Ratio, Proportion and Averages Explained for the CSE

Ratio, proportion and averages sit behind a large share of the word problems in the Civil Service Exam’s Numerical Ability section, and they also appear in data interpretation for the Professional level. The good news is that two simple ideas solve almost all of them: find the value of one part, and work with totals. This guide shows both, with worked examples.

Part 1: Ratios

A ratio compares two or more quantities. The ratio 4 : 5 means that for every 4 of the first, there are 5 of the second.

Simplifying a ratio

Divide every term by the greatest common factor.

48 : 60 → divide by 12 → 4 : 5

Sharing an amount in a ratio

1. Add the parts.
2. Divide the total by the number of parts to get the value of one part.
3. Multiply.

Share ₱24,000 in the ratio 2 : 3 : 3. Parts: 2 + 3 + 3 = 8. One part = 24,000 ÷ 8 = ₱3,000. Shares: ₱6,000, ₱9,000 and ₱9,000. Check: they add up to ₱24,000.

Working back from one share

Ana and Ben’s ages are in the ratio 3 : 4. Ana is 18. One part = 18 ÷ 3 = 6. Ben = 4 × 6 = 24.

Ratio and a total from geometry

A rectangle’s length and width are in the ratio 3 : 2, and its perimeter is 50 m. Length + width = 50 ÷ 2 = 25 m. One part = 25 ÷ 5 = 5 m. Length = 15 m, width = 10 m.

Part 2: Proportion

A proportion says two ratios are equal: 3 : 5 = x : 40. Solve by cross-multiplying: 5x = 120, so x = 24.

Direct proportion: more means more

If one quantity increases, the other increases at the same rate. Find the value for one unit first.

4 printers print 1,200 pages an hour. How many pages do 7 printers print? One printer: 1,200 ÷ 4 = 300. Seven printers: 2,100 pages.

12 kilos of rice cost ₱660. What do 5 kilos cost? One kilo: ₱55. Five kilos: ₱275.

Inverse proportion: more means less

When more workers share a job, it takes fewer days. Multiply first to find the total amount of work.

6 workers finish a repair in 10 days. How long for 15 workers? Total work = 6 × 10 = 60 worker-days. 60 ÷ 15 = 4 days.

Quick check: ask whether the answer should go up or down. More workers must mean fewer days, so if your answer is larger than 10, you used the wrong kind of proportion.

Map scales

A scale of 1 : 50,000 means 1 cm on the map is 50,000 cm in reality.

Two towns are 6 cm apart on the map. 6 × 50,000 = 300,000 cm = 3 km (since 100,000 cm = 1 km).

Unit conversion is where most scale mistakes happen.

Part 3: Averages

Average = total ÷ number of items

The total is the key

Most average problems become easy when you turn the average back into a total.

Finding a missing value

The average of four numbers is 25. Three are 20, 30 and 22. Total needed = 4 × 25 = 100. Known total = 72. Missing = 28.

Reaching a target average

A clerk encoded 120, 135, 150 and 115 records. How many on day 5 to average 130? Needed total = 5 × 130 = 650. Current total = 520. Day 5 = 130.

When someone joins or leaves

Five employees average 30 years old. A sixth joins and the average becomes 31. Totals: 5 × 30 = 150 and 6 × 31 = 186. New employee = 36.

Weighted averages

When groups are of different sizes, you cannot simply average the averages.

A student scored 80 in 2 tests and 90 in 3 tests. Total = 2 × 80 + 3 × 90 = 430. Average = 430 ÷ 5 = 86, not 85.

The same trap appears in data interpretation: the overall passing rate of several groups must be calculated from the combined totals, not by averaging their percentages.

A shortcut for evenly spaced numbers

For numbers that increase by the same amount, the average is the middle number.

12, 15, 18, 21, 24 → average = 18.

Common mistakes to avoid

• Mixing up the order of a ratio. If the question asks for “education to health”, write education first.
• Using direct proportion when it is inverse, or the reverse. Ask whether the answer should grow or shrink.
• Averaging averages for groups of different sizes.
• Forgetting units, especially in map scales and speed problems.

Practise

Work through our ratio, proportion and averages set, where every answer was calculated and checked by script, then apply the same ideas in word problems and data interpretation.