Percentage for AFCAT: multipliers, successive change and word problems

If you learn one topic well in AFCAT Numerical Ability, make it percentage. Profit and loss is percentage of cost price. Interest is percentage of principal. Discount, population growth, depreciation, election results and exam marks are all percentage questions with a different story attached. A candidate who is quick with percentage finds half the section easier.

This guide gives you the three tools that do most of the work (fraction values, multipliers and the successive-change rule), explains why each works, and then solves the question types AFCAT likes.

Tool 1: percentages as fractions

"Per cent" means "per hundred", so r% is r/100. The fast route is to know the common ones as simple fractions.

So 37.5% of 480 is 480 ÷ 8 × 3 = 180, and 16⅔% of 912 is 912 ÷ 6 = 152. No decimals, no long multiplication.

Tool 2: multipliers

An increase of r% multiplies a quantity by (1 + r/100). A decrease multiplies it by (1 − r/100).

Why this matters: a quantity that rises by 25% becomes 125% of itself, which is × 1.25. Once you think this way, "a number increased by 15% becomes 460" is simply 460 ÷ 1.15 = 400, with no equation to set up.

Percentage change is always (change ÷ original) × 100. The base is the value you are changing from. That one rule prevents most percentage mistakes.

Tool 3: successive changes

Two changes of a% and b% in a row give a net change of

a + b + (a × b)/100

with decreases entered as negative numbers.

Why it works: the multipliers are (1 + a/100) and (1 + b/100). Multiplying them gives 1 + a/100 + b/100 + ab/10,000, and turning that back into a percentage gives a + b + ab/100.

• Increases of 10% and 20%: 10 + 20 + 2 = 32% increase.
• A 10% rise then a 10% fall: 10 − 10 − 1 = 1% decrease. The fall is taken on a bigger number, so it removes more than the rise added.

For three or more changes, just multiply the multipliers.

"More than" and "less than"

If A is r% more than B, then B is less than A by

r ÷ (100 + r) × 100%

If A is r% less than B, then B is more than A by

r ÷ (100 − r) × 100%

Why it works: the difference is the same number of rupees or marks either way, but it is divided by a different base. Take B = 100 and A = 125 (25% more). The difference, 25, is 25/125 = 20% of A.

The same formula handles price and consumption. If the price rises by r%, consumption must fall by r ÷ (100 + r) × 100% to keep spending the same. If the price falls by r%, consumption can rise by r ÷ (100 − r) × 100%.

Worked examples

Example 1 (three changes). A salary is raised by 10%, raised again by 10%, and then cut by 20%. What is the net change?

• Multipliers: 1.1 × 1.1 × 0.8 = 1.21 × 0.8 = 0.968.
• Net change = 0.968 − 1 = −0.032, a 3.2% decrease.

Example 2 (price and spending). The price of petrol rises by 20%, and a driver cuts consumption by 10%. By what percentage does the driver's spending change?

• Spending = price × quantity, so multiply: 1.2 × 0.9 = 1.08.
• Spending rises by 8%. (By the formula: 20 − 10 − 200/100 = 8.)

Example 3 (election). In an election between two candidates, 10% of the voters on the list did not vote. The winner got 60% of the votes cast and won by 2,160 votes. How many voters were on the list?

• Let the voters on the list be V. Votes cast = 0.9V.
• The loser got 40% of the votes cast, so the margin is 20% of 0.9V = 0.18V.
• 0.18V = 2,160, so V = 12,000.

Example 4 (pass marks). One student scores 30% and fails by 30 marks. Another scores 40% and gets 20 marks more than the pass mark. Find the maximum marks and the pass mark.

• The gap between the two students is 30 + 20 = 50 marks, and it equals 40% − 30% = 10% of the maximum.
• Maximum marks = 50 × 10 = 500.
• Pass mark = 30% of 500 + 30 = 150 + 30 = 180. Check: 40% of 500 = 200 = 180 + 20.

Example 5 (population). A town has 50,000 people. The population grows by 10% in the first year and 8% in the second. What is it after two years?

• 50,000 × 1.1 = 55,000.
• 55,000 × 1.08 = 59,400.

Example 6 (comparing two quantities). If 20% of A equals 30% of B, find A : B.

• 0.2A = 0.3B, so A/B = 0.3/0.2 = 3/2.
• A : B = 3 : 2. The quantity with the smaller percentage must be the larger one.

Practice set

1. Find 40% of 650.
2. A's income is 20% more than B's. By what percentage is B's income less than A's?
3. The price of a commodity falls by 20%. By what percentage can a family increase consumption without spending more?
4. A number increased by 15% becomes 460. Find the number.
5. What percentage of 1.5 kg is 120 g?
6. A price is increased by 10% and then decreased by 10%. What is the net change?
7. A machine worth ₹1,00,000 loses 10% of its value every year. What is it worth after 2 years?
8. In an exam, 65% of students passed English, 70% passed Maths and 20% failed both. What percentage passed both?

Answers:

1. 260. 650 × 2/5 = 260.
2. 16⅔%. 20 ÷ 120 × 100 = 16⅔.
3. 25%. 20 ÷ 80 × 100 = 25.
4. 400. 460 ÷ 1.15 = 400.
5. 8%. 1.5 kg = 1,500 g, and 120 ÷ 1,500 × 100 = 8.
6. 1% decrease. 10 − 10 − 100/100 = −1.
7. ₹81,000. 1,00,000 × 0.9 × 0.9 = 81,000.
8. 55%. 35% failed English and 30% failed Maths. Failed at least one = 35 + 30 − 20 = 45%, so 100 − 45 = 55% passed both.

Common mistakes

• Using the wrong base. Percentage change is always on the original value. A rise from 80 to 100 is 25%; a fall from 100 to 80 is 20%.
• Adding successive changes. A 20% rise and a 20% fall do not cancel; they leave a 4% fall.
• Mixing units. Convert kg to g or hours to minutes before taking a percentage.
• Stopping one step early. In election and pass-mark questions, check whether the question asks for total votes, votes cast, maximum marks or pass marks.

What to do next

• Learn the fraction table until you can answer any row in two seconds.
• Solve 20 questions on successive change and "more than / less than".
• Move on to ratio and proportion, then profit and loss, which is percentage applied to cost price.