Percentages

Percentage is the backbone of arithmetic in CDS — profit/loss, interest and data all rest on it. Fluency with fraction equivalents makes these questions almost instant.

Core ideas

• x% means x/100. To find x% of N, compute (x/100) x N.
• Percentage change = (change / original) x 100.
• Successive change of a% then b% = a + b + (ab/100) percent (use a minus sign for a decrease).

Must-know fraction equivalents

• 1/2 = 50%, 1/3 = 33.33%, 1/4 = 25%, 1/5 = 20%, 1/6 = 16.67%, 1/8 = 12.5%, 1/9 = 11.11%, 1/16 = 6.25%.

Handy relationships

• If A is x% more than B, then B is [x/(100+x)] x 100 percent less than A.
• If A is x% less than B, then B is [x/(100-x)] x 100 percent more than A.
• A rise of x% followed by a fall of x% gives a NET fall of (x^2/100) percent.

Exam Tricks & Tips

• 🎯 Convert percentages to fractions (37.5% = 3/8) to multiply mentally and skip long arithmetic.
• 🎯 For successive changes use a + b + ab/100 in one step instead of two separate calculations.
• 🎯 A +x% then -x% is never zero — it is always a net LOSS of x^2/100 percent.
• 🎯 'More than / less than' is not symmetric: 20% more one way is not 20% less back — use the x/(100 plus or minus x) formula.
• 🎯 Population/price after n years of r% growth = P(1 + r/100)^n (same form as compound interest).
• ❌ Common mistake: taking the two percentages in a successive-change problem as simply additive (ignoring the ab/100 term).

Expected exam pattern

Percentage-of, percentage-change, successive-change and 'more/less than' questions; the base skill for the whole arithmetic section.

Quick recap

Memorise the fraction table, use a + b + ab/100 for successive changes, apply the x/(100 plus or minus x) rule for more/less, and note the x^2/100 net-loss result.

Cover the answer, recall, then check. 11 cards on percentages.

Q1. Formula for percentage change?
A1. (change / original value) x 100.

Q2. Successive changes of a% and b% combine to?
A2. a + b + (ab/100) percent.

Q3. 37.5% as a fraction?
A3. 3/8.

Q4. 1/6 and 1/8 as percentages?
A4. 16.67% and 12.5%.

Q5. A number rises 10% then falls 10%. Net change?
A5. A net fall of 1% (x^2/100 = 100/100).

Q6. If A is 25% more than B, how much less is B than A?
A6. [25/125] x 100 = 20% less.

Q7. 15% of 240?
A7. 36.

Q8. Price grows r% per year for n years from P. Final value?
A8. P(1 + r/100)^n.

Q9. Convert 0.2 and 0.75 to percentages.
A9. 20% and 75%.

Q10. If A is 20% less than B, how much more is B than A?
A10. [20/80] x 100 = 25% more.

Q11. How do you find x% of N?
A11. (x/100) x N.

Percentages are the master key of CDS arithmetic — profit/loss, interest, data interpretation, and population problems all reduce to percentage reasoning. Fluency here speeds up half the maths paper.

Core idea / what this tests

This tests conversion between percentages, fractions, and decimals, and the calculation of percentage increase/decrease and successive changes. A percentage is simply "per hundred," so x% of N = (x/100) × N.

Deep explanation

Beginner — the basics

• x% = x/100. To find x% of N, compute (x/100) × N.
• Fraction ↔ percentage conversions worth memorising: 1/2 = 50%, 1/3 ≈ 33⅓%, 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5%, 1/6 ≈ 16⅔%, 1/7 ≈ 14²/₇%. These turn "16⅔% of 300" into "1/6 of 300 = 50" instantly.

Intermediate — increase, decrease and finding the base

• Percentage change = (change ÷ original) × 100.
• Increase by x%: multiply by (1 + x/100); decrease by x%: multiply by (1 − x/100).
• To find the original from a changed value, divide (don't just subtract the percentage): if a price after a 20% rise is 120, original = 120 ÷ 1.20 = 100.

Advanced — successive changes and the net-change formula

For two successive percentage changes of a% and b%, the net percentage change = a + b + (ab/100), using signs (decrease is negative). Example: a 20% rise then a 20% fall gives 20 + (−20) + (20 × −20)/100 = −4%, a net 4% decrease — not zero. This formula also handles population growth, discount-then-tax, and length-breadth area changes.

Worked example

A number is first increased by 25% and then decreased by 20%. What is the net percentage change?

Step 1 — set up successive changes: a = +25, b = −20.
Step 2 — apply the net-change formula: net = a + b + (ab/100) = 25 + (−20) + (25 × −20)/100.
Step 3 — compute: 25 − 20 + (−500/100) = 5 − 5 = 0.
Check with numbers: 100 → +25% → 125 → −20% → 125 × 0.8 = 100. Same start and end.
Answer: net change is 0% (no change). The formula and the numeric check agree.

CDS relevance

Percentages appear directly and inside profit/loss, interest, and data-interpretation questions. Memorised fraction-percentage equivalents and the successive-change formula turn multi-step problems into quick mental arithmetic — a decisive time advantage under CDS pressure.

Exam tricks & shortcuts

• Memorise fraction ↔ percentage values to convert instantly.
• Successive changes: a + b + ab/100 (with signs) — never just add the percentages.
• To reverse a percentage change, divide by the multiplier, don't subtract.
• A x% more than B ⇒ B is [x/(100+x)]×100% less than A (the "more/less" asymmetry).

Key formulas

• x% of N = (x/100)·N; percentage change = (change/original)×100.
• Increase by x% then decrease by x%: net = −x²/100 % (a loss).
• Successive changes a% and b%: net = a + b + ab/100 %.
• If A is x% more than B, B is [x/(100+x)]×100 % less than A.
• Fraction ↔ percent: 1/2=50%, 1/3≈33.33%, 1/4=25%, 1/8=12.5%, 1/6≈16.67%.
• Population/price after n years at r%: P(1 ± r/100)ⁿ.

Usage: memorise fraction–percent equivalents to convert instantly.

Worked Example

Problem: A's salary is 20% more than B's salary. By what percentage is B's salary less than A's?

Solution:
The trap here is to assume the answer is also 20% — it is not, because the base of comparison changes.

Step 1 — Assign a convenient value to B. Let B's salary = 100.
Then A's salary = 100 + 20% of 100 = 120.

Step 2 — Now compare B with A. We want how much less B is than A, as a percentage of A (A is the new base):
Difference = 120 − 100 = 20.
Percentage = (difference / A) × 100 = (20 / 120) × 100 = 100/6 = 16⅔%.

So B's salary is 16⅔% (about 16.67%) less than A's.

Answer: B's salary is 16⅔% (≈ 16.67%) less than A's.