Approximation for AFCAT: round safely, check the options, save time

AFCAT gives you 100 questions in two hours, with no calculator and a mark taken away for every wrong answer. That works out to about 72 seconds a question across the whole paper. Long multiplication of decimals eats that time fast.

Approximation is the skill of getting close enough, quickly enough, to pick the right option. Some questions ask for it directly ("? is approximately equal to"). Many more can be solved or checked faster by estimating first. The skill is not guessing. It is controlled rounding: you know roughly how big your error is and whether the options leave room for it.

Step 1: look at the options before you calculate

The options tell you how accurate you need to be. Find the two closest options and compare their gap with their size.

• Options 800, 1,000, 1,200, 1,400: the gap is 200 on numbers near 1,000, which is 20%. A rough estimate is enough.
• Options 224, 231, 238, 245: the gap is 7 on numbers near 230, about 3%. You need to be more careful, or exact.

A good working rule: if your estimate could be off by less than half the gap between neighbouring options, the estimate is safe.

Step 2: round to friendly numbers

Round each number to something easy to work with: a multiple of 10, a whole number, or a value with a simple fraction.

• 49.9 → 50, 20.2 → 20, 601 → 600, 39.9 → 40.
• 12.5% → 1/8, 33.3% → 1/3, 14.28% → 1/7.

Rounding 49.9 to 50 changes it by only 0.2%. Rounding 20.2 to 20 changes it by 1%. Always think of rounding errors as percentages of the number, because that is how they carry through multiplication and division.

Step 3: compensate, and know why it works

This is where most candidates lose accuracy without noticing.

In a product, round one number up and the other down. If you raise a by 1% and b by 1%, the product rises by about 2%, because the percentage errors add. If you raise one and lower the other, the errors roughly cancel.

In a quotient, round both numbers the same way. In a ÷ b, raising both a and b by about the same percentage leaves the ratio almost unchanged. Rounding them in opposite directions doubles the error.

In a sum, errors add as plain amounts, so rounding every term up is a bad idea when there are many terms.

Friendly fractions to know

Questions often use 14.28%, 16.66% or 33.33% precisely because the fraction makes the division neat. When you see one of these, switch to the fraction at once.

Square roots and cube roots

Find the nearest perfect square. For a closer value, use this refinement:

√(n² + d) ≈ n + d/(2n) when d is small compared with n².

It works because (n + d/2n)² = n² + d + (d/2n)², and the last term is tiny. For example, √630: 25² = 625, so d = 5 and √630 ≈ 25 + 5/50 = 25.1. The true value is 25.0998.

For cube roots, know the cubes up to 15 (for example 11³ = 1,331, 12³ = 1,728, 15³ = 3,375). Then ∛3,370 ≈ 15.

Worked examples

Example 1 (compensating product). 49.9 × 20.2 ≈ ? Options: 900, 1,000, 1,100, 1,200.

• 49.9 → 50 (up 0.2%) and 20.2 → 20 (down 1%). One up, one down.
• 50 × 20 = 1,000. The exact value is 1,007.98, within 1%. The options are 10% apart, so this is safe.

Example 2 (percentages). 29.8% of 601 + 15.2% of 399 ≈ ?

• 30% of 600 = 180.
• 15% of 400 = 60.
• Total ≈ 240. The exact value is 239.75.

Example 3 (quotient and root). 7,998 ÷ 39.9 + √1,443 ≈ ?

• Both parts of the quotient go up: 8,000 ÷ 40 = 200. The ratio barely moves.
• 38² = 1,444, so √1,443 ≈ 38.
• Total ≈ 238. The exact value is 238.44.

Example 4 (friendly fraction). 14.28% of 3,493 ≈ ?

• 14.28% is almost exactly 1/7.
• 3,493 is close to 3,500, and 3,500 ÷ 7 = 500. The exact value is 498.8.

Example 5 (powers). (4.98)² + (3.02)³ ≈ ?

• 5² + 3³ = 25 + 27 = 52. The exact value is 52.34.
• Note that the cube magnifies errors: 3.02 is 0.7% above 3, but its cube is about 2% above 27. Powers multiply percentage errors by the power.

Example 6 (when rounding is not enough). 18.5 × 12.5 = ? Options: 224, 231, 238, 245.

• Rounding one up and one down gives 18 × 13 = 234, which sits between 231 and 238. That is useless here.
• Calculate exactly instead. Multiplying by 12.5 is the same as multiplying by 100 and dividing by 8: 18.5 × 100 ÷ 8 = 1,850 ÷ 8 = 231.25, so 231.

When not to approximate

• Close options, as in Example 6.
• "Exact value" questions, such as simplification with fractions that cancel neatly.
• Many steps, where small errors compound. Round once, near the end, not at every line.
• Differences of nearly equal numbers. In 501 − 498, rounding both to 500 gives 0 instead of 3. Subtraction can turn a tiny rounding error into a huge relative error.
• Powers and remainders, where the last digit or the exact value matters.

Practice set

1. 19.9 × 30.1 ≈ ? Options: 500, 600, 700, 800.
2. 24.8% of 399 ≈ ?
3. √899 ≈ ?
4. 1,201 ÷ 29.9 ≈ ?
5. 12.5% of 1,601 + 33.3% of 902 ≈ ?
6. √1,610 to one decimal place ≈ ?
7. 3,589 ÷ 11.9 × 4.02 ≈ ? Options: 1,100, 1,200, 1,300, 1,400.
8. 999.8 ÷ 24.9 + 16.1 × 5.02 ≈ ?

Answers:

1. 600. 20 × 30, one rounded up and one down. Exact: 598.99.
2. 100. 25% of 400 = 100. Exact: 98.95.
3. 30. 30² = 900, and the refinement gives 30 − 1/60 ≈ 29.98.
4. 40. 1,200 ÷ 30, both rounded the same way. Exact: 40.17.
5. 500. 1/8 of 1,600 = 200, and 1/3 of 900 = 300. Exact: 500.49.
6. 40.1. 40² = 1,600, d = 10, so 40 + 10/80 = 40.125. Exact: 40.12.
7. 1,200. 3,600 ÷ 12 = 300, and 300 × 4 = 1,200. Exact: 1,212.4, closest to 1,200.
8. 120. 1,000 ÷ 25 = 40, and 16 × 5 = 80. Exact: 120.97.

What to do next

• Learn the friendly-fraction table until 1/7, 1/9 and 1/11 come without thinking.
• Learn squares to 30 and cubes to 15, so root estimates are instant.
• In your next mock, glance at the options before every calculation and note which questions an estimate would have solved.
• Pair this with calculation speed and simplification, which use the same habits.