Percentage, Ratio, and Average Test: IQ and Logical Reasoning Questions, Tricks, and Practice

The Percentage, Ratio, and Average Test is a fundamental topic in Numerical Reasoning that evaluates your understanding of proportional relationships, comparative analysis, and central tendency calculations. These questions test your ability to calculate percentages, compare quantities using ratios, and determine averages in various scenarios. The key skill is to quickly establish the relationship between the given quantities and apply the correct formula with speed and accuracy.

For Officer Cadet Entrance Exam preparation, mastering this topic is essential because it combines arithmetic skills with logical reasoning across a wide range of applications.

In this guide, we’ll explore what percentage, ratio, and average problems involve, the key formulas you need, the common question types you’ll encounter, proven methods to solve them, and practical tricks to boost your speed and accuracy.

What is Percentage, Ratio, and Average in Numerical Reasoning?

Percentage, Ratio, and Average are three interconnected concepts used to compare quantities, express proportions, and find central values in a data set.

• Percentage – a number expressed as a fraction of 100, denoted by “%,” where “percent” literally means “per hundred”
• Ratio – a comparison of two or more quantities of the same kind, expressed as a : b
• Average (arithmetic mean) – the sum of all observations divided by the number of observations, representing the central value of a data set

Example: What is 25% of 200? Taking 25 parts out of every 100 from a total of 200 gives the part directly. Answer: 50

These concepts show up constantly in real life — calculating discounts and profit margins, comparing population or demographic data, working out average scores or expenditures, and analyzing data in economics and statistics.

What Skills Do These Questions Measure?

Percentage, ratio, and average questions test your ability in:

• Arithmetic calculation – working accurately with fractions, decimals, and whole numbers
• Understanding of proportional relationships – seeing how parts relate to wholes and to each other
• Comparing quantities – using ratios to express relative size
• Applying formulas quickly – matching the right formula to what’s given
• Speed and accuracy – solving multi-step problems under exam time pressure

These topics are high-scoring in the Numerical Reasoning section because questions follow predictable patterns with straightforward formulas. Percentage scenarios include finding a percentage of a number, finding what percentage one number is of another, successive percentage changes, and population or depreciation problems. Ratio scenarios include dividing quantities in a given ratio, comparing ratios, and solving proportions. Average scenarios include basic averages, finding a missing number from a given average, weighted averages, and age-based average problems.

Important Terms and Definitions

• Percentage – a number expressed as a fraction of 100: Percentage = (Part / Whole) × 100
• Fraction to percentage – multiply the fraction by 100; for example, 3/4 × 100 = 75%
• Percentage to fraction – divide by 100 and simplify; for example, 25% = 25/100 = 1/4
• Ratio – a comparison of two quantities of the same kind, written a : b, where 2 : 3 means for every 2 units of the first quantity there are 3 units of the second
• Proportion – an equation stating two ratios are equal; if a : b = c : d, then a, b, c, and d are in proportion (for example, 2 : 3 = 4 : 6)
• Average (arithmetic mean) – Average = Sum of all observations / Number of observations
• Weighted average – used when observations carry different weights: Weighted Average = (w₁x₁ + w₂x₂ + …) / (w₁ + w₂ + …)

Fundamental Formulas

Percentage Formulas

Successive Percentage Change Formula

For two successive percentage changes of x% and y%:

Net % Change = x + y + (xy / 100)

This works for two successive increases, two successive decreases (using negative values), or one increase and one decrease (using the appropriate signs).

Ratio Formulas

Average Formulas

Quick Conversion Table

How to Solve These Questions

Follow this step-by-step method during your exam.

Step 1: Read the Question Carefully

Identify what’s given — the part and whole for percentage problems, the quantities being compared for ratio problems, or the observations for average problems.

Step 2: Identify the Type of Question

Determine whether it’s a percentage calculation, a ratio comparison, an average calculation, or a combination of concepts.

Step 3: Select the Correct Formula

Choose the formula that matches what’s given and what you’re solving for.

Step 4: Set Up the Equation

Write out the equation using the formula you selected.

Step 5: Solve Step by Step

Work through the calculation carefully, especially when fractions or decimals are involved.

Step 6: Verify Your Answer

Check that the result is logical: a percentage should fall between 0% and 100% unless it represents an increase or decrease, a ratio should be in its simplest form, and an average should fall between the minimum and maximum values in the data set.

Types of Percentage, Ratio, and Average Questions

These problems generally fall into a handful of recurring patterns:

• Basic percentage calculation – finding a percentage of a given number
• Percentage of one number to another – expressing one value as a percentage of another
• Successive percentage change – combining two percentage changes into one net figure
• Basic ratio problems – dividing a quantity according to a given ratio
• Finding a missing number in a ratio – solving a proportion for an unknown term
• Basic average calculation – finding the mean of a set of values
• Finding a missing number from a given average – working backward from a known mean
• Weighted average – combining groups with different sizes and averages
• Combined percentage and ratio problems – applying both concepts in a single scenario
• Age-based average problems – a common variation involving people joining or leaving a group

Basic Percentage Calculation

Example: What is 25% of 200? Part = (25 × 200) / 100. Answer: 50

Finding Percentage of One Number to Another

Example: 40 is what percent of 160? Percentage = (40 / 160) × 100. Answer: 25%

Successive Percentage Change

Example: A number is increased by 10% and then decreased by 10%. What is the net percentage change? Using the successive change formula, Net % Change = 10 + (−10) + (10 × −10)/100 = 0 − 1. Answer: 1% decrease

Basic Ratio Problems

Example: Divide 360 in the ratio 2 : 3 : 4. The sum of the ratio terms is 9, so each part is found by multiplying 360 by its share of that total: 2/9, 3/9, and 4/9. Answer: 80, 120, 160

Finding a Missing Number in a Ratio

Example: If 3 : 5 = x : 20, find x. Cross-multiplying gives 3/5 = x/20, so x = (3 × 20) / 5. Answer: 12

Basic Average Calculation

Example: Find the average of 10, 20, 30, 40, 50. The sum of the five values is 150, divided by the count of 5. Answer: 30

Finding a Missing Number from a Given Average

Example: The average of 5 numbers is 20. If four numbers are 15, 18, 22, 25, find the fifth number. The total sum of all five numbers must be 100 (20 × 5), so subtracting the known four numbers’ sum of 80 leaves the fifth number. Answer: 20

Weighted Average

Example: A class has 20 boys with average weight 50 kg and 30 girls with average weight 45 kg. Find the average weight of the whole class. The boys’ total weight is 1000 kg and the girls’ total weight is 1350 kg, giving a combined total of 2350 kg across all 50 students. Answer: 47 kg

Combined Percentage and Ratio Problems

Example: A bag contains coins in the ratio 2 : 3 : 4 of denominations 10, 5, and 2 rupees respectively. If the total value is Rs. 430, find the number of 10-rupee coins. Letting the coin counts be 2x, 3x, and 4x, their combined value works out to 43x, and setting that equal to 430 gives x = 10. Answer: 20 coins

Age-Based Average Problems

Example: The average age of 20 students is 15 years. If the teacher’s age is included, the average becomes 16 years. Find the teacher’s age. The students’ combined age is 300 years, and the combined age of all 21 people (students plus teacher) must be 336 years, so the difference is the teacher’s age. Answer: 36 years

Important Tricks

Trick 1: Use Percentage Equivalents

Memorize common percentage-fraction equivalents for faster calculation: 10% = 1/10, 12.5% = 1/8, 20% = 1/5, 25% = 1/4, 33⅓% = 1/3, 50% = 1/2, 66⅔% = 2/3, 75% = 3/4.

Trick 2: Use Multiplication Factors

Instead of calculating percentages step by step, multiply the number directly — 25% is the same as multiplying by 0.25 (or dividing by 4), 20% as multiplying by 0.20 (or dividing by 5), and 50% as multiplying by 0.50 (or dividing by 2).

Trick 3: Always Simplify Ratios

Reduce every ratio to its simplest form by dividing both terms by their highest common factor.

Trick 4: Use Cross-Multiplication for Proportions

For a : b = c : d, use ad = bc to solve for a missing value quickly.

Trick 5: Use the Shortcut for a Changing Average

When a new value is added and the average increases, the relationship between the old average, the increase, and the new value can be worked out directly without recalculating the whole sum from scratch.

Trick 6: Use the Unitary Method for Ratios

Find the value of a single unit first, then multiply it out to get each part of the ratio.

Trick 7: Apply the Successive Percentage Formula

Use x + y + (xy/100) whenever you’re combining two percentage changes into one net figure.

Trick 8: Use Assumed Values for Complex Problems

For problems without concrete numbers, assume a convenient value like 100 or 100x to simplify the algebra.

Common Mistakes to Avoid

• Taking a percentage of the wrong base — always confirm whether the percentage applies to the original value or a changed one
• Adding percentage changes directly — two successive percentage changes rarely add up simply; use the net % change formula instead
• Leaving ratios unsimplified — an unreduced ratio can look like a wrong answer even when the underlying value is correct
• Confusing simple and weighted averages — a weighted average is not the same as averaging the group averages directly; the group sizes matter
• Forgetting to update the count in average problems — when someone joins or leaves a group, both the sum and the number of observations change

Quick Solving Checklist

Before selecting your answer, use this checklist:

• Read the question carefully
• Identified the type of problem (percentage, ratio, average, or mixed)
• Selected the correct formula
• Converted percentages to fractions where helpful
• Simplified ratios to their lowest terms
• Performed the calculations carefully
• Verified the answer makes logical sense
• Avoided spending too much time on one question

Practice Questions

Conclusion

Mastering the Percentage, Ratio, and Average Test is essential for the Nepal Army Officer Cadet Entrance Examination. Success requires systematic application of formulas and concepts — not guesswork — across scenarios involving proportional relationships, comparative analysis, and central tendency calculations.

Follow this disciplined approach:

Read → Identify → Apply Formula → Calculate → Verify → Answer

Regular practice enhances pattern recognition, speed, logic, and accuracy. Percentage, Ratio, and Average form the foundation of many numerical reasoning problems, and with the right formulas, conversion shortcuts, and the unitary method, these questions can be solved quickly and accurately.

This resource is part of the Officer Cadet Entrance Exam Preparation series, focusing on IQ and Logical Reasoning for the Nepal Army Officer Cadet Entrance Exam. For structured lessons, comprehensive practice banks, and mock tests, explore the Percentage, Ratio, and Average Test course on the Pratima Academy LMS.