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Calculating with decimals

Numbers · level 4, lesson 4 of 7

A question to start

Work these out in your head:

\[0.5 \times 0.5 = \;? \qquad\qquad 3 \div 0.5 = \;?\]

Many people say 2.5 for the first one and 1.5 for the second. The real answers are 0.25 and 6. Half of a half is a quarter. And how many halves fit into 3? Six.

Decimals follow exactly the same rules as fractions, because they are fractions (tenths, hundredths, ...). This lesson shows where the decimal point goes in every kind of calculation, and why, so that you can check every calculator answer you ever get.

The idea

Adding and subtracting: line up the places

To add 12.5 and 3.75, you must add tenths to tenths and hundredths to hundredths, just as you add tens to tens with whole numbers. Lining up the decimal points lines up the places:

    1 2 . 5 0
 +    3 . 7 5
 -----------
    1 6 . 2 5

The 0 after 12.5 is not needed, but it helps: 12.5 is 12.50 (five tenths is fifty hundredths). Carrying works as usual. Here 5 tenths + 7 tenths = 12 tenths, which is 1 one and 2 tenths, so a 1 is carried into the ones.

Subtracting: \(7 - 2.48\). Write 7 as 7.00 so every place has a digit, then subtract with regrouping:

    7 . 0 0
 -  2 . 4 8
 ---------
    4 . 5 2

Check by adding: \(4.52 + 2.48 = 7.00\). ✓

Why line up the points and not the right-hand ends? With whole numbers, the right-hand ends are the ones places, so lining up the ends lines up the places. With decimals, the right-hand ends can be different places (tenths in 12.5, hundredths in 3.75). The point is what marks the ones place, so the point is what must line up.

Multiplying and dividing by 10, 100, 1000

Multiplying by 10 moves every digit one place left, because each place is worth ten times the one to its right. \(3.47 \times 10 = 34.7\): the 3 ones become 3 tens, the 4 tenths become 4 ones, the 7 hundredths become 7 tenths.

Dividing by 10 moves every digit one place right: \(3.47 \div 10 = 0.347\).

By 100 or 1000, move two or three places. It often looks as if the point is moving, and that is a fine way to do it, but remember it is really the digits that move to bigger or smaller places.

Multiplying decimals

A small one first. \(0.3 \times 0.4\). As fractions, this is \(\tfrac{3}{10} \times \tfrac{4}{10} = \tfrac{12}{100} = 0.12\). On a hundred square, 3 tenths (columns) by 4 tenths (rows) overlap in 12 small squares, each one a hundredth:

0.3 0.4 whole square = 1 green: 12 hundredths 0.3 × 0.4 = 0.12

Notice: tenths times tenths make hundredths, so the answer has two decimal places, even though each number had only one.

The general method. To work out \(2.4 \times 1.35\):

  1. Ignore the points and multiply whole numbers: \(24 \times 135 = 3240\).
  2. Count decimal places in the question: 2.4 has 1, 1.35 has 2. Total: 3.
  3. Put that many decimal places in the answer: 3.240, which is 3.24.
  4. Estimate to check: \(2.4 \times 1.35\) is between \(2 \times 1.35 = 2.7\) and \(3 \times 1.35 = 4.05\). 3.24 fits. ✓

Why does counting places work? Because 2.4 is \(24 \div 10\) and 1.35 is \(135 \div 100\). So

\[2.4 \times 1.35 = \frac{24}{10} \times \frac{135}{100} = \frac{24 \times 135}{10 \times 100} = \frac{3240}{1000} = 3.240\]

Each decimal place in the question is one "divide by 10". One place and two places make three "divide by 10"s together, which is dividing by 1000. That is three places in the answer.

The rule counts places before you tidy up. In \(0.25 \times 0.6\): \(25 \times 6 = 150\), with \(2 + 1 = 3\) places: 0.150, which is 0.15. If you dropped the 0 from 150 first you would get the wrong place.

Dividing a decimal by a whole number

\(7.2 \div 4\): long division works exactly as before. Keep the point in the same place in the answer.

  • 7 ones ÷ 4 = 1 one, 3 left over.
  • Trade the 3 ones for 30 tenths, plus the 2 tenths: 32 tenths ÷ 4 = 8 tenths.

So \(7.2 \div 4 = 1.8\). Check: \(1.8 \times 4 = 7.2\). ✓

If there is a remainder, add zeros and keep going: \(6.3 \div 4\). 6 ÷ 4 = 1 r 2; 23 tenths ÷ 4 = 5 r 3; 30 hundredths ÷ 4 = 7 r 2; 20 thousandths ÷ 4 = 5. So \(6.3 \div 4 = 1.575\).

Dividing by a decimal: scale both numbers

\(4.5 \div 0.15\) looks hard. But dividing by a whole number is easy. So make the divisor a whole number by multiplying both numbers by 100:

\[4.5 \div 0.15 = 450 \div 15 = 30\]

Why is it allowed to multiply both numbers by the same thing? A division is a fraction: \(4.5 \div 0.15 = \tfrac{4.5}{0.15}\). Multiplying the top and bottom of a fraction by the same number gives an equivalent fraction, the same amount: \(\tfrac{4.5}{0.15} = \tfrac{450}{15}\).

Or think of money: "how many 15-cent stickers can you buy with 4 dollars 50 cents?" is the same question as "how many 15s fit in 450?" Changing from dollars to cents multiplies both numbers by 100, and the answer does not care which unit you count in.

Multiply by 10 if the divisor has one decimal place, 100 if it has two, and so on. The dividend does not need to become whole: \(9.45 \div 0.7 = 94.5 \div 7 = 13.5\).

Now the opening question: \(3 \div 0.5 = 30 \div 5 = 6\). ✓

Worked examples

1. Add three numbers. \(6.04 + 3.9 + 12\). Write as 6.04, 3.90, 12.00 and line up the points. Total 21.94.

2. Subtract. \(15.3 - 8.76\). Write 15.30 − 8.76. Regroup: 6.54. Check: \(6.54 + 8.76 = 15.30\). ✓

3. Multiply. \(0.07 \times 0.8\). \(7 \times 8 = 56\). Places: \(2 + 1 = 3\). So 0.056 (a zero is needed to make 3 places). Check with fractions: \(\tfrac{7}{100} \times \tfrac{8}{10} = \tfrac{56}{1000}\). ✓

4. Multiply. \(5.4 \times 2.5\). \(54 \times 25 = 1350\). Places: 2. So 13.50 = 13.5. Estimate: \(5 \times 2.5 = 12.5\), a bit less. ✓

5. Sols and days. A sol (a Mars day) is about 1.03 Earth days. About how many Earth days are 30 sols? \(1.03 \times 30 = 30.9\) days. (A 30-sol "month" is almost 31 Earth days.)

6. Divide by a decimal. \(20 \div 0.4\). Multiply both by 10: \(200 \div 4 = 50\). Check: \(50 \times 0.4 = 20\). ✓

Try it

  1. \(8.25 + 13.7\) and \(10 - 3.65\).
  2. \(4.06 \times 100\) and \(4.06 \div 100\).
  3. \(0.6 \times 0.7\) and \(0.06 \times 0.7\).
  4. \(1.2 \times 1.2\). Is the answer more or less than 1.2? Why?
  5. \(6.3 \div 4\), giving the full decimal answer.
  6. \(3.6 \div 0.12\) and \(3.5 \div 0.05\).
  7. A bottle holds 1.5 litres. How many 0.25-litre cups can it fill?
  8. Harder: Without calculating, put in order from smallest to biggest: \(8 \times 0.9\), \(8 \div 0.9\), \(8\).
Answers
  1. \(8.25 + 13.70 = 21.95\). \(10.00 - 3.65 = 6.35\).
  2. \(4.06 \times 100 = 406\). \(4.06 \div 100 = 0.0406\).
  3. \(6 \times 7 = 42\) with 2 places: 0.42. \(6 \times 7 = 42\) with 3 places: 0.042.
  4. \(12 \times 12 = 144\) with 2 places: 1.44. More than 1.2, because we multiplied 1.2 by a number bigger than 1.
  5. 6 ÷ 4 = 1 r 2; 23 tenths ÷ 4 = 5 r 3; 30 hundredths ÷ 4 = 7 r 2; 20 thousandths ÷ 4 = 5. So 1.575. Check: \(1.575 \times 4 = 6.3\). ✓
  6. \(3.6 \div 0.12 = 360 \div 12 = 30\). \(3.5 \div 0.05 = 350 \div 5 = 70\).
  7. \(1.5 \div 0.25 = 150 \div 25 = 6\) cups.
  8. \(8 \times 0.9\) is less than 8 (multiplying by less than 1); \(8 \div 0.9\) is more than 8 (dividing by less than 1). Order: \(8 \times 0.9\) (7.2), then 8, then \(8 \div 0.9\) (about 8.89).

Common mistake

Lining up the right-hand ends. Adding 12.5 and 3.75 by lining up the 5s adds tenths to hundredths, and gives nonsense. Line up the points (and fill gaps with zeros).

Putting the point where it "looks" right in multiplication. People often write \(0.3 \times 0.4 = 1.2\), because "3 times 4 is 12 and there is a point somewhere". Count the places (one plus one makes two): 0.12. Or estimate: 0.3 and 0.4 are both less than a half, so the answer must be less than a quarter. 0.12 is sensible; 1.2 is not.

Where it leads