Arithmetical Reasoning

Arithmetical Reasoning develops the skill of converting word statements into mathematical operations and logical answers.

Reasoning Ability Arithmetical Reasoning Competitive Exams

Arithmetical Reasoning is an important reasoning ability topic that tests your ability to understand numerical statements, convert them into mathematical operations, and solve them logically. It combines basic arithmetic with reasoning-based interpretation.

What is Arithmetical Reasoning?

Arithmetical Reasoning refers to word-based numerical problems where the question must be understood carefully before applying arithmetic operations. These questions may involve age, ratio, percentage, profit and loss, average, time and work, speed, distance, money, or simple number logic.

Unlike direct calculation questions, arithmetical reasoning questions require you to identify what is given, what is asked, and which operation or formula is suitable. The main skill is to convert a sentence into a mathematical relation.

Quick idea: Read the question slowly, underline the given numbers, identify the relation, and then decide whether to use addition, subtraction, multiplication, division, ratio, percentage, average, or equation method.

Question Type	Core Logic	Example Focus
Number Problems	Use equations or arithmetic operations.	Find a number from clues.
Age Problems	Past, present, and future ages are connected.	Age ratio, age difference.
Percentage	Convert percentage into fraction or decimal.	Increase, decrease, marks, discount.
Average	Total value divided by number of items.	Average marks, age, speed, salary.

“Arithmetical reasoning is not only calculation; it is understanding what calculation is needed.”

Reasoning Tip

Key points

Read the question carefully.
Identify given and required values.
Convert statements into equations.
Use correct arithmetic operation.
Check units such as rupees, years, km, hours.
Verify the answer with the question condition.

numbers logic word problems calculation

Common Types of Arithmetical Reasoning Questions

Competitive exams ask arithmetical reasoning questions in different forms. The following types are commonly seen.

Number-Based Problems

Find a number using given clues.

Twice a number
Sum or difference
Product or quotient
Equation method

Age Problems

Compare ages in past, present, or future.

Present age
Age after years
Age before years
Age ratio

Percentage Problems

Use percentage to compare values.

Marks percentage
Increase/decrease
Discount
Profit/loss percent

Average Problems

Use total and number of values.

Average marks
Average age
Average salary
Missing value

Rule: Do not start calculation immediately. First classify the problem type, then choose the correct method.

Formula and Method Bank

Average

Average = Total / Number of items
Total = Average × Number of items

Percentage

Percentage = (Part / Whole) × 100
Part = Percentage × Whole / 100

Profit and Loss

Profit = SP - CP
Loss = CP - SP
Profit % = Profit / CP × 100

Speed, Time, Distance

Speed = Distance / Time
Distance = Speed × Time
Time = Distance / Speed

Tip: Most arithmetical reasoning questions become simple after writing the correct relation.

Arithmetical reasoning concept — **Arithmetical Reasoning** develops the skill of converting word statements into mathematical operations and logical answers.

Step-by-Step Solving Method

Step	Action	Example
Step 1	Read the question and identify what is asked.	Find a number, age, percentage, average, or time.
Step 2	List the given values.	Example: total = 250, percentage = 20%.
Step 3	Choose the correct formula or operation.	20% of 250 = 20 × 250 / 100.
Step 4	Calculate carefully with correct units.	20 × 250 / 100 = 50.
Step 5	Check whether the answer satisfies the question.	20% of 250 is 50, so answer is correct.

Important: In arithmetical reasoning, understanding the statement is more important than lengthy calculation.

Solved Examples

Question	Method	Answer
A number is increased by 12 and becomes 35. Find the number.	Let the number be x. x + 12 = 35. x = 35 - 12.	23
Twice a number is 48. Find the number.	Let the number be x. 2x = 48. x = 48 / 2.	24
The average of 5 numbers is 18. Find their total.	Total = Average × Number of items. Total = 18 × 5.	90
Find 25% of 240.	25% of 240 = 25 × 240 / 100. Or 25% = 1/4, so 240 / 4.	60
A shopkeeper buys an item for ₹500 and sells it for ₹650. Find the profit.	Profit = Selling Price - Cost Price. Profit = 650 - 500.	₹150
A person travels 120 km in 3 hours. Find the speed.	Speed = Distance / Time. Speed = 120 / 3.	40 km/h
A father is 4 times as old as his son. If the son is 8 years old, find the father’s age.	Father’s age = 4 × son’s age. Father’s age = 4 × 8.	32 years
A student scored 72 marks out of 90. Find the percentage.	Percentage = Marks obtained / Total marks × 100. Percentage = 72 / 90 × 100.	80%

Note: Always check whether the question asks for value, percentage, total, average, profit, speed, or age.

Common Traps and Shortcuts

Common Traps

Starting calculation without understanding the question.
Using the wrong formula for average or percentage.
Confusing cost price and selling price.
Forgetting units such as km/h, rupees, years, or marks.
Not converting percentage into proper fraction or decimal.
Ignoring words like more than, less than, twice, half, and difference.

Useful Shortcuts

Write unknown quantity as x.
Convert word statements into equations.
Use 25% = 1/4, 50% = 1/2, 75% = 3/4.
For average, remember total = average × number of items.
For speed problems, keep units consistent.
Verify the final answer with the original question.

Exam approach: Identify whether the question is based on number relation, average, percentage, profit/loss, age, or speed-time-distance.

Practice

A) Multiple Choice Questions

A number is increased by 15 and becomes 50. Find the number.
25 30 35 40
The average of 6 numbers is 20. Find their total.
100 110 120 130
Find 20% of 350.
60 70 80 90
An item is bought for ₹800 and sold for ₹950. Find the profit.
₹100 ₹120 ₹150 ₹180
A vehicle travels 180 km in 3 hours. Find the speed.
50 km/h 55 km/h 60 km/h 65 km/h

B) Solve the Higher-Order Problems

The sum of two numbers is 75. One number is 25. Find the other number. (Hint: Other number = total sum - given number.)
A student scored 84 marks out of 120. Find the percentage. (Hint: Percentage = obtained marks / total marks × 100.)
The average age of 4 persons is 30 years. Find their total age. (Hint: Total = average × number of persons.)
A man spends ₹450 out of ₹900. What percentage of money did he spend? (Hint: Spent percentage = spent amount / total amount × 100.)
A train covers 240 km in 4 hours. Find its speed. (Hint: Speed = distance / time.)

C) Match the Concept with the Correct Rule

Concept	Correct Rule / Meaning
Average	Total divided by number of items
Percentage	Part divided by whole multiplied by 100
Profit	Selling price minus cost price
Loss	Cost price minus selling price
Speed	Distance divided by time
Equation Method	Using x to represent an unknown value

Reasoning Reminder

Arithmetical reasoning questions are solved by understanding the statement and selecting the correct arithmetic operation. The most common question areas are number relations, average, percentage, profit and loss, age, and speed-time-distance.

Task: Create five arithmetical reasoning questions using percentage, average, number relation, profit/loss, and speed-time-distance.

Show Suggested Answers

Multiple Choice

35
Let the number be x.
x + 15 = 50, so x = 50 - 15 = 35.
120
Total = Average × Number of values.
Total = 20 × 6 = 120.
70
20% of 350 = 20 × 350 / 100 = 70.
₹150
Profit = Selling Price - Cost Price.
Profit = 950 - 800 = ₹150.
60 km/h
Speed = Distance / Time.
Speed = 180 / 3 = 60 km/h.

Higher-Order Problems

Sum of two numbers = 75.
One number = 25.
Other number = 75 - 25 = 50.
Percentage = 84 / 120 × 100.
= 70%.
Answer = 70%.
Total age = Average age × Number of persons.
Total age = 30 × 4 = 120 years.
Spent percentage = 450 / 900 × 100.
= 50%.
Answer = 50%.
Speed = Distance / Time.
Speed = 240 / 4 = 60 km/h.

Concept Matching

Average → Total divided by number of items
Percentage → Part divided by whole multiplied by 100
Profit → Selling price minus cost price
Loss → Cost price minus selling price
Speed → Distance divided by time
Equation Method → Using x to represent an unknown value

Clue Explanation

Arithmetical reasoning depends on converting statements into mathematical relations. Once the relation is clear, the calculation usually becomes simple and direct.

Exam tips

Underline numbers and keywords.
Identify what is asked before solving.
Use x for unknown values.
Convert percentage into fraction where possible.
Keep units consistent in speed and time questions.
Check the answer with the original statement.

Practice MCQs

Learning Modules