6.
3. Boys = 2 × 6 = 12.

There are 12 boys.

Inverse Proportion

In an inverse proportion, one quantity increases as the other decreases.

Common examples:

• Number of workers vs. completion time
• Speed vs. travel time

Formula:

a₁b₁ = a₂b₂

Example:

If 6 workers can finish a job in 10 days, how long will it take 15 workers?

6 × 10 = 15 × x
60 = 15x
x = 4

The job will take 4 days.

Direct Proportion

In direct proportion, quantities increase or decrease together at the same rate.

Example:
If 5 kg of rice cost 250 pesos, how much will 8 kg cost?

Set up a proportion:

5/250 = 8/x

Cross multiply:

• 5x = 2000
• x = 400

Thus, 8 kg costs 400 pesos.

Unit Rate Problems

These questions involve finding “per unit” values.

Example:

A car travels 180 km in 3 hours. What is the speed?

Unit rate:

180 ÷ 3 = 60 km/h

This value may later be used in ratio-based speed problems in the exam.

Mixture Problems Using Ratios

Mixture problems combine quantities while maintaining proportional relationships.

Example:

A mixture contains water and alcohol in the ratio 3 : 2. If there are 18 liters of water, how much alcohol is in the mixture?

Set up proportion:

3/2 = 18/x

Cross multiply:

• 3x = 36
• x = 12

There are 12 liters of alcohol.

Scaling and Similar Figures (Geometry Ratio Problems)

These sometimes appear in the exam.

Example:

Two similar rectangles have lengths in the ratio 4 : 7. If the smaller rectangle is 12 cm long, find the length of the larger.

4/7 = 12/x

Cross multiply:

• 4x = 84
• x = 21

The larger rectangle is 21 cm long.

Shortcut Techniques for Ratio and Proportion

1. Use the Total Parts Method

This is essential for distribution problems. Add ratio parts, divide the total by this sum, then multiply by each part.

2. Identify Ratio Multipliers

If a ratio is 2 : 5 and you see 10 as one value, the multiplier is 5. You can quickly scale the other term using this multiplier.

3. Use Cross Multiplication Strategically

Cross multiplication avoids unnecessary long division and is very efficient for exam-type proportions.

4. Memorize Common Ratio Sets

Common sets include relationships such as 1:2, 2:3, 3:4, and 4:5. These often appear in word problems.

5. Check Reasonableness

A common exam pitfall is selecting numbers that do not make sense. Always estimate to verify your calculated answer.

Common Errors in Ratio and Proportion Questions

Be aware of typical mistakes that lead to wrong answers:

• Forgetting to convert total parts into actual values
• Misinterpreting which ratio corresponds to which quantity
• Failing to simplify ratios before solving
• Mixing direct and inverse proportion concepts
• Incorrectly setting up proportions (flipping ratios)

To avoid mistakes, carefully read labels, write ratios clearly, and double-check your setup before solving.

Practice Examples With Solutions

Example 1

The ratio of men to women in a group is 7 : 9. If there are 63 men, how many women?

Use the proportion:

7/9 = 63/x

Cross multiply:

• 7x = 567
• x = 81

There are 81 women.

Example 2

Divide 900 into the ratio 5 : 1.

Total parts = 5 + 1 = 6
Each part = 900 ÷ 6 = 150

Values:

• 5 parts → 5 × 150 = 750
• 1 part → 1 × 150 = 150

So the numbers are 750 and 150.

Example 3

Two numbers are in the ratio 4 : 6. Their sum is 150. Find the numbers.

Total parts = 4 + 6 = 10
Each part = 150 ÷ 10 = 15

Numbers:

• 4 × 15 = 60
• 6 × 15 = 90

The two numbers are 60 and 90.

Example 4

A machine makes 120 items in 8 hours. How many can it make in 20 hours (direct proportion)?

Set the proportion:

8/120 = 20/x

Cross multiply:

• 8x = 2400
• x = 300

The machine can make 300 items in 20 hours.

Example 5

If 4 painters finish a job in 12 days, how long will 6 painters take (inverse proportion)?

Use inverse proportion:

4 × 12 = 6 × x
48 = 6x
x = 8

It will take 8 days with 6 painters.

Strategies for Civil Service Exam Success

• Master cross multiplication because many exam questions rely on it.
• Write ratios clearly before using them in calculations.
• Draw tables for complex word problems to organize given data.
• Estimate answers to avoid selecting trap options in multiple-choice questions.
• Practice ratio-based reasoning in mixture, work, and speed questions to build flexibility.

Final Tips

Ratio and proportion form the foundation of measurable comparisons in mathematics. For the Civil Service Exam, you need strong computational skills and an understanding of how ratios appear in real-world scenarios.

With consistent practice and awareness of common mistakes, you can improve your accuracy and solve questions quickly. Mastering ratio and proportion helps you solve related topics more effectively, such as percentages, work problems, and mixture questions.

A thorough understanding of these concepts ensures a strong performance in the Numerical Ability section of the Civil Service Exam.

Problem Sets

1. Simplify the ratio 36 : 63 to its lowest terms.
2. Write the ratio 35 : 63 in simplest form.
3. Solve for x in the proportion: 3/8 = x/64.
4. Solve for x in the proportion: 5/x = 10/36.
5. Divide 1,200 pesos between A and B in the ratio 3 : 2. How much does A receive?
6. The ages of Peter and John are in the ratio 3 : 2. If the sum of their ages is 60 years, find the age of each.
7. In a class of 40 students, the ratio of boys to girls is 3 : 5. How many boys and how many girls are there?
8. The ratio of red marbles to blue marbles in a box is 5 : 9. If there are 20 red marbles, how many blue marbles are there?
9. A mixture contains milk and water in the ratio 3 : 2. If the total mixture is 10 liters, how many liters are milk and how many are water?
10. A car travels 180 kilometers in 2.5 hours. What is its average speed in kilometers per hour?
11. The time taken to complete Job A and Job B is in the ratio 3 : 4. If Job B takes 16 days to finish, how many days will Job A take?
12. Nine workers can finish a job in 16 days. Assuming the amount of work stays the same and the work rate is proportional to the number of workers, in how many days can 12 workers finish the same job?
13. The ratio of men to women in a team is 5 : 3. If there are 56 people in the team, how many are men and how many are women?
14. In a recipe, the ratio of sugar to flour is 6 : 7. If the amount of sugar is increased by 3 equal parts while the amount of flour stays the same, what is the new ratio of sugar to flour?
15. The product of two numbers is 540, and their ratio is 3 : 5. Find the two numbers.
16. Twenty-four laborers can complete a project in 12 days. If the amount of work is the same and work rate is inversely proportional to the number of laborers, how many laborers are needed to complete the project in 16 days?
17. The cost of printing books is directly proportional to the number of copies. If printing 500 copies costs 2,500 pesos, how much will it cost to print 750 copies?
18. The monthly salaries of two employees are 25,000 pesos and 40,000 pesos. Express their salaries in simplest ratio form.
19. The length and width of a rectangle are in the ratio 5 : 3. If the perimeter of the rectangle is 80 cm, find the length and the width.
20. An amount of 1,200 pesos is to be divided among P, Q, and R in the ratio 4 : 6 : 5. How much does each person receive?

Answer Key

1. 36 : 63 = 4 : 7
2. 35 : 63 = 5 : 9
3. 3/8 = x/64 → x = 24
4. 5/x = 10/36 → 5 × 36 = 10x → x = 18
5. Total parts = 3 + 2 = 5; each part = 1,200 ÷ 5 = 240.
   A’s share = 3 × 240 = 720 pesos
6. 3k + 2k = 60 → 5k = 60 → k = 12
   Peter = 3 × 12 = 36 years
   John = 2 × 12 = 24 years
7. 3 + 5 = 8 parts; each part = 40 ÷ 8 = 5
   Boys = 3 × 5 = 15
   Girls = 5 × 5 = 25
8. 5/9 = 20/x → 5x = 180 → x = 36 blue marbles
9. 3 + 2 = 5 parts; each part = 10 ÷ 5 = 2 liters
   Milk = 3 × 2 = 6 liters
   Water = 2 × 2 = 4 liters
10. Speed = distance ÷ time = 180 ÷ 2.5 = 72 km/h
11. Time ratio A : B = 3 : 4
    If B = 16 days, then A = (3/4) × 16 = 12 days
12. Workers × days is constant: 9 × 16 = 12 × x
    144 = 12x → x = 12 days
13. 5 + 3 = 8 parts; each part = 56 ÷ 8 = 7
    Men = 5 × 7 = 35
    Women = 3 × 7 = 21
14. Original ratio sugar : flour = 6 : 7
    Increased sugar by 3 parts → new sugar = 6 + 3 = 9
    New ratio = 9 : 7
15. Let the numbers be 3k and 5k.
    Product: 3k × 5k = 15k² = 540 → k² = 36 → k = 6
    Numbers: 3 × 6 = 18 and 5 × 6 = 30
16. Workers × days is constant: 24 × 12 = n × 16
    288 = 16n → n = 18 laborers
17. Direct proportion: cost per copy = 2,500 ÷ 500 = 5 pesos per copy
    For 750 copies: 750 × 5 = 3,750 pesos
18. 25,000 : 40,000 → divide both by 5,000 → 5 : 8
19. Let length = 5k and width = 3k.
    Perimeter = 2(l + w) = 80 → l + w = 40 → 5k + 3k = 8k = 40 → k = 5
    Length = 5 × 5 = 25 cm
    Width = 3 × 5 = 15 cm
20. Ratio 4 : 6 : 5 → total parts = 4 + 6 + 5 = 15
    Each part = 1,200 ÷ 15 = 80 pesos
    P = 4 × 80 = 320 pesos
    Q = 6 × 80 = 480 pesos
    R = 5 × 80 = 400 pesos

Civil Service Exam Philippines: Complete Preparation and Passing Guide