Simplification and approximation for LIC AAO

In a 20-minute Quant section with 35 questions, you have about 34 seconds a question. Simplification and approximation are where you buy time for everything else. They need no reading, no diagram and no formula beyond school arithmetic. A candidate who takes 20 seconds on each of these, instead of 50, saves several minutes for a DI set or two word problems. The speed comes from a small set of facts you know by heart and a few methods you trust.

The rules and why they work

BODMAS, including "of"

Work in this order: Brackets, Orders (powers and roots) and "of", Division and Multiplication from left to right, then Addition and Subtraction from left to right. "Of" means multiplication, but it is done before a plain division. So 48 ÷ 1/4 of 16 means 48 ÷ (1/4 × 16) = 48 ÷ 4 = 12, not (48 ÷ 1/4) × 16.

Division and multiplication have equal rank, so they are done left to right. 72 ÷ 6 × 3 is 12 × 3 = 36, not 72 ÷ 18 = 4.

Algebraic identities

• a² − b² = (a − b)(a + b). It turns two squares into one easy product: 47² − 43² = 4 × 90 = 360.
• (a ± b)² = a² ± 2ab + b². Use it for squares near a round number: 47² = (50 − 3)² = 2,500 − 300 + 9 = 2,209.
• Squares ending in 5. For a number ending in 5, multiply the tens part by the next number and write 25 after it. 65²: 6 × 7 = 42, so 4,225. This is the identity (10n + 5)² = 100n(n + 1) + 25.

Square roots of perfect squares

Look at the last digit, then bracket the number between two round squares.

For √1,849: the last digit 9 means the root ends in 3 or 7. 40² = 1,600 and 45² = 2,025, so the root is between 40 and 45. It must be 43.

Cube roots of perfect cubes

Every digit 0 to 9 gives a different last digit when cubed, so the last digit of a cube fixes the last digit of its root. Cubes ending in 2 have roots ending in 8, and cubes ending in 8 have roots ending in 2. Cubes ending in 3 and 7 swap in the same way. The other digits stay the same.

For ∛74,088: the last digit 8 means the root ends in 2. Drop the last three digits to leave 74. Since 4³ = 64 ≤ 74 < 125 = 5³, the tens digit is 4. The root is 42.

Fractions and percentages

Multiplying by 25 is dividing by 4 and multiplying by 100. Multiplying by 125 is dividing by 8 and multiplying by 1,000.

Missing-value questions: work backwards

When the unknown sits inside the expression, undo each operation in reverse order. If ? ÷ 12 × 7 = 336, then ? = 336 ÷ 7 × 12.

A quick check: the digit sum

A number leaves the same remainder on division by 9 as the sum of its digits. So you can check a product by comparing digit sums. For 138 × 27 = 3,726: the digit sum of 138 is 12, which reduces to 3. The digit sum of 27 is 9. 3 × 9 = 27, which reduces to 9. The digit sum of 3,726 is 18, which reduces to 9. They match. The check cannot catch swapped digits, but it catches most slips in seconds.

Approximation: how to round

1. Round each number to the nearest convenient value. 24.97% becomes 25%, 1,279.6 becomes 1,280 and 7.99 becomes 8.
2. Keep percentages and fractions clean. Round 33.2% to one-third and 12.4% to one-eighth when the options allow.
3. Look at the gap between options. If the options are 30 or more apart, rough rounding is safe. If two options are within a few units, round less aggressively.
4. Balance the rounding. If you round one factor up, avoid rounding the other factor up as well in a large product.

Worked examples

Example 1. 35% of 840 + 3/7 of 1,029 − ? = 18²
35% of 840 = 294, since 10% = 84, 30% = 252 and 5% = 42.
3/7 of 1,029 = 147 × 3 = 441.
18² = 324.
294 + 441 = 735, so ? = 735 − 324 = 411.

Example 2. √1,849 + ∛74,088 = ?
From the methods above, √1,849 = 43 and ∛74,088 = 42.
43 + 42 = 85.

Example 3. 47² − 43² + 65² = ?
47² − 43² = (47 − 43)(47 + 43) = 4 × 90 = 360.
65² = 4,225.
360 + 4,225 = 4,585.

Example 4. (? ÷ 12) × 7 = 37.5% of 896
37.5% = 3/8, and 3/8 of 896 = 112 × 3 = 336.
? = 336 ÷ 7 × 12 = 48 × 12 = 576.

Example 5. 24.97% of 1,279.6 + 17.02 × 22.98 − 399.8 ÷ 7.99 ≈ ?
Options: 600, 640, 661, 690, 720.
≈ 25% of 1,280 + 17 × 23 − 400 ÷ 8
= 320 + 391 − 50 = 661.

Example 6. √1,443.8 × 12.03 + 59.97% of 849.9 ≈ ?
√1,444 = 38, since 38² = 1,444.
38 × 12 = 456, and 60% of 850 = 510.
456 + 510 = 966.

Practice set

1. 45% of 560 + 5/9 of 1,062 = ? 842. 45% of 560 = 252. 1,062 ÷ 9 = 118, and 118 × 5 = 590. 252 + 590 = 842.
2. 58² − 42² = ? 1,600. (58 − 42)(58 + 42) = 16 × 100.
3. √2,209 + ∛13,824 = ? 71. √2,209 ends in 3 or 7 and lies between 45 and 50, so it is 47. ∛13,824 ends in 4, and 13 lies between 2³ and 3³, so it is 24. 47 + 24 = 71.
4. 62.5% of 1,440 − 125 × 16 ÷ 5 = ? 500. 5/8 of 1,440 = 900. 125 × 16 = 2,000, and 2,000 ÷ 5 = 400. 900 − 400 = 500.
5. ? × 14 + 12.5% of 2,400 = 1,000. Find ?. 50. 12.5% of 2,400 = 300, so ? × 14 = 700.
6. 29.98% of 2,399.7 + 44.03 × 8.97 ≈ ? About 1,116. 30% of 2,400 = 720, and 44 × 9 = 396.
7. (3,199.8 ÷ 15.97) × 4.02 − 11.99² ≈ ? About 656. 3,200 ÷ 16 = 200. 200 × 4 = 800, and 800 − 144 = 656. The exact value is nearer 662, because 3,199.8 ÷ 15.97 is slightly above 200. That gap is harmless when options are 30 or more apart, but it is why you should pick the nearest option, not look for an exact match.
8. √3,135 + 6.98 × 21.03 ≈ ? About 203. 56² = 3,136, so the root is about 56. 7 × 21 = 147, and 56 + 147 = 203.

A seven-day drill plan

What to do next

• Write squares to 50 and cubes to 25 from memory until both are error-free.
• Do one timed set of 10 simplification or approximation questions every day.
• Move on to number series and quadratic comparisons, the other quick-scoring prelims blocks.
• See where this fits in the wider quant plan for LIC AAO.