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.
Why this topic matters
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%.
| Fraction | Percent | Fraction | Percent |
|---|---|---|---|
| 1/2 | 50% | 1/8 | 12.5% |
| 1/3 | 33.33% | 1/9 | 11.11% |
| 1/4 | 25% | 1/11 | 9.09% |
| 1/5 | 20% | 1/12 | 8.33% |
| 1/6 | 16.67% | 1/16 | 6.25% |
| 1/7 | 14.28% | 1/20 | 5% |
- 37.5% = 3/8.From the table.
- 1288 / 8 = 161.Divide first.
- 161 × 3 = 483.
- 16.67% = 1/6.
- 3954 / 6 = 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%.
- a + b + ab/100 = 10 + 10 + 100/100 = 21.The second rise is on the bigger price, so there is an extra 1%.
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.
- Unit digit: 7 × 6 = 42 → ends in 2. All three end in 2, so no help here.Try the fastest check first.
- Digit sums: 487 → 19 → 1. 36 → 9 → 0. 1 × 0 = 0.Product's digit sum must be 0 (or 9).
- 17532 → 18 → 0 passes. 17542 → 19 → 1 fails. 17432 → 17 → 8 fails.
- (a).
Formula summary
Shortcuts and tricks
Digit-sum check
Use it when: After any long multiplication, before you mark the option.
- Reduce each factor to a single digit by adding digits (9 counts as 0).
- Multiply those, reduce again.
- Reduce your answer. They must match.
- 234 → 9 → 0. 567 → 18 → 0. 0 × 0 = 0.
- 132678 → 27 → 0. Match, so it passes.
Percent as fraction in a question
Use it when: Whenever a % in the question is on the table.
- Replace 37.5% by 3/8, then the number cancels cleanly.
- 1288 / 8 = 161, x 3 = 483.
Before you move on, you should be able to...
- 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
Practice questions (5)
Try each one before opening the solution. Or practise them one by one so your score is saved.
easy 62.5% of 1,352
- 5/8 × 1352 = 169 × 5 = 845.
Answer: 845
easy 14.28% of 2,891 (approx 1/7)
- 2891 / 7 = 413.
Answer: 413
medium Which is the correct value of 487 × 36? (a) 17,532 (b) 17,542 (c) 17,432
- Digit sums: 487 → 19 → 1, 36 → 9 → 0, product 0.
- 17532 → 18 → 0 passes; 17542 → 19 → 1 fails; 17432 → 17 → 8 fails.
Answer: (a) 17,532
medium 83.33% of 2,454
- 5/6 × 2454 = 409 × 5 = 2045.
Answer: 2045
medium 11.11% of 5,868 + 9.09% of 4,741
- 5868/9 = 652. 4741/11 = 431. Sum 1083.
Answer: 1083
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 min | Quiz: say each fraction as a percent, fast. |
| 25 min | The table, with ten '% of' questions done by division. |
| 20 min | x% of y = y% of x, and successive changes. |
| 25 min | Checks: digit sum, unit digit, size. |
| 30 min | Practice set, timed. |
| 10 min | Recap. |