Percent-fraction table and quick checks

Half of percentage, profit, interest and DI questions get easy once 12.5% reads as 1/8 without thinking. The digit-sum check catches a wrong product in 3 seconds.

Start with this

12.5% is 1/8. Once your brain reads it that way, '12.5% of 648' is just 648 / 8 = 81.

Percentage, profit and loss, interest and data interpretation all turn into fractions. The table on the cheat-sheet page is worth learning cold. The checks at the end of this lesson catch wrong answers before you mark them.

Read percentages as fractions

Percent means 'out of 100'. 25% is 25/100 = 1/4. Most exam percentages are one of about twenty fractions, and dividing is faster than multiplying by a decimal.

Learn the family: halves, thirds, quarters, fifths, sixths, eighths, ninths, elevenths, twelfths. 1/8 = 12.5%, 3/8 = 37.5%, 5/8 = 62.5%, 7/8 = 87.5%.

Read percentages as fractions: reference

Worked example

37.5% of 1,288

1. 37.5% = 3/8.
   Why: From the table.
2. 1288 / 8 = 161.
   Why: Divide first.
3. 161 × 3 = 483.

Answer: 483

Worked example

16.67% of 3,954

1. 16.67% = 1/6.
2. 3954 / 6 = 659.

Answer: 659

Two tricks that swap hard for easy

x% of y = y% of x. 18% of 50 is the same as 50% of 18 = 9. Flip whenever the flipped version is rounder.

Successive percentage changes a% then b%: the total change is a + b + ab/100 (use minus signs for falls). +20% then -10%: 20 - 10 - 200/100 = +8%.

Worked example

A price rises 10% and then another 10%. Total rise?

1. a + b + ab/100 = 10 + 10 + 100/100 = 21.
   Why: The second rise is on the bigger price, so there is an extra 1%.

Answer: 21%

Check your answer in three seconds

Digit-sum check (casting out nines): add the digits of each number until one digit is left (9 counts as 0). The digit sum of a product equals the digit sum of the two digit sums multiplied. If they differ, your answer is wrong.

Unit-digit check: the last digit of a product is the last digit of the two last digits multiplied. 487 × 36 must end in 7 × 6 = 42 → 2.

Size check: round both numbers and multiply. 487 × 36 is about 500 × 36 = 18,000, so 1,753 or 1,75,320 are impossible.

Worked example

Which is 487 × 36: (a) 17,532 (b) 17,542 (c) 17,432?

1. Unit digit: 7 × 6 = 42 → ends in 2. All three end in 2, so no help here.
   Why: Try the fastest check first.
2. Digit sums: 487 → 19 → 1. 36 → 9 → 0. 1 × 0 = 0.
   Why: Product's digit sum must be 0 (or 9).
3. 17532 → 18 → 0 passes. 17542 → 19 → 1 fails. 17432 → 17 → 8 fails.
4. (a).

Answer: 17,532

Shortcut: Digit-sum check

Use it when: After any long multiplication, before you mark the option.

1. Reduce each factor to a single digit by adding digits (9 counts as 0).
2. Multiply those, reduce again.
3. Reduce your answer. They must match.

Digit-sum check: try it

Is 234 × 567 = 132678?

1. 234 → 9 → 0. 567 → 18 → 0. 0 × 0 = 0.
2. 132678 → 27 → 0. Match, so it passes.

Answer: Passes

Shortcut: Percent as fraction in a question

Use it when: Whenever a % in the question is on the table.

1. Replace 37.5% by 3/8, then the number cancels cleanly.

Percent as fraction in a question: try it

37.5% of 1,288

1. 1288 / 8 = 161, x 3 = 483.

Answer: 483

Your turn medium

Which is the correct value of 487 × 36? (a) 17,532 (b) 17,542 (c) 17,432

1. Digit sums: 487 → 19 → 1, 36 → 9 → 0, product 0.
2. 17532 → 18 → 0 passes; 17542 → 19 → 1 fails; 17432 → 17 → 8 fails.

Answer: (a) 17,532

Your turn medium

83.33% of 2,454

1. 5/6 × 2454 = 409 × 5 = 2045.

Answer: 2045

Your turn medium

11.11% of 5,868 + 9.09% of 4,741

1. 5868/9 = 652. 4741/11 = 431. Sum 1083.

Answer: 1083

Recap

• read 12.5%, 16.67%, 37.5%, 83.33% as fractions without thinking
• combine two successive % changes with a + b + ab/100
• reject wrong options with digit-sum and unit-digit checks

Now practise

5 questions with full solutions.

Open practice Back to the lesson