Lessons / Quantitative aptitude
Quantitative aptitude · Day 1

Percentage

A percentage is a fraction with 100 underneath. Swap it for the fraction from the table and most questions become one division.

Practise (16 questions) Present as slidesCheat sheet: Fraction to percentage table

Why this topic matters

Percent means 'per hundred'. 30% is 30 out of every 100. That is the whole topic; everything else is choosing which number is the 100.

Percentage is inside profit and loss, discount, interest, population, elections and every DI chart. The one skill that decides most answers: knowing what the percentage is OF. The base changes from question to question.

What a percentage is, and finding x% of a number

x% of a number = x/100 × the number. 30% of 250 = 30/100 × 250 = 75.

Quicker: 10% is one tenth, 1% is one hundredth. 30% of 250 = 3 × 25 = 75. 15% = 10% + 5% (half of 10%).

To write a number as a percentage of another: (part / whole) × 100. 45 out of 60 = 45/60 × 100 = 75%.

30% = 7570% = 1750100%
30% of 250: split the whole into 100 equal parts, take 30
Example 1What is 35% of 640?
  1. 10% of 640 = 64.Move the decimal point one place.
  2. 30% = 3 × 64 = 192. 5% = half of 64 = 32.Build 35% from easy pieces.
  3. 192 + 32 = 224.
Example 2A student scores 234 out of 360. What percentage is that?
  1. 234/360 × 100.Part over whole, times 100.
  2. 234/360 = 13/20 = 0.65.Divide both by 18.
  3. 65%.

Increase, decrease, and the multiplier

A 20% increase means the new value is 120% of the old one: multiply by 1.2. A 20% decrease means 80%: multiply by 0.8.

Percentage change = (change / ORIGINAL value) x 100. The original is always the base.

Going back: if a price after a 25% rise is 500, the original is 500 / 1.25 = 400, not 500 - 25% of 500.

New = old × (1 + r/100)
Increase by r%.
New = old × (1 - r/100)
Decrease by r%.
% change = (new - old)/old × 100
Always divide by the old value.
Example 1After a 25% increase a price is Rs 750. What was it before?
  1. New = old × 1.25.A 25% increase is 125% of the old price.
  2. Old = 750 / 1.25 = 600.Divide to go back; do not subtract 25% of 750.

'More than' and 'less than' are not symmetric

If A is 25% more than B, B is NOT 25% less than A. Take B = 100, then A = 125. B is 25 less than A, and 25/125 = 20%.

Rule: if A is r% more than B, then B is r/(100 + r) × 100 % less than A. If A is r% less than B, B is r/(100 - r) × 100 % more than A.

B100A100+25
A = 125 when B = 100. The same 25 is 25% of B but only 20% of A
Example 1A's income is 20% less than B's. By what percent is B's income more than A's?
  1. Let B = 100. Then A = 80.B is the base in the sentence.
  2. B is 20 more than A.
  3. As a percentage of A: 20/80 × 100 = 25%.'More than A' means A is the base now.

Successive changes

Two changes one after the other: the second applies to the NEW value, so you cannot just add them.

Net change for a% then b% = a + b + ab/100 (use minus for decreases). +10% then +10% = 21%. +20% then -20% = -4%.

Population questions are successive changes over years: P × (1 + r/100)n.

Example 1A town's population of 50,000 grows 10% in the first year and falls 10% in the second. Population after two years?
  1. After year 1: 50000 × 1.1 = 55,000.Increase applies to the starting population.
  2. After year 2: 55000 × 0.9 = 49,500.The fall applies to 55,000, not 50,000.
  3. 49,500 (a 1% net fall: 10 - 10 - 100/100 = -1).

Price and consumption, income and spending

Spending = price × quantity. If the price rises and you want to spend the same, you must use less.

If price rises by r%, consumption must fall by r/(100 + r) × 100 % to keep spending the same. For a 25% rise: 25/125 = 20% cut.

Same idea for income, expenditure and savings: savings = income - expenditure. Take income = 100 and follow each change.

Example 1The price of sugar rises by 25%. By how much must a family cut consumption to keep spending the same?
  1. Take price 100, quantity 100, spend 10,000.Take 100 for everything.
  2. New price 125. To spend 10,000: quantity = 10000/125 = 80.
  3. Cut = 20%.

Elections, marks and passing

Election questions: votes for the winner minus votes for the loser = winning margin. If 20% did not vote, only 80% of the voters are counted.

Pass-mark questions: 'A gets 30% and fails by 15 marks, B gets 40% and gets 35 more than the pass mark'. The difference between them, 10% of the maximum, is 15 + 35 = 50 marks. So the maximum is 500.

Example 1In an election between two candidates, the winner got 58% of the valid votes and won by 1,120 votes. How many valid votes were there?
  1. Loser got 42%. Margin = 58 - 42 = 16% of valid votes.The margin is the gap between their shares.
  2. 16% = 1120, so 1% = 70.
  3. Valid votes = 7,000.
Example 2A student needs 40% to pass. He gets 178 marks and fails by 22 marks. What is the maximum mark?
  1. Pass mark = 178 + 22 = 200.Failing by 22 means 22 short of the pass mark.
  2. 40% of max = 200.
  3. Max = 500.

Formula summary

x% of y
x/100 × y. Use the fraction table: 12.5% = 1/8, 16.67% = 1/6.
A is r% more than B → B is less than A by
r / (100 + r) × 100. (25% more → 20% less.)
A is r% less than B → B is more than A by
r / (100 - r) × 100. (20% less → 25% more.)
Successive changes a% then b%
a + b + ab/100 (use minus signs for decreases).
Price up r%, keep spend same
Cut consumption by r / (100 + r) × 100.
Population / depreciation
P(1 + r/100)n, or (1 - r/100)n for decline.
Election, 2 candidates
Winning margin % of valid votes = winner% - loser%.

Shortcuts and tricks

Successive change in one line

Use it when: Any two changes stacked on each other.

  1. Net = a + b + ab/100.
  2. +20 then -20: 20 - 20 - 400/100 = -4%.
ExamplePrice raised 25% then cut 20%. Net change?
  1. 25 - 20 - 500/100 = 0.

Pass/fail marks pair

Use it when: 'Scores x% and fails by m; scores y% and gets n more than pass'.

  1. The gap in % covers the gap in marks: (y - x)% of Max = m + n.
Example30% fails by 30; 40% passes by 30. Max marks?
  1. 10% = 60.
  2. Max = 600.

Common mistakes

Undoing a 25% rise by taking 25% off the new price.
Divide by 1.25.
Adding successive changes: +10% and +10% = +20%.
It is 21%: the second 10% is on 110.
Using the wrong base: 'B is less than A by ...' divided by B.
Divide by the value after 'than'.

Before you move on, you should be able to...

  • find any percentage using 10% and 1% pieces
  • go back from a new value by dividing by the multiplier
  • explain why 25% more is not 25% less
  • combine two changes with a + b + ab/100
  • solve election and pass-mark questions

Practice questions (16)

Try each one before opening the solution. Or practise them one by one so your score is saved.

easy If A's income is 25% more than B's, by what % is B's income less than A's?
  1. 25/125 × 100.

Answer: 20%

easy Sugar price rises 25%. By what % must a family cut consumption to keep spending the same?
  1. 25/125 × 100.

Answer: 20%

easy A town of 1,00,000 grows 10% in year 1 and falls 10% in year 2. Population after 2 years?
  1. 100000 × 1.1 × 0.9.

Answer: 99,000

easy What is 45% of 360?
  1. 10% = 36, so 40% = 144, 5% = 18.Build it from easy pieces.
  2. 144 + 18 = 162.

Answer: 162

easy A number increased by 15% becomes 460. Find the number.
  1. Number × 1.15 = 460.115% of the number.
  2. 460 / 1.15 = 400.

Answer: 400

medium Ravi spends 80% of his income. Income rises 20% and spending rises 10%. By what % do his savings rise?
  1. Income 100, spend 80, save 20.
  2. New: 120, 88, 32.
  3. 12/20 = 60%.

Answer: 60%

medium A number is raised 20% and then cut 25%, giving 540. Find the number.
  1. x × 1.2 × 0.75 = 0.9x = 540.

Answer: 600

medium If A is 50% more than B, by what percent is B less than A?
  1. B = 100, A = 150.Take 100.
  2. 50/150 × 100 = 33.33%.

Answer: 33.33%

hard In an election between 2 candidates, 10% of votes were invalid. The winner got 58% of valid votes and won by 1,440. Total votes cast?
  1. Margin = 58 - 42 = 16% of valid.
  2. Valid = 1440/0.16 = 9000.
  3. That is 90% of total → 10000.

Answer: 10,000

hard Price of petrol rises 20% and then another 10%. By how much must consumption fall (overall) to keep the bill the same? (2 d.p.)
  1. Net price rise = 20 + 10 + 2 = 32%.
  2. Cut = 32/132 × 100 = 24.24%.

Answer: 24.24%

hard The value of a machine depreciates 10% a year. It is worth Rs 72,900 after 3 years. Original value?
  1. 72900 / 0.93 = 72900 / 0.729.

Answer: Rs 1,00,000

hard 65% passed English, 60% passed Maths, 25% failed both. What % passed both?
  1. Failed English 35, failed Maths 40, failed both 25.
  2. Failed at least one = 35 + 40 - 25 = 50.
  3. Passed both = 50%.

Answer: 50%

hard A salary is cut 20% and then raised 20%, giving Rs 9,600. Original salary?
  1. 0.8 × 1.2 = 0.96.
  2. 9600/0.96.

Answer: Rs 10,000

hard A earns 50% more than B; B earns 20% less than C. A's salary is what % of C's?
  1. A = 1.5B = 1.5 × 0.8C = 1.2C.

Answer: 120%

hard A population rises 10%, then 20%, then falls 10%, reaching 11,880. Starting population?
  1. 1.1 × 1.2 × 0.9 = 1.188.

Answer: 10,000

hard A student needs 36% to pass. She scores 150 and fails by 30 marks. Maximum marks?
  1. Pass mark 180 = 36%.
  2. 180/0.36.

Answer: 500

Class plan (2 hours, for trainers)

A tested order for teaching this topic in one 2-hour session. Present mode follows the same order.

10 minSpeed warm-up: fraction-percent table.
15 minWhat % means; x% of y with 10% and 1% pieces.
15 minIncrease, decrease, multiplier, going back.
15 minMore than / less than asymmetry with the bar picture.
15 minSuccessive change and population.
15 minPrice and consumption, income and savings.
15 minElections and marks.
15 minPractice set, timed.
5 minRecap.
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