SSC CGL Preparation: Concepts and Practice

Learning outcome

Make useful estimates, measure their limitations and check numerical answers without confusing plausibility with proof.

Concepts and reasons

An exact answer is the original expression’s true value. An estimate replaces some numbers with nearby convenient values. Write ≈, meaning “approximately equal to,” between an original expression and its estimate. Use = only where equality is exact, including arithmetic performed on the rounded numbers themselves.

To round a positive integer to the nearest ten, inspect its units digit: 0–4 rounds down and 5–9 rounds up. For the nearest hundred, inspect the last two digits: below 50 rounds down; 50 or more rounds up. Choose useful precision.

Check the expected sign and size first. A product of two negative numbers must be positive. A positive number divided by an integer greater than 1 must become smaller. Passing these checks does not prove correctness.

For addition and subtraction, the total error cannot exceed the sum of the individual rounding errors’ magnitudes. Errors may reinforce or cancel each other. Products need different care because input errors can be amplified by the other factor; do not reuse the sum-based bound automatically.

Check exact division by multiplying the proposed quotient by the nonzero divisor: it must reproduce the dividend.

Worked examples

Example 1 — Estimate with an error bound. Round to the nearest ten: 498 + 703 + 196 ≈ 500 + 700 + 200 = 1400. The individual errors have magnitudes 2, 3 and 4, totalling 9. Thus the exact sum lies from 1391 to 1409. Exact addition gives 1397, only 3 below the estimate.

Example 2 — Check a product’s size. Estimate 198 × 31 ≈ 200 × 30 = 6000. A stronger check uses 190 < 198 < 200: multiplying by positive 31 gives 5890 < 198 × 31 < 6200. Exactly, (200 - 2) × 31 = 6200 - 62 = 6138.

Example 3 — Reject a plausible quotient. A calculation claims 5916 ÷ 29 = 214. The estimate 5916 ÷ 29 ≈ 6000 ÷ 30 = 200 alone cannot reliably reject every nearby wrong answer. However, 29 × 214 = 6206, not 5916. Since 5916 = 29 × 200 + 116 and 116 = 29 × 4, the correct quotient is 204.

Common mistakes

Do not label estimates exact. Coarse rounding of nearly equal numbers can erase their small difference. A correct sign or unit digit is necessary, not sufficient; an incorrect one can reject an answer immediately.

Practice questions

1. Estimate 602 + 297 - 198 by rounding to the nearest ten, then calculate exactly.
2. Estimate 48 × 203 using 50 and 200. Find the exact product and how far the estimate differs.
3. Check whether 8576 ÷ 32 = 268 using multiplication.
4. Is the claim 497 × 18 = 8948 correct? Check the unit digit and give the exact answer.

Answers and explanations

1. Estimate 700; exact answer 701. We have 602 + 297 - 198 ≈ 600 + 300 - 200 = 700. Exactly, 899 - 198 = 701. The estimate is 1 too small.
2. Estimate 10000; exact answer 9744. Here 48 × 203 ≈ 50 × 200 = 10000. Exactly, 48 × (200 + 3) = 9600 + 144 = 9744. The estimate is too large by 10000 - 9744 = 256.
3. Correct. The inverse check gives 32 × 268 = 32 × (260 + 8) = 8320 + 256 = 8576. It reproduces the dividend, proving the exact quotient.
4. Incorrect; the exact answer is 8946. The unit-digit product is 7 × 8 = 56, so the answer must end in 6, not 8. Exactly, (500 - 3) × 18 = 9000 - 54 = 8946.