SSC CGL Reasoning · Arithmetic Reasoning

Equations from Word Problems

Turn each fact in the question into an equation in one or two unknowns, solve, and check the answer against every fact, or put the options straight into the facts.

All Arithmetic Reasoning notes

Why this matters

Fifteen questions in the bank, twelve EASY and three MODERATE, all from the 2025 shifts. Six give two facts about two numbers, two ask which relation follows from two equations, two are about ages, and five share a fixed total among a new number of people or ask for a simple average.

Concept 1 of 4: Two facts about two numbers

Two unknowns need two facts. Write each fact as an equation, and the pair settles both numbers. Several kinds come up so often that a one-line route exists for each.

Definition

Sum S and difference D: larger $= \frac{S + D}{2}$ , smaller $= \frac{S - D}{2}$ .
Consecutive numbers: call the middle one $x$ . Three consecutive even (or odd) numbers are $x - 2$ , $x$ , $x + 2$ , and the middle one is the sum divided by 3.
A two-digit number with tens digit $a$ and units digit $b$ is $10a + b$ . Reversing the digits changes it by $9(b - a)$ .
Heads and legs: pretend every animal has 2 legs. The legs left over, divided by 2, give the number of 4-legged animals.
Option test (fast route): put each option into both facts and keep the one that fits both. It is safe when the options are the unknowns themselves; when the question asks for something built from them (a product, a sum of squares), find the numbers first.

Two numbers from their sum and difference

\text{Larger} = \frac{S + D}{2} \qquad \text{Smaller} = \frac{S - D}{2}

Sthe sum of the two numbers
Dtheir difference

Worked example

A farmer has hens and goats: 30 heads and 84 legs in all. How many goats are there?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 24 Sep 2025, Shift 2 · Q22Easy

Example 1 · Arithmetic Reasoning · Equations from Word Problems

The sum of two numbers is 40 and their difference is 4. What is the product of the numbers?

Stopping at the middle number

When the question asks for the largest of three consecutive even numbers, the sum divided by 3 gives the middle one. Add 2 to answer what was asked.

Taking the digit difference as the change

Reversing a two-digit number changes it by 9 times the difference of its digits, not by the difference itself. A change of 36 means the digits differ by 4.

Drill 5 more on two facts about two numbers

Concept 2 of 4: Which relation follows from two equations

You are not asked to solve for each letter, only to find a relation that must be true. Adding or subtracting the two equations usually cancels the letters you do not need.

Definition

Add the two equations, or subtract one from the other, then cancel any letter that appears on both sides.
Rearrange what is left and compare it with each option.
Number test (fast route): choose values for two letters, use the given equations to work out the rest, and test every option. An option that fails is out.
The number test can only rule options out. If two options survive, try a second set of numbers, or prove the survivor by adding the equations.

Worked example

P - Q = R

and

R + S = 2P

, which of these must be true:

S = P + Q

S = P - Q

Q = P + S

R = S

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 23 Sep 2025, Shift 2 · Q6Moderate

Example 2 · Arithmetic Reasoning · Equations from Word Problems

X + Y = 2Z

and

Z + W = 2X

, then:

Trusting one number test

One set of numbers can let a wrong option pass by chance. Use numbers to rule options out, then confirm the last one by adding or subtracting the equations.

Picking numbers that break the given equations

Choose only two letters freely and work out the others from the equations. Numbers that do not satisfy the given equations prove nothing.

Drill 1 more on which relation follows from two equations

Concept 3 of 4: Ages: the gap never changes

Everyone grows older at the same rate, so the difference between two people's ages is the same in every year. Write each age in terms of one unknown and move it forward or back by the same number of years for everyone.

Definition

Call the present age $x$ . Then $t$ years ago it was $x - t$ , and in $t$ years it will be $x + t$ .
In $t$ years both people are $t$ years older.
The age gap stays fixed; the ratio of the ages changes.
"Now k times what it was t years ago": $x = k(x - t)$ .
Option test: ages are small whole numbers, so trying each option in both statements is often the fastest route.

An age now and an age t years ago

x = k(x - t)

xthe present age
tthe number of years ago
khow many times the old age the present age is

Worked example

A father is 24 years older than his son. In 6 years he will be three times as old as his son. How old is the son now?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 26 Sep 2025, Shift 1 · Q24Easy

Example 3 · Arithmetic Reasoning · Equations from Word Problems

A man is 4 times as old as his daughter. In 20 years, he will be twice as old as she will be then. How old is the daughter now?

Ageing only one person

"In 20 years" adds 20 to every age in the question. Adding it to the father's age only breaks the equation.

Keeping the ratio fixed

A mother three times her son's age now will not be three times his age later. The difference stays the same; the ratio shrinks towards 1.

Drill 1 more on ages: the gap never changes

Concept 4 of 4: Fixed totals and equal shares

When people drop out of a shared bill, the bill does not shrink: the same total is split among fewer people. Find the total first, then divide it by the new count.

Definition

Total $=$ number of people $\times$ share. When people leave or join and the total stays the same, the new share is the total divided by the new count.
Fast route: the share scales by old count over new count. With 12 people becoming 8, each pays $\frac{12}{8} = 1.5$ times as much.
Average of two numbers: their sum divided by 2.
Evenly spaced numbers (consecutive, consecutive odd, consecutive even): the average is the middle term, or (first $+$ last) $\div 2$ .
The last of $n$ consecutive odd numbers starting at $a$ is $a + 2(n - 1)$ .

New share and the average of evenly spaced numbers

\text{New share} = \frac{\text{Old count} \times \text{Old share}}{\text{New count}} \qquad \text{Average} = \frac{\text{First} + \text{Last}}{2}

Worked example

Twelve people agree to pay ₹30 each for a gift. Four of them drop out. How much must each of the others pay so that the gift can still be bought?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 16 Sep 2025, Shift 2 · Q25Easy

Example 4 · Arithmetic Reasoning · Equations from Word Problems

10 friends went for dinner and planned to split Rs. 200 equally. 2 didn't pay. How much did each of the rest pay?

Dividing the old share instead of the total

When people leave, first rebuild the total (count times share), then divide by the people who remain. Dividing the old share by anything gives a smaller amount, which cannot be right.

Finding the last term by adding the count

Five consecutive odd numbers starting at 15 end at

15 + 2 \times 4 = 23

, not

15 + 5 = 20

. Their average is the middle one, 19.

Drill 4 more on fixed totals and equal shares

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Two facts about two numbers
Two numbers from their sum and difference
$\text{Larger} = \frac{S + D}{2} \qquad \text{Smaller} = \frac{S - D}{2}$
Ages: the gap never changes
An age now and an age t years ago
$x = k(x - t)$
Fixed totals and equal shares
New share and the average of evenly spaced numbers
$\text{New share} = \frac{\text{Old count} \times \text{Old share}}{\text{New count}} \qquad \text{Average} = \frac{\text{First} + \text{Last}}{2}$

Watch out for (8)

Stopping at the middle number
→ Two facts about two numbers
Taking the digit difference as the change
→ Two facts about two numbers
Trusting one number test
→ Which relation follows from two equations
Picking numbers that break the given equations
→ Which relation follows from two equations
Ageing only one person
→ Ages: the gap never changes
Keeping the ratio fixed
→ Ages: the gap never changes
Dividing the old share instead of the total
→ Fixed totals and equal shares
Finding the last term by adding the count
→ Fixed totals and equal shares

Test yourself on Arithmetic Reasoning

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Arithmetic Reasoning chapter test Drill all 15 questions