Year 7 · Unit 1 Number systems and place value · Lesson 7 of 7

How close is close enough?

Rounding finds the nearest friendly number, and estimating uses it to check an answer before you trust it. Move a number along the line and see which way it goes.

Number and AlgebraScientific calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. What is the value of the 6 in 3.64?
    Answer0.6, six tenths.
  2. Which is closer to 70: 64 or 77?
    Answer64, because 70 − 64 = 6 and 77 − 70 = 7.
  3. What is 6 × 30?
    Answer180. 6 × 3 = 18, then a zero for the ten.
Words used in this lesson
round
replace a number by a nearby friendly number. For example: 365 rounds to 370 to the nearest 10.
estimate
an approximate answer found by rounding. For example: 49 × 21 is about 1000.
nearest
the closest of the possible choices. For example: the nearest 10 to 43 is 40.
decimal place
a place to the right of the decimal point. For example: 7.46 has two decimal places.
reasonable
close to what you would expect. For example: 1871.8 is not reasonable for 38.2 × 4.9.

Which way does it go?

To round a number, find the two multiples it sits between and ask which is nearer. The halfway point decides: a number exactly halfway goes up. In practice you look at the digit just below the place you are rounding to: 5 or more rounds up, less than 5 rounds down.

Predict first: 365 to the nearest 10 and to the nearest 100. Which way does each go? Then try the buttons, and a number such as 349.

365
370rounds to
5deciding digit

365 is exactly halfway between 360 and 370, and halfway rounds up, so 365 rounds to 370 to the nearest 10.

Show me the steps
  1. 365 to the nearest 10: between 360 and 370; the ones digit is 5, so round up to 370.
  2. 364 to the nearest 10: the ones digit is 4, so round down to 360.
  3. 365 to the nearest 100: between 300 and 400; the tens digit is 6, so round up to 400.
  4. 349 to the nearest 100: the tens digit is 4, so round down to 300, even though 349 is almost 350.
  5. 950 to the nearest 100: the tens digit is 5, so round up to 1000. 9 hundreds and 1 more hundred make 10 hundreds, which is 1000.
  6. Nearest tenth (the slider counts hundredths): 3.65 is between 3.6 and 3.7. The hundredths digit is 5, so round up to 3.7. And 3.64 rounds down to 3.6.

Look one place to the right. To round to a place, look at the next digit: 5 or more, round up; 4 or less, round down. For decimals it is the same: to one decimal place (nearest tenth), look at the hundredths.

Round once, from the original number. 6.349 to one decimal place is 6.3, not 6.35 and then 6.4: only the digit just to the right decides.

Estimating to check

An estimate rounds each number to something friendly, then calculates. It is not the exact answer. It tells you the size, so you can see at once if an answer is wildly wrong. Try it: the estimate is usually closer when one number rounds up and the other down, but rounding a small number such as 14 can still leave it well out.

49
21
1000estimate
1029exact answer

49 × 21: round to 50 × 20 = 1000. The exact answer is 1029, which is 29 more than the estimate.

Show me the steps
  1. 49 × 21: round to 50 × 20 = 1000. Exact 1029, only 29 away.
  2. 35 × 35: round to 40 × 40 = 1600. Exact 1225, well over: both numbers rounded up.
  3. 43 × 43: round to 40 × 40 = 1600. Exact 1849, well under: both numbers rounded down.
  4. 14 × 86: round to 10 × 90 = 900. Exact 1204, a long way off, because 14 lost a lot when it went down to 10.

A common mistake: “349 rounds to 400 because it is nearly 350.” Only the tens digit decides to the nearest hundred, and it is 4, so 349 rounds down to 300.

Worked example: estimate, then calculate

Estimate 62 × 48, then work it out.

  1. Round each to the nearest 10: 62 → 60 and 48 → 50.
  2. Estimate: 60 × 50 = 3000.
  3. Exact: 62 × 48 = 62 × 50 − 62 × 2 = 3100 − 124 = 2976, which is close to 3000, so it is reasonable.

Your go. The first step is given. Estimate 4.1 × 29.6.

  1. 4.1 rounds to 4.
  2. Round 29.6 to the nearest 10, then multiply, then check against the exact answer, 121.36.
Check the answer

29.6 rounds to 30, and 4 × 30 = 120. The exact answer, 121.36, is close, so it is reasonable.

With a calculator

A scientific calculator gives 4.1 × 29.6 = 121.36 at once, and it will give 1871.8 for a mistyped 38.2 × 4.9 just as quickly. It does what you type, not what you meant. Estimate first (about 40 × 5 = 200), then check that the calculator's answer is close. If it is not, a key was slipped, usually a decimal point.

A calculator also shows many digits. Deciding where to round is your job: look back at what the question asks for.

Check before you go on

1. What is 3847 rounded to the nearest hundred?

Your turn

Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. Round 465 to the nearest 10.

3. Round 7.46 to one decimal place.

Level 2 · Building

4. Estimate 49 × 21 by rounding each number to the nearest 10.

5. Estimate 612 ÷ 29 by rounding to numbers that divide neatly.

6. Mei types 38.2 × 4.9 and her calculator shows 1871.8. Which estimate shows that this is not reasonable?

Level 3 · Stretch

7. Round 7.996 to two decimal places.

8. A whole number rounds to 450 to the nearest 10. What is the greatest whole number it could be?

Something to think about

A shop rounds every price to the nearest 10 before adding them up. Will the total of the rounded prices always be the same as the exact total rounded to the nearest 10? Test it with two items that each cost 14 dirhams.

One way to think about it

No. 14 rounds to 10, and 10 + 10 = 20, but the exact total is 28, which rounds to 30. Each rounding loses a little, and the losses can add up. That is why an estimate checks the size of an answer and does not replace it.

Exit ticket

9. Bilal rounds 6.349 to one decimal place like this: 6.349 → 6.35 → 6.4. Which reply is right?