Chapter 7 of 8

All 8 chapters
  1. 1Multiplying and dividing in your head, readunderstood
  2. 2Short multiplication, readunderstood
  3. 3Long multiplication, readunderstood
  4. 4Short division, readunderstood
  5. 5Long division, readunderstood
  6. 6Order of operations, readunderstood
  7. 7Estimate, then check, readunderstood
  8. 8Choosing what to do, readunderstood

M3.2.7 Chapter 7

Estimate, then check

How can I tell if 389 × 52 = 20 228 is about right?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Round to one digit to estimate 389 × 52
  • Check a division by multiplying back

Kofi works out 389 × 52 on paper and gets 20 228. Is that about right? One slip with a carry could be a long way out. A quick estimate will tell him.

An estimate is a rough answer, worked out in your head. To make one, round each number to one digit. Keep the first digit, and fill the rest with zeros. If the next digit is 5 or more, round up.

Draw it first

Round 389 to one digit

  1. A number line from 300 to 400 with a tick every 10. The first picture marks 300, 350 and 400. The next adds a dot for 389, with the label 389 under it, close to 400. The last has the words 389 rounds to 400.300350400

    1 389 is between 300 and 400. Half way between them is 350. Mark 300, 350 and 400.

  2. A number line from 300 to 400 with a tick every 10. The first picture marks 300, 350 and 400. The next adds a dot for 389, with the label 389 under it, close to 400. The last has the words 389 rounds to 400.300350400389

    2 Mark 389. It is past 350.

  3. A number line from 300 to 400 with a tick every 10. The first picture marks 300, 350 and 400. The next adds a dot for 389, with the label 389 under it, close to 400. The last has the words 389 rounds to 400.300350400389389 rounds to 400

    3 389 is nearer to 400 than to 300, so it rounds to 400.

Do the same with 52. It is between 50 and 60, and the half way mark is 55. 52 is before 55, so it rounds down to 50. Then 389 × 52 is about 400 × 50.

A grid method table with one box. The column is 400 and the row is 50. The box holds 20 000. Under the grid are the words total 20 000 and not to scale.4005020 000total 20 000not to scale
The estimate, as a grid with one box.

Worked example

Estimate 389 × 52.

  1. Round each number to one digit. 389 is about 400. 52 is about 50.
  2. Work out 4 × 5 = 20. There are two zeros in 400 and one in 50, so put back three zeros.
  3. 400 × 50 = 20 000.
  4. So 389 × 52 is about 20 000.

Answer: 20 000

See it three ways

With things

A farmer plants 52 rows of trees, with 389 trees in each row. Say the numbers roughly: about 50 rows, and about 400 trees in each row.

As a picture

A table with the columns number and rounded. The first row says 389 and 400. The second row says 52 and 50.numberrounded3894005250

In symbols

389 × 52 is about 400 × 50 = 20 000.

An estimate catches big slips, such as a lost zero. It cannot catch a small slip. Multiplying back catches those. Division and multiplying are opposites. If 7 870 ÷ 23 = 342 remainder 4, then 342 × 23 + 4 must be 7 870.

Worked example

Check that 7 870 ÷ 23 = 342 remainder 4 by multiplying back.

  1. Multiply the answer by the divisor. Work out 342 × 23.
  2. Row 1: 342 × 3 = 1 026.
  3. Row 2: 342 × 20 = 6 840.
  4. Add the rows. 1 026 + 6 840 = 7 866.
  5. Add the remainder. 7 866 + 4 = 7 870. This is the number we started with, so the division is right.

Answer: 7 870

Worked example

Estimate 7 870 ÷ 23. Is 342 about right?

  1. Round 7 870 to one digit. It is 8 000.
  2. Round 23 to one digit. It is 20.
  3. Cross out one zero from each number. 8 000 ÷ 20 is the same as 800 ÷ 2.
  4. Divide. 800 ÷ 2 = 400.
  5. The estimate is 400. Both numbers were rounded so that the estimate grew: 7 870 went up and 23 went down. So the true answer should be a little under 400. 342 is in the hundreds, like 400, so no digit is in the wrong place.

Answer: 400

Try it

Multiply back

You need: paper and a pencil.

  1. Work out 7 870 ÷ 23 on paper. You should get 342 remainder 4.
  2. Now multiply your answer by the divisor. Work out 342 × 23 in columns.
  3. Add the remainder to what you get.

What you will see: 342 × 23 = 7 866. Adding the 4 left over gives 7 870, the number you began with. If it does not match, one of the two calculations has a slip.

If you cannot try it, this is what happens.
The calculation 342 times 23 written in columns. Row 1 is 1 026. Row 2 is 6 840. The total is 7 866.24332×62010486+6687

A wrong idea many people have

To check a division, just divide again.

Why it feels right: Doing the same calculation twice feels like a fair test.

The test that breaks it: If you made a slip the first time, you may make the same slip again. Multiplying back is a different move. 342 × 23 + 4 = 7 870 proves the first answer.

What is true: Check with the opposite operation. Multiply the answer by the divisor, add the remainder, and you must get the number you began with.

Key words

estimate
a rough answer, worked out quickly from rounded numbers.
round
to change a number to a nearby number that is easier to use, such as 389 to 400.

Back to your guess

The answer is C: Round each number to one digit: 400 × 50 = 20 000. The answer is close, so it is about right.

Rounding to one digit gives 400 × 50 = 20 000. The answer 20 228 is close to that, so it is about right. Adding is the wrong operation, and the length of an answer proves nothing.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Round 52 to one digit. Which number is it?

    A number line from 50 to 60 with a tick for each whole number. The marked points are 50, 52, 55 and 60.50525560
    Tap your answer.

    1. Not this one. That is the number itself. Rounding changes it to a nearby number with zeros. Go to M3.2.7. Try another.

    2. Not this one. 52 is only 2 above 50 but 8 below 60. It is nearer to 50, so it rounds down. Go to M3.2.7. Try another.

    3. Not this one. 55 is the half way mark, not a rounded number. Rounding to one digit gives 50 or 60. Go to M3.2.7. Try another.

    4. Yes, that is right. 52 is between 50 and 60. It is before the half way mark 55, so it rounds down to 50.

    See the answer

    50

    52 is between 50 and 60. It is before the half way mark 55, so it rounds down to 50.

    • If you chose A: That is the number itself. Rounding changes it to a nearby number with zeros. Go to M3.2.7.
    • If you chose B: 52 is only 2 above 50 but 8 below 60. It is nearer to 50, so it rounds down. Go to M3.2.7.
    • If you chose C: 55 is the half way mark, not a rounded number. Rounding to one digit gives 50 or 60. Go to M3.2.7.

  2. Which estimate is best for 389 × 52?

    Tap your answer.

    1. Not this one. 389 is nearer to 400 than to 300, because it is past the half way mark 350. It rounds up. Go to M3.2.7. Try another.

    2. Yes, that is right. 389 rounds to 400 and 52 rounds to 50. 4 × 5 = 20, and three zeros go back. 400 × 50 = 20 000.

    3. Not this one. 52 is about 50, not 5. You lost a zero, so the estimate is ten times too small. Go to M3.2.7. Try another.

    4. Not this one. The calculation is a multiplication. Round the numbers, but keep the × sign. Go to M3.2.7. Try another.

    See the answer

    400 × 50 = 20 000

    389 rounds to 400 and 52 rounds to 50. 4 × 5 = 20, and three zeros go back. 400 × 50 = 20 000.

    • If you chose A: 389 is nearer to 400 than to 300, because it is past the half way mark 350. It rounds up. Go to M3.2.7.
    • If you chose C: 52 is about 50, not 5. You lost a zero, so the estimate is ten times too small. Go to M3.2.7.
    • If you chose D: The calculation is a multiplication. Round the numbers, but keep the × sign. Go to M3.2.7.

  3. Ana works out 1 000 ÷ 7 and gets 142 remainder 6. How can she check?

    Tap your answer.

    1. Not this one. 142 × 7 = 994, not 1 000. The remainder must be added back to reach 1 000. Go to M3.2.7. Try another.

    2. Not this one. Dividing again does not undo the first calculation. The opposite of dividing is multiplying. Go to M3.2.7. Try another.

    3. Not this one. Adding 142 and 7 does not undo the division. Multiply 142 by 7, then add the 6. Go to M3.2.7. Try another.

    4. Yes, that is right. 142 × 7 = 994, and 994 + 6 = 1 000. That is the number she started with, so her division is right.

    See the answer

    Multiply 142 by 7, then add 6. The result should be 1 000.

    142 × 7 = 994, and 994 + 6 = 1 000. That is the number she started with, so her division is right.

    • If you chose A: 142 × 7 = 994, not 1 000. The remainder must be added back to reach 1 000. Go to M3.2.7.
    • If you chose B: Dividing again does not undo the first calculation. The opposite of dividing is multiplying. Go to M3.2.7.
    • If you chose C: Adding 142 and 7 does not undo the division. Multiply 142 by 7, then add the 6. Go to M3.2.7.

  4. The picture shows 745 × 7 = 5 215. Ravi divided 5 217 by 7 and got 745 remainder 2. Is his division right?

    The calculation 745 times 7 written in columns, with the carried digits small. The answer is 5 215.5477×512533
    Tap your answer.

    1. Yes, that is right. 745 × 7 = 5 215. Adding the remainder 2 gives 5 217, the number he started with.

    2. Not this one. The remainder 2 is the gap. 5 215 and the 2 left over make 5 217, so the numbers do match. Go to M3.2.7. Try another.

    3. Not this one. A remainder does not have to be 0. Many divisions leave something over. Go to M3.2.7. Try another.

    4. Not this one. Close is not enough for a check. The check must give the number exactly, and here it does after adding the 2. Go to M3.2.7. Try another.

    See the answer

    Yes, because 5 215 + 2 = 5 217.

    745 × 7 = 5 215. Adding the remainder 2 gives 5 217, the number he started with.

    • If you chose B: The remainder 2 is the gap. 5 215 and the 2 left over make 5 217, so the numbers do match. Go to M3.2.7.
    • If you chose C: A remainder does not have to be 0. Many divisions leave something over. Go to M3.2.7.
    • If you chose D: Close is not enough for a check. The check must give the number exactly, and here it does after adding the 2. Go to M3.2.7.

  5. Which is a good estimate for 7 870 ÷ 23?

    Tap your answer.

    1. Not this one. 23 is about 20, not 2. Using 2 makes the estimate ten times too big. Go to M3.2.7. Try another.

    2. Not this one. 7 870 is about 8 000, not 800. You lost a zero, so the estimate is ten times too small. Go to M3.2.7. Try another.

    3. Yes, that is right. 7 870 rounds to 8 000 and 23 rounds to 20. 8 000 ÷ 20 is 800 ÷ 2, which is 400.

    See the answer

    8 000 ÷ 20 = 400

    7 870 rounds to 8 000 and 23 rounds to 20. 8 000 ÷ 20 is 800 ÷ 2, which is 400.

    • If you chose A: 23 is about 20, not 2. Using 2 makes the estimate ten times too big. Go to M3.2.7.
    • If you chose B: 7 870 is about 8 000, not 800. You lost a zero, so the estimate is ten times too small. Go to M3.2.7.

  6. Round each number to one digit to estimate 612 × 38. Give the estimate.

    See the answer

    24 000

    612 rounds to 600 and 38 rounds to 40. 6 × 4 = 24, and three zeros go back. The estimate is 24 000.

  7. Dividing by 7 gave 814 remainder 3. Multiply back to find the number that was divided.

    See the answer

    5 701

    814 × 7 = 5 698. Adding the remainder, 5 698 + 3 = 5 701. This is the number that was divided.

  8. From an earlier chapter

    Work out 9 + 4 × 6.

    See the answer

    33

    Multiply first. 4 × 6 = 24. Then 9 + 24 = 33.

Keep it

Copy this into your exercise book.

Estimate: round each number to one digit, then work it out. 389 × 52 is about 400 × 50 = 20 000. Check a division by multiplying back, then add the remainder.

Draw: Draw a number line from 300 to 400. Mark 350 in the middle and 389 near the end. Write "rounds to 400".

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (792 KB). It opens on any phone.