Approximation for IBPS Clerk

Approximation questions look like simplification with ugly decimals: 44.9% of 880.3 − 12.02 × 14.97 ≈ ? The trap is to treat them like simplification and calculate every decimal. You are not expected to. The question says "approximately", and the five options are spread out so that sensible rounding lands you on the right one.

Done well, an approximation question takes 15–25 seconds. Done exactly, it takes a minute and a half, and you still have to match it to an option. The skill is knowing how much to round, and when to be careful.

How to round

The rule behind all of these: a small change in a number makes a small change in the answer, as long as you do not multiply that change by something large.

Why some rounding is dangerous

Rounding the base of a square or cube can move the answer much more than rounding a plain number. For a number a + d, where d is small:

• (a + d)² = a² + 2ad + d². The error is roughly 2ad.
• So 15.02² ≈ 225 + 2 × 15 × 0.02 = 225.6. Rounding to 225 is fine.
• But 15.4² ≈ 225 + 2 × 15 × 0.4 + 0.16 = 225 + 12 + 0.16 = 237.16. Rounding to 225 is off by 12.

So round the base of a square only when it is very close to a whole number. When it is not, use (a + d)² ≈ a² + 2ad.

Square roots work the other way. For a number just above or below a perfect square n²:

• √(n² + k) ≈ n + k ÷ (2n).
• √440 = √(441 − 1) ≈ 21 − 1/42 ≈ 20.98.
• √530 = √(529 + 1) ≈ 23 + 1/46 ≈ 23.02.

In practice this means: a number near a perfect square has a root very near the whole number. Round to it with confidence.

Using the options

Look at the options before you calculate.

• If they are far apart, for example 150, 200, 250, 300, 350, rough rounding is safe.
• If they are close together, for example 408, 412, 420, 428, 436, round more carefully and keep track of which way each rounding goes.

Balancing errors

• In a product, if you round one number up and the other down, the errors partly cancel.
• In a quotient, round the top and bottom in the same direction. 1,199.8 ÷ 39.97 becomes 1,200 ÷ 40: both went up slightly, so the ratio hardly changes.
• If you round everything up, your answer will be a little high. With close options, pick the option just below your figure rather than just above.

Percentages from 10%

Most percentages are fastest built from 10%.

• 45% of 880 = 4 × 88 + 44 = 352 + 44 = 396.
• 65% of 720 = 6 × 72 + 36 = 432 + 36 = 468.
• 35% of 1,200 = 3 × 120 + 60 = 420.

Worked examples

Example 1. 24.98% of 399.7 + 15.02² ≈ ?

1. Round: 25% of 400 + 15².
2. 100 + 225 = 325.
3. The exact value is about 325.4, so the rounding cost almost nothing.

Example 2. 1,199.8 ÷ 39.97 × 5.01 ≈ ?

1. Round: 1,200 ÷ 40 × 5.
2. Left to right: 30 × 5 = 150.

Example 3. √624 + √1,023 ≈ ?

1. Round to the nearby perfect squares: √625 + √1,024.
2. 25 + 32 = 57.

Example 4. 44.9% of 880.3 − 12.02 × 14.97 ≈ ?

1. Round: 45% of 880 − 12 × 15.
2. 396 − 180 = 216.
3. The exact value is about 215.3. Any option near 216 is right.

Example 5 (the trap). 15.4² + 29.9% of 610.2 ≈ ? Options: 408, 412, 420, 428, 436.

1. The options are close, so be careful with the square. 15.4 is not close to 15.
2. 15.4² ≈ 225 + 12 + 0.16 = 237.16.
3. 29.9% of 610.2 ≈ 30% of 610 = 183.
4. 237 + 183 = 420.
5. A candidate who rounded 15.4² to 225 gets 408, which is also an option. That is why it is there.

Example 6 (missing value). ? ÷ 7.98 + 19.98² ≈ 25.01% of 2,399.6

1. Right side: 25% of 2,400 = 600.
2. 19.98² ≈ 20² = 400.
3. ? ÷ 8 = 600 − 400 = 200, so ? ≈ 1,600.

Common mistakes

• Calculating with every decimal. Slow, and the options do not need it.
• Rounding the base of a square or cube that is not close to a whole number, as in example 5.
• Rounding the top of a fraction up and the bottom down, which pushes the answer the wrong way twice.
• Picking the first option that "looks about right" without finishing the rough calculation.

Practice

Set a 3-minute timer.

1. 19.97% of 750.4 + 8.99 × 12.01 ≈ ?
2. 3,599.7 ÷ 59.9 ≈ ?
3. √440 × 4.98 ≈ ?
4. 34.99% of 1,200.2 − 199.9 ≈ ?
5. 17.02² − 9.99² ≈ ?
6. 64.95% of 1,599.8 + 11.98 × 7.03 ≈ ?
7. 4,801.2 ÷ 39.9 ÷ 3.02 ≈ ?
8. 12.5% of 7,999.6 + ∛1,330 ≈ ?

Answers (approximate):

1. 258. 20% of 750 = 150; 9 × 12 = 108.
2. 60. 3,600 ÷ 60.
3. 105. √440 ≈ 21, so 21 × 5. (More precisely 20.98 × 4.98 ≈ 104.5, still nearest 105.)
4. 220. 35% of 1,200 = 420; 420 − 200.
5. 190. 289 − 100 = 189; the exact value is about 189.9, so pick the option nearest 189–190.
6. 1,124. 65% of 1,600 = 1,040; 12 × 7 = 84.
7. 40. 4,800 ÷ 40 = 120; 120 ÷ 3.
8. 1,011. 12.5% is 1/8, and 8,000 ÷ 8 = 1,000; ∛1,331 = 11.

What to do next

• Do five approximation questions a day with a 2-minute timer.
• Before each one, glance at the options and decide how carefully to round.
• Revise squares to 50 and cubes to 20 from the simplification guide, because near-squares are half of this topic.
• Follow the order in the numerical ability plan.