Simplification for AFCAT: BODMAS, identities and fast multiplication

Simplification questions give you an expression and ask for its value. There is no word problem to decode, so they should be among the fastest marks in the paper. They go wrong in two ways: an operation done in the wrong order, or a long calculation where a one-line identity would have done.

This guide covers the order of operations, including the "of" rule that trips many candidates, the identities that turn ugly numbers into easy ones, quick multiplication, the laws of indices (exponents are in the AFCAT syllabus) and approximation. Each method comes with the reason it works, and the worked examples are at the level AFCAT asks.

The order of operations

Work in this order:

1. Brackets, innermost first: ( ), then { }, then [ ]. A bar over part of an expression (a vinculum) is treated as the innermost bracket.
2. Of: "3/5 of 250" means 3/5 × 250, and in these exams "of" is done before division and multiplication.
3. Division and Multiplication, from left to right, whichever comes first.
4. Addition and Subtraction, from left to right.

Powers and roots are worked out before division and multiplication.

The two rules that decide most wrong answers:

• Left to right. 48 ÷ 6 × 2 is 8 × 2 = 16, not 48 ÷ 12 = 4. Division does not beat multiplication, or the other way round; whichever is on the left goes first.
• "Of" before division. 48 ÷ 4 of 3 is 48 ÷ 12 = 4, because "4 of 3" is worked out first. Compare 48 ÷ 4 × 3 = 12 × 3 = 36.

Identities that save time

The skill is recognising the pattern. When you see two numbers that are equally far from a round number, think a² − b². When you see cubes on top and squares with a middle term below, think of the cube identities.

Fast multiplication

Numbers near 100. For 97 × 96: each number is below 100 by 3 and 4. Take either number minus the other's gap (97 − 4 = 93), then multiply the gaps (3 × 4 = 12) and write it as two digits: 9,312.

Why it works: (100 − a)(100 − b) = 100(100 − a − b) + ab. The first part gives the hundreds; ab fills the last two places. That is why 98 × 97 gives 95 and then 06, not 6: the answer is 9,506.

Above 100 it works the same way with plus signs: 104 × 107 → 104 + 7 = 111, and 4 × 7 = 28 → 11,128.

One above and one below: 103 × 96 → 100(103 − 4) + 3 × (−4) = 9,900 − 12 = 9,888. The product of the gaps is negative, so subtract it.

Squares ending in 5. For n5², multiply n by (n + 1) and write 25 after it. 65² → 6 × 7 = 42, then 25: 4,225. Why it works: (10n + 5)² = 100n² + 100n + 25 = 100n(n + 1) + 25.

Multiplying by 25 or 125. 25 = 100 ÷ 4 and 125 = 1,000 ÷ 8. So 36 × 25 = 3,600 ÷ 4 = 900, and 48 × 125 = 48,000 ÷ 8 = 6,000.

Laws of indices

The trick in index questions is to rewrite every number with the same base first: 4 = 2², 8 = 2³, 9 = 3², 27 = 3³, 125 = 5³.

Approximation

When the options are far apart, round each number and estimate. 49.8 × 20.1 ÷ 9.97 ≈ 50 × 20 ÷ 10 = 100. If the options are close together, approximation is risky; calculate properly.

Worked examples

Example 1. Simplify 48 ÷ 4 of 3 + 2 × 5 − 6.

• "Of" first: 4 of 3 = 12.
• Division and multiplication: 48 ÷ 12 = 4; 2 × 5 = 10.
• Left to right: 4 + 10 − 6 = 8.

Example 2. Simplify 36 − [18 − {14 − (15 − 4 ÷ 2 × 2)}].

• Innermost bracket, left to right: 4 ÷ 2 = 2, then 2 × 2 = 4, so 15 − 4 = 11.
• Curly bracket: 14 − 11 = 3.
• Square bracket: 18 − 3 = 15.
• Finally: 36 − 15 = 21.

Example 3. Simplify (0.8³ + 0.2³) ÷ (0.8² − 0.8 × 0.2 + 0.2²).

• This is (a³ + b³) ÷ (a² − ab + b²) with a = 0.8 and b = 0.2.
• The value is a + b = 1. No cubing needed.

Example 4. Find 998 × 1,002.

• Both are 2 away from 1,000: (1,000 − 2)(1,000 + 2) = 1,000² − 2².
• 10,00,000 − 4 = 9,99,996.

Example 5. Simplify 2⁵ × 4³ ÷ 8².

• Same base: 4³ = (2²)³ = 2⁶, and 8² = (2³)² = 2⁶.
• 2⁵ × 2⁶ ÷ 2⁶ = 2⁵ = 32.

Example 6. Find the approximate value of 19.98% of 601 + 7.99 × 12.02.

• Round: 20% of 600 = 120, and 8 × 12 = 96.
• 120 + 96 ≈ 216. Pick the option closest to 216.

Practice set

1. 64 ÷ 8 × 4 − 10
2. 72 ÷ 3 of 4 + 5
3. 105 × 95
4. 98 × 97
5. 85²
6. (5.4² − 4.6²) ÷ 0.8
7. 7.5² − 2.5²
8. 3⁴ × 9² ÷ 27²

Answers:

1. 22. Left to right: 64 ÷ 8 = 8, 8 × 4 = 32, 32 − 10 = 22.
2. 11. "Of" first: 3 of 4 = 12; 72 ÷ 12 = 6; 6 + 5 = 11.
3. 9,975. (100 + 5)(100 − 5) = 10,000 − 25.
4. 9,506. 98 − 3 = 95; 2 × 3 = 06.
5. 7,225. 8 × 9 = 72, then 25.
6. 10. (5.4 − 4.6)(5.4 + 4.6) = 0.8 × 10 = 8, and 8 ÷ 0.8 = 10.
7. 50. (7.5 − 2.5)(7.5 + 2.5) = 5 × 10.
8. 9. 9² = 3⁴ and 27² = 3⁶, so 3⁴ × 3⁴ ÷ 3⁶ = 3² = 9.

What to do next

• Learn squares to 30 and cubes to 15; they turn up inside simplification all the time.
• Solve 20 BODMAS questions that include "of", and check each one for left-to-right slips.
• Practise the near-100 method until a product like 96 × 93 takes under ten seconds.
• Revise decimals and fractions, since most simplification questions mix them in, and then try approximation.