Percentage

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

Start with this

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

Worked example

What is 35% of 640?

1. 10% of 640 = 64.
   Why: Move the decimal point one place.
2. 30% = 3 × 64 = 192. 5% = half of 64 = 32.
   Why: Build 35% from easy pieces.
3. 192 + 32 = 224.

Answer: 224

Worked example

A student scores 234 out of 360. What percentage is that?

1. 234/360 × 100.
   Why: Part over whole, times 100.
2. 234/360 = 13/20 = 0.65.
   Why: Divide both by 18.
3. 65%.

Answer: 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.

Increase, decrease, and the multiplier: the rules

Worked example

After a 25% increase a price is Rs 750. What was it before?

1. New = old × 1.25.
   Why: A 25% increase is 125% of the old price.
2. Old = 750 / 1.25 = 600.
   Why: Divide to go back; do not subtract 25% of 750.

Answer: 600

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

Worked example

A'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.
   Why: B is the base in the sentence.
2. B is 20 more than A.
3. As a percentage of A: 20/80 × 100 = 25%.
   Why: 'More than A' means A is the base now.

Answer: 25%

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.

Worked example

A 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.
   Why: Increase applies to the starting population.
2. After year 2: 55000 × 0.9 = 49,500.
   Why: The fall applies to 55,000, not 50,000.
3. 49,500 (a 1% net fall: 10 - 10 - 100/100 = -1).

Answer: 49500

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.

Worked example

The 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.
   Why: Take 100 for everything.
2. New price 125. To spend 10,000: quantity = 10000/125 = 80.
3. Cut = 20%.

Answer: 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.

Worked example

In 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.
   Why: The margin is the gap between their shares.
2. 16% = 1120, so 1% = 70.
3. Valid votes = 7,000.

Answer: 7000

Worked example

A 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.
   Why: Failing by 22 means 22 short of the pass mark.
2. 40% of max = 200.
3. Max = 500.

Answer: 500

Shortcut: 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%.

Successive change in one line: try it

Price raised 25% then cut 20%. Net change?

1. 25 - 20 - 500/100 = 0.

Answer: 0%

Shortcut: 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.

Pass/fail marks pair: try it

30% fails by 30; 40% passes by 30. Max marks?

1. 10% = 60.
2. Max = 600.

Answer: 600

Common mistakes

Common mistakes

Your turn 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

Your turn 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%

Your turn 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

Your turn 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%

Recap

• 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

Now practise

16 questions with full solutions.

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