Math Detours · Grade 5 · Sharpen · 16

Good enough answers

Round first, then calculate. Estimating tells you whether an exact answer is sensible, and often it is all you need.

Name ____________________Date ____________

Tonight's idea

Some sums are hard to do in your head. A nearby sum is often easy. To estimate is to swap the hard numbers for easy ones close by, then calculate.

Swapping a number for an easy neighbour is called rounding. Pick the nearest ten, hundred or thousand.

47 away 53 away 347 300 nearest hundred 350 nearest ten 400
347 is nearer to 300 than to 400. To the nearest ten it is 350.

What if a number is exactly half way, like 350 between 300 and 400? The usual rule is to round it up.

Now estimate \(347 + 289\). Round each to the nearest hundred: \(300 + 300 = 600\). The exact answer is 636. That is close, and it took two seconds.

Round first, then calculate. Write ≈ for "is about": \(347 + 289 \approx 600\).

An estimate does two jobs. It catches mistakes: an exact answer far from your estimate is probably wrong. And often it is all you need. "Do I have enough money?" needs a yes or no.

Watch one: 48 × 21

Round each number to the nearest ten. 48 becomes 50 and 21 becomes 20.

\[48 \times 21 \approx 50 \times 20 = 1000.\]

Ana works it out exactly and gets 1008. Ben gets 10 080. The estimate says the answer is near 1000. So Ana's answer is sensible. Ben's is ten times too big, so he has slipped a place somewhere.

The estimate did not tell us the answer. It told us which answer to trust.

Your turn

  1. Round: (a) 62 to the nearest ten; (b) 285 to the nearest hundred; (c) 4499 to the nearest thousand; (d) 750 to the nearest hundred.
  2. Estimate each one by rounding first. Do not work out the exact answer.
    (a) \(398 + 207\)   (b) \(61 \times 29\)   (c) \(812 \div 4\)   (d) \(9.8 \times 5.1\)
  3. Only one answer in each row is right. Use an estimate to pick it.
    (a) \(52 \times 38\):   876,   1976,   19 760
    (b) \(603 - 298\):   205,   305,   405
    (c) \(7.9 \times 6.2\):   4.898,   48.98,   489.8
  4. You have $20. You want four things that cost $3.85, $6.90, $4.75 and $2.95. (a) Round every price up to the next dollar and add. (b) Are you sure you have enough? (c) Why is rounding up safer here than rounding to the nearest dollar?

Stretch: which way is it out?

Estimate \(49 \times 49\) as \(50 \times 50 = 2500\). Without working out \(49 \times 49\), say whether 2500 is too big or too small, and why. Then find the exact answer. How far out was the estimate?

Answers

  1. (a) 60. (b) 300. (c) 4000, since 4499 is just under half way. (d) 800, since half way rounds up.
  2. (a) \(400 + 200 = 600\). (b) \(60 \times 30 = 1800\). (c) \(800 \div 4 = 200\). (d) \(10 \times 5 = 50\). The exact answers are 605, 1769, 203 and 49.98.
  3. (a) \(50 \times 40 = 2000\), so 1976. (b) \(600 - 300 = 300\), so 305. (c) \(8 \times 6 = 48\), so 48.98.
  4. (a) \(4 + 7 + 5 + 3 = 19\) dollars. (b) Yes. Every real price is less than its rounded price, so the real total is less than $19. It is $18.45. (c) Rounding to the nearest dollar can make a price smaller: $3.20 would become $3. Then the real total could be more than the estimate, and you could be caught short.

Stretch. Too big. Both numbers were rounded up, so the product went up. The exact answer is \(49 \times 49 = 2401\), so the estimate was 99 too big. That is tiny next to 2500: still a good estimate.

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