SSC CGL Reasoning · Arithmetic Reasoning

Percentages and Interest

A percentage is a fraction out of 100: change it to a simple fraction to find a part, divide by it to find the whole, and multiply the factors when two changes come one after the other.

All Arithmetic Reasoning notes

Why this matters

Twelve questions in the bank, all EASY and all from the 2025 shifts. Five find a percentage of a number or work back from one, three apply two changes one after the other, and four are about simple or compound interest.

Concept 1 of 3: A percentage as a fraction

Most SSC percentages are friendly fractions in disguise. 25% is a quarter, 20% is a fifth, 12.5% is an eighth. Change to the fraction and the arithmetic is one step.

Definition

Fractions to know: 10% $= \frac{1}{10}$ , 12.5% $= \frac{1}{8}$ , 20% $= \frac{1}{5}$ , 25% $= \frac{1}{4}$ , 50% $= \frac{1}{2}$ , 60% $= \frac{3}{5}$ , 75% $= \frac{3}{4}$ .
A part: p% of N $= \frac{p}{100} \times N$ .
Working back: if p% of a number is A, the number is $A \times \frac{100}{p}$ .
What is left: if a share is used up, the rest is 100% minus that share. Work with the share the question gives you a number for.
Profit on cost: selling price $=$ cost price $\times \frac{100 + \text{profit}\%}{100}$ .
The fraction route is exact for these friendly percentages; for a percentage such as 37% simply multiply.

A part, the whole, and selling price

p\% \text{ of } N = \frac{p}{100} \times N \qquad N = \frac{A \times 100}{p} \qquad SP = CP \times \frac{100 + p}{100}

Athe amount that is p% of the number
SP, CPselling price and cost price

Worked example

A student spends 35% of her pocket money and has ₹520 left. What is her pocket money?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 23 Sep 2025, Shift 1 · Q20Easy

Example 1 · Arithmetic Reasoning · Percentages and Interest

A person spends 80% of his income and saves ₹2000. What is his income?

Using the share spent when the amount given is what is left

If the number you are given is the money left over, it matches 100% minus the share spent. Dividing it by the spent share gives a wrong total.

Taking profit on the selling price

A profit percentage is on the cost price unless the question says otherwise. A 20% profit on a cost of ₹150 is ₹30, giving ₹180.

Drill 4 more on a percentage as a fraction

Concept 2 of 3: Two changes one after the other

The second change is taken on the new value, not the original. So a rise and an equal fall do not cancel: the fall is taken on a bigger number and wins.

Definition

Factors: a rise of a% multiplies by $1 + \frac{a}{100}$ ; a fall of b% multiplies by $1 - \frac{b}{100}$ . Multiply the two factors.
Shortcut: the net change is $a + b + \frac{ab}{100}$ percent, with a fall written as a negative number.
Area: a change in length and a change in width combine by the same rule.
Equal rise and fall of x%: the result is always a fall of $\frac{x^2}{100}$ percent.
Check: start from 100 and apply each change in turn.

Net effect of two changes

\text{Net change} = a + b + \frac{ab}{100}

athe first change in per cent (negative for a fall)
bthe second change in per cent (negative for a fall)

Worked example

A price rises by 25% and then falls by 10%. What is the net change?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 19 Sep 2025, Shift 3 · Q6Easy

Example 2 · Arithmetic Reasoning · Percentages and Interest

A number is increased by 20% and then decreased by 20%. What is the net change in the number?

Adding the two percentages

A 30% rise followed by a 30% fall is not zero change. The fall is taken on the bigger price, so the result is a 9% fall.

Dropping the sign of a fall

a + b + \frac{ab}{100}

, a fall is negative, so the product term is negative too. A 10% rise then a 10% fall is

10 - 10 - 1 = -1

, a 1% fall.

Drill 2 more on two changes one after the other

Concept 3 of 3: Simple and compound interest

Simple interest is the same every year, worked out on the original sum. Compound interest is earned on the amount at the end of the year before, so two consecutive years' amounts give the rate directly.

Definition

Simple interest: $SI = \frac{P \times R \times T}{100}$ . Rearranged, $R = \frac{SI \times 100}{P \times T}$ .
A sum that becomes k times itself at simple interest has earned $(k - 1)$ times the principal, so $R \times T = 100(k - 1)$ . Doubling: $R \times T = 100$ .
Compound interest from two consecutive years: the gap between the amounts is one year's interest on the earlier amount, so the rate is that gap divided by the earlier amount.

Simple interest, k times, and compound rate

SI = \frac{PRT}{100} \qquad R \times T = 100(k - 1) \qquad R = \frac{A_{n+1} - A_n}{A_n} \times 100

Pthe principal
Rthe rate in per cent a year
Tthe time in years
Aₙthe compound amount after n years

Worked example

A sum becomes 3 times itself in 16 years at simple interest. What is the rate?

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 · Q21Easy

Example 3 · Arithmetic Reasoning · Percentages and Interest

A principal amount of ₹1,000 yields a simple interest of ₹180 over a period of 2 years. What is the annual rate of simple interest?

Counting the whole amount as interest

A sum that doubles has earned interest equal to the principal, not twice it. A sum that becomes 2.5 times has earned 1.5 times the principal.

Dividing by the later amount

For a compound rate from two consecutive years, divide the gap by the earlier year's amount. Dividing by the later amount gives a rate that is too small.

Drill 3 more on simple and compound interest

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)

A percentage as a fraction
A part, the whole, and selling price
$p\% \text{ of } N = \frac{p}{100} \times N \qquad N = \frac{A \times 100}{p} \qquad SP = CP \times \frac{100 + p}{100}$
Two changes one after the other
Net effect of two changes
$\text{Net change} = a + b + \frac{ab}{100}$
Simple and compound interest
Simple interest, k times, and compound rate
$SI = \frac{PRT}{100} \qquad R \times T = 100(k - 1) \qquad R = \frac{A_{n+1} - A_n}{A_n} \times 100$

Watch out for (6)

Using the share spent when the amount given is what is left
→ A percentage as a fraction
Taking profit on the selling price
→ A percentage as a fraction
Adding the two percentages
→ Two changes one after the other
Dropping the sign of a fall
→ Two changes one after the other
Counting the whole amount as interest
→ Simple and compound interest
Dividing by the later amount
→ Simple and compound interest

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.

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