Decimals
Every question with a fully worked solution.
- Decimals - Video - Adding and Subtracting Decimals (How to) Watch
Theory
To round a decimal, look at the digit to the right of the place you want: \(5\) or more rounds up. To add or subtract, line up the decimal points. A decimal can be negative: work out the sign first, then work with the sizes. This Year 8 maths page covers both.
Rounding. Go to the decimal place asked for, then look at the next digit to the right. If it is \(5\) or more (\(5, 6, 7, 8, 9\)), increase the digit in your place by \(1\); if it is \(4\) or less, leave it. Then drop the digits after it: \(7.5468 \approx 7.55\) to \(2\) decimal places.
Adding and subtracting. Line up the decimal points and fill gaps with zeros, then work as with whole numbers.
The size of a decimal is how far it is from zero, so the size of \(-6.4\) is \(6.4\). Different signs: subtract the smaller size from the larger and take the sign of the larger: \(-0.8 + 3.5 = 2.7\). A positive from a negative: add the sizes and keep the minus sign: \(-2.5 - 1.2 = -3.7\). Subtracting a negative: add the opposite: \(3.4 - (-1.6) = 5\).
Rounding a negative decimal. Round the size, then put the minus sign back: \(-3.18 \approx -3.2\).
Dividing. By a whole number, add zeros and keep going: \(9.4 \div 4 = 2.35\). By a decimal, first make the divisor whole by multiplying both numbers by \(10\), \(100\) or \(1000\): \(4.5 \div 0.9 = 45 \div 9 = 5\).
Missing numbers. Undo adding by subtracting: \(-1.5 + {?} = 4.2\) gives \({?} = 4.2 - (-1.5) = 5.7\). Estimate first by rounding each number to a whole number. The order of operations does not change.
Totals, means and money. The mean is the total divided by how many values, and can be negative; the range is highest \(-\) lowest. The nearest cent is \(2\) decimal places of a dollar; the nearest gram is \(3\) decimal places of a kilogram. Rounded shares may not add back exactly.
Words that set the sign of a decimal:
| Situation | Decimal |
|---|---|
| \(\$12.50\) overdrawn | \(-12.50\) |
| \(2.4^\circ\)C below zero | \(-2.4\) |
| a drop of \(3.25^\circ\)C | \(-3.25\) |
| \(0.6\) m below sea level | \(-0.6\) |
| \(1.85\) s slower than the leader (a longer time) | \(+1.85\) |
How to round to a number of decimal places
- Find the last digit you keep.
- Look at the next digit: \(5\) or more rounds up, \(4\) or less stays.
- For a negative, round the size and put the minus sign back.
How to add or subtract signed decimals
- Change double signs.
- Decide the sign from the sizes.
- Line up the decimal points and work with the sizes.
(i) \(1\) decimal place
(ii) \(2\) decimal places
(i) The next digit after \(7.5\) is \(4\), so the \(5\) stays.
(ii) The next digit after \(7.54\) is \(6\), so the \(4\) becomes \(5\).
(i) \(7.5\)
(ii) \(7.55\)
(i) \(42.63 + 4.25\)
(ii) \(7.21 - 0.43\)
| \(42.63 + 4.25\) | \(=\) | \(46.88\) |
| \(7.21 - 0.43\) | \(=\) | \(6.78\) |
(i) \(46.88\)
(ii) \(6.78\)
(i) \(-6.4 + 2.35\)
(ii) \(-3.05 - 1.7\)
(iii) \(4.2 - (-1.65)\)
| \(6.40 - 2.35\) | \(=\) | \(4.05\) |
| \(3.05 + 1.70\) | \(=\) | \(4.75\) |
| \(4.20 + 1.65\) | \(=\) | \(5.85\) |
(i) \(6.4\) is the bigger size, so \(-4.05\)
(ii) \(-4.75\)
(iii) \(5.85\)
(i) \(-8.372\) to \(2\) decimal places
(ii) \(-0.96\) to \(1\) decimal place
Round the size, then put the minus sign back.
(i) The next digit is \(2\), so \(8.37\) stays.
(ii) The next digit is \(6\), so \(0.9\) rounds up to \(1.0\).
(i) \(-8.37\)
(ii) \(-1.0\)
(ii) Estimate \(31.6 - 8.45 - 2.9\), then work it out exactly.
| \(7.2 \div 0.8\) | \(=\) | \(72 \div 8 = 9\) |
| \(14.5 - 9\) | \(=\) | \(5.5\) |
| \(32 - 8 - 3\) | \(=\) | \(21\) |
| \(31.60 - 8.45 - 2.90\) | \(=\) | \(20.25\) |
(i) \(5.5\) (divide first)
(ii) Estimate \(21\); exact \(20.25\), which is close.
(i) What is the balance?
(ii) What is the mean change per transaction?
| \(18.40 - 27.15\) | \(=\) | \(-8.75\) |
| \(-8.75 + 12.50\) | \(=\) | \(3.75\) |
| \(3.75 - 9.80\) | \(=\) | \(-6.05\) |
| \(-24.45 \div 3\) | \(=\) | \(-8.15\) |
(i) \(-\$6.05\) (overdrawn by \(\$6.05\))
(ii) The changes total \(-27.15 + 12.50 - 9.80 = -24.45\), so the mean is \(-\$8.15\).
(ii) A \(1.9\) kg cake is cut into \(6\) equal pieces. Find the mass of each piece to the nearest gram.
| \(40 \div 3\) | \(=\) | \(13.333\ldots\) |
| \(3 \times 13.33\) | \(=\) | \(39.99\) |
| \(1.9 \div 6\) | \(=\) | \(0.3166\ldots\) |
(i) \(\$13.33\) each; \(\$39.99\) is collected, \(1\) cent short.
(ii) \(0.317\) kg (\(3\) decimal places is the nearest gram).
Common pitfalls
Frequently asked questions
How do you round a decimal?
Look at the digit after the last place you keep: \(5\) or more rounds up. \(7.5468 \approx 7.5\) to \(1\) decimal place.
How do you add decimals?
Line up the decimal points, fill gaps with zeros, then add as with whole numbers.
How do you subtract a negative decimal?
Add its opposite: \(4.2 - (-1.65) = 5.85\).
How do you divide by a decimal?
Multiply both numbers by \(10\), \(100\) or \(1000\) to make the divisor whole: \(4.5 \div 0.9 = 45 \div 9 = 5\).
How do you round a negative decimal?
Round its size, then put the minus sign back: \(-0.96 \approx -1.0\).