In this guide
DILR is a reasoning section, but every DI set and many LR sets end in arithmetic. An aspirant who needs thirty seconds for 63 ÷ 360 loses several minutes across a paper, and in a 40-minute section that can be the difference between finishing a set and abandoning it.
Speed here does not come from tricks you look up on exam day. It comes from a small body of facts you know without thinking, plus a few methods you have used so often that they are automatic. This guide lists those facts and methods, with worked examples, and ends with a drill. Every figure has been re-computed.
The fraction–percentage table
This is the single most useful table in DI. Learn it both ways: see 14.29% and think 1/7; see 1/7 and think 14.29%.
| Fraction | % | Fraction | % | Fraction | % |
|---|---|---|---|---|---|
| 1/2 | 50 | 1/8 | 12.5 | 1/14 | 7.14 |
| 1/3 | 33.33 | 1/9 | 11.11 | 1/15 | 6.67 |
| 1/4 | 25 | 1/10 | 10 | 1/16 | 6.25 |
| 1/5 | 20 | 1/11 | 9.09 | 1/17 | 5.88 |
| 1/6 | 16.67 | 1/12 | 8.33 | 1/19 | 5.26 |
| 1/7 | 14.29 | 1/13 | 7.69 | 1/20 | 5 |
Multiples follow directly: 3/8 = 37.5%, 5/8 = 62.5%, 2/7 = 28.57%, 3/7 = 42.86%, 5/6 = 83.33%, 4/9 = 44.44%, 5/12 = 41.67%, 3/11 = 27.27%.
Using it. 63/360: divide both by 9 to get 7/40, which is 7 × 2.5% = 17.5%. 58/290 = 1/5 = 20%. 45/315 = 1/7 = 14.29%.
Squares from 11 to 30
| n | n² | n | n² | n | n² | n | n² |
|---|---|---|---|---|---|---|---|
| 11 | 121 | 16 | 256 | 21 | 441 | 26 | 676 |
| 12 | 144 | 17 | 289 | 22 | 484 | 27 | 729 |
| 13 | 169 | 18 | 324 | 23 | 529 | 28 | 784 |
| 14 | 196 | 19 | 361 | 24 | 576 | 29 | 841 |
| 15 | 225 | 20 | 400 | 25 | 625 | 30 | 900 |
Squares matter in DILR for area and growth questions, and for estimating square roots. For numbers ending in 5, multiply the leading part by one more than itself and write 25: 65² = 6 × 7 then 25 = 4,225; 115² = 11 × 12 then 25 = 13,225.
Multiplication shortcuts
| Shortcut | How | Example |
|---|---|---|
| × 5 | × 10, then halve | 348 × 5 = 3,480 ÷ 2 = 1,740 |
| × 25 | × 100, then ÷ 4 | 64 × 25 = 6,400 ÷ 4 = 1,600 |
| × 125 | × 1,000, then ÷ 8 | 36 × 125 = 36,000 ÷ 8 = 4,500 |
| × 11 (two digits) | Write the digits apart and put their sum between | 43 × 11 = 4 (4+3) 3 = 473; 68 × 11 = 6 (14) 8 = 748 with a carry |
| Near 100 | Subtract each from 100, cross-subtract, multiply the gaps | 96 × 93: gaps 4 and 7; 96 − 7 = 89; 4 × 7 = 28; answer 8,928 |
| Above 100 | Same with excesses | 103 × 107: 107 + 3 = 110; 3 × 7 = 21; answer 11,021 |
| Around a middle number | (a + b)(a − b) = a² − b² | 41 × 39 = 1,600 − 1 = 1,599 |
Percentage shortcuts
Percentage of a number. Build from 10%, 5% and 1%. 15% of 640 = 64 + 32 = 96. 35% of 180 = 3 × 18 + 9 = 63.
Successive changes. Two changes of a% and b% in a row give a net change of a + b + ab/100 percent. Use negative signs for falls.
| Changes | Net | Check by multipliers |
|---|---|---|
| +10%, then +20% | 10 + 20 + 2 = +32% | 1.1 × 1.2 = 1.32 |
| +20%, then −20% | 20 − 20 − 4 = −4% | 1.2 × 0.8 = 0.96 |
| +25%, then −20% | 25 − 20 − 5 = 0% | 1.25 × 0.8 = 1.00 |
| +30%, then −30% | 30 − 30 − 9 = −9% | 1.3 × 0.7 = 0.91 |
| +10% three times | 1.1 × 1.1 × 1.1 | 1.331, that is +33.1% |
Undoing a change. After a rise of r%, the fall needed to return to the start is r/(100 + r). After a 25% rise, you need 25/125 = 20% off. After a 20% rise, 20/120 = 16.67%. After a 30% rise, 30/130 ≈ 23.08%.
Growth to fraction. A growth of 12.5% means the new value is 9/8 of the old. If a value grew 12.5% to 540, the old value was 540 × 8/9 = 480.
Division by estimation
Round the divisor to a friendly number, then correct.
- 437 ÷ 19: 19 × 23 = 437 exactly, and 19 × 20 = 380 gets you close enough to see it.
- 333 ÷ 2,570: 10% is 257, the remaining 76 is about 3% (77.1): about 13%.
- 101 ÷ 880: 10% is 88, 12.5% is 110, so the answer is between, about 11.5%.
A checking trick: digit sums
The digit sum of a product matches the product of the digit sums, after reducing each to a single digit (dropping multiples of 9). It catches most slips in seconds.
Example. Is 54 × 552 = 29,808? Digit sum of 54 is 9, which reduces to 0. So the product's digit sum must also reduce to 0. 2 + 9 + 8 + 0 + 8 = 27, which reduces to 0. Consistent. (The check cannot catch swapped digits, so treat it as a filter, not a proof.)
Common mistakes
- Adding successive percentages. +20% then −20% is not zero; it is −4%.
- Using the wrong base to undo a change. Undoing a 25% rise needs 20% off, not 25%.
- Calculating when the options are far apart. Read the options first.
- Practising only in long sessions. Ten minutes daily beats one hour weekly for recall.
A ten-minute daily drill
Do these in your head, then check. A fresh set each day, same format.
- 72/480 as a percentage.
- 36 × 125.
- 97 × 96.
- A price rises 20% and then falls 25%. Net change?
- 17/68 as a percentage.
- 85².
- 52 × 48.
- After a 40% rise, what percentage fall restores the original?
- 15% of 348.
- Which is larger, 5/12 or 3/7?
Answers:
- 15%. 72/480 = 3/20.
- 4,500.
- 9,312. Gaps 3 and 4; 97 − 4 = 93; 3 × 4 = 12.
- −10%. 20 − 25 − 5 = −10; check 1.2 × 0.75 = 0.9.
- 25%. 17/68 = 1/4.
- 7,225. 8 × 9 = 72, then 25.
- 2,496. 50² − 2² = 2,500 − 4.
- About 28.57%. 40/140 = 2/7.
- 52.2. 10% is 34.8, 5% is 17.4.
- 3/7. 42.86% against 41.67%; or cross-multiply, 5 × 7 = 35 against 3 × 12 = 36.
What to do next
- Write the fraction table from memory every day this week until it takes under two minutes.
- Run the ten-question drill daily for a month, and time it.
- Apply the shortcuts in calculation-heavy DI and pie charts.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Indian Institutes of Management website .
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