Questions: Estimation in Multiplication and Division

Question 1

A student multiplies 4 × 38 on paper and gets 242. She quickly estimates: 4 × 40 = 160. What should she conclude?

• A. Her estimate is wrong because 4 × 40 is not the same as 4 × 38
• B. The estimate of 160 confirms that 242 is a reasonable answer
• C. The gap between 160 and 242 is too large — she should recheck her calculation
• D. Estimation always gives a smaller number, so 242 must be the correct exact answer

Answer: (C)

The actual answer is 4 × 38 = 152, very close to the estimate of 160. A computed answer of 242 is far from the estimate — this gap signals a calculation error. Estimation's power as a checking tool works exactly like this: when your estimate and your computed answer are far apart, something went wrong in the computation. The estimate doesn't confirm 242; it reveals 242 is unreasonable.

Question 2

Which is the best estimation strategy for 9 × 53?

• A. Round 53 to 50, then compute 9 × 50 = 450
• B. Round 53 to 60, then compute 9 × 60 = 540
• C. Round both numbers: 10 × 50 = 500
• D. Round 53 to 55, then compute 9 × 55 = 495

Answer: (A)

The simplest, most efficient approach is to round 53 to the nearest ten (50) and multiply: 9 × 50 = 450. The actual answer is 477, so the estimate is off by about 6% — close enough to check reasonableness. Rounding to 60 overshoots; 55 is not a clean round number and doesn't simplify mental math. Rounding both numbers (10 × 50 = 500) is also valid but introduces more rounding error.

Question 3

An estimate of 7 × 30 = 210 for the problem 7 × 28 is useful even though it is not exactly correct.

• T. True
• F. False

Answer: True

The value of an estimate is not exactness — it's being close enough to serve its purpose. The actual answer is 196. An estimate of 210 correctly tells you the answer should be around 200, not 20 or 2,000. That's enough to check that a computed answer is in the right range. Estimation is intentionally approximate — the goal is approximately right on purpose, not accidentally wrong.

Question 4

When estimating products and quotients, you should usually round up so your estimate is seldom smaller than the real answer.

• T. True
• F. False

Answer: False

Rounding should go toward the nearest ten (or whichever place makes the arithmetic easy) — not always up. 28 rounds up to 30, but 42 rounds down to 40. Always rounding up produces estimates that are systematically too high and less useful as a sanity check. The goal is the closest, cleanest number, not a guaranteed overestimate.

Question 5

Explain how you would use estimation to check whether the answer to 8 × 46 = 408 is reasonable, and what you would conclude.

Model answer: Round 46 to the nearest ten: 50. Estimate: 8 × 50 = 400. The computed answer of 408 is very close to 400 (off by less than 2%), so 408 is a reasonable answer. If the computed answer had been 48 or 4,080, the estimate of 400 would immediately reveal those as errors worth investigating.

Estimation sets the expected magnitude of the answer. As long as the computed answer is close to the estimate, you can be confident the calculation is probably correct. A large gap between estimate and computation is a red flag — not proof of error, but a clear signal to recheck. This is estimation as a sanity check: fast, rough, and powerful.