Decimals and Significant Figures
Every question with a fully worked solution.
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Theory
A decimal uses place value after the decimal point, and significant figures count the digits that carry meaning, from the first non-zero digit. This Year 9 maths page covers the four operations with decimals, rounding to decimal places and to significant figures, and giving calculator answers in scientific notation.
Every digit after the decimal point has a place value: tenths, then hundredths, then thousandths. Each place is ten times smaller than the one to its left.
To add or subtract, line up the decimal points and fill any gaps with zeros. To multiply two decimals, multiply the digits and then count the decimal places in the question. To divide by a decimal, multiply both numbers by \(10\), \(100\) or \(1000\) until you are dividing by a whole number.
Rounding to decimal places keeps a set number of digits after the point. Rounding to significant figures counts from the first non-zero digit, wherever it is. Zeros in front of it are only placeholders.
Every non-zero digit is significant. Leading zeros, as in \(0.0045\), never count. Zeros between other digits, as in \(4.07\), always count. Trailing zeros after a decimal point, as in \(6.30\), count because they were written on purpose; trailing zeros in a whole number, as in \(7900\), are usually placeholders.
A number in scientific notation is written as \(a \times 10^n\), where \(1 \le a \lt 10\). All its significant figures sit in the front number \(a\).
Write \(\approx\) for a rounded answer. Always round once, from the original number.
An estimate rounds each number to something easy. It tells you roughly what size the real answer should be.
Multiplying or dividing by a power of ten moves the point:
Dividing by a decimal: multiply both numbers by the same power of ten first:
| Number | Significant figures | Why |
|---|---|---|
| \(0.00560\) | \(3\) | leading zeros do not count; the last \(0\) does |
| \(2.005\) | \(4\) | the zeros sit between significant digits |
| \(7900\) | \(2\) | the zeros are placeholders |
How to work with decimals
- Estimate first: round each number to something easy and work it out in your head.
- Add or subtract with the points lined up. Multiply the digits and count the places. Divide by making the divisor whole first.
- Round once, from the full answer, to the places or significant figures asked for. Use \(\approx\).
- Check the answer against the estimate. It should be about the same size.
How to round to significant figures
- Find the first non-zero digit. That is the first significant figure.
- Count along to the number of figures needed and mark a cut.
- Look at the next digit: \(5\) or more, round up; otherwise leave it. A \(9\) that rounds up carries to the left.
- Fill any whole-number places you removed with zeros, so the size stays the same.
(i) \(12.4 + 3.75\)
(ii) \(8 - 2.36\)
(iii) Find the missing number: \(5.3 + {?} = 9.04\)
(i) Line up the decimal points. Write \(12.4\) as \(12.40\) so both numbers have two places:
| \(12.40 + 3.75\) | \(=\) | \(\mathbf{16.15}\) |
(ii) Write \(8\) as \(8.00\), then subtract:
| \(8.00 - 2.36\) | \(=\) | \(\mathbf{5.64}\) |
Check (ii) by adding back: \(5.64 + 2.36 = 8.00\).
(iii) Undo the adding by subtracting. Write \(5.3\) as \(5.30\):
| \({?}\) | \(=\) | \(9.04 - 5.30\) |
| \(\) | \(=\) | \(\mathbf{3.74}\) |
Check: \(5.3 + 3.74 = 9.04\).
(i) \(0.7 \times 0.04\)
(ii) \(5.6 \div 0.8\)
(iii) \(0.0581 \times 100\)
(i) Multiply the digits, then count the decimal places in the question:
| \(7 \times 4\) | \(=\) | \(28\) |
| \(\text{places}\) | \(=\) | \(1 + 2 = 3\) |
| \(0.7 \times 0.04\) | \(=\) | \(\mathbf{0.028}\) |
(ii) Multiply both numbers by \(10\) so the divisor is a whole number. The answer does not change:
| \(5.6 \div 0.8\) | \(=\) | \(56 \div 8\) |
| \(\) | \(=\) | \(\mathbf{7}\) |
(iii) Multiplying by \(100\) moves the point two places to the right:
| \(0.0581 \times 100\) | \(=\) | \(\mathbf{5.81}\) |
(i) \(3.0472\) to two decimal places
(ii) \(0.0725\) to two significant figures
(iii) \(864\,520\) to two significant figures
(iv) \(29.96\) to three significant figures
(i) Cut after the second decimal place. The next digit is \(7\), so round up:
| \(3.04\,|\,72\) | \(\approx\) | \(\mathbf{3.05}\) |
(ii) The zeros in front are placeholders. The first significant figure is the \(7\). Count two figures and cut. The next digit is \(5\), so round up:
| \(0.072\,|\,5\) | \(\approx\) | \(\mathbf{0.073}\) |
(iii) Count two figures from the \(8\). The next digit is \(4\), so leave it, and fill the removed places with zeros to keep the size:
| \(86\,|\,4520\) | \(\approx\) | \(\mathbf{860\,000}\) |
(iv) The next digit is \(6\), so the \(9\) rounds up and carries: \(29.9\) becomes \(30.0\).
| \(29.9\,|\,6\) | \(\approx\) | \(\mathbf{30.0}\) |
Keep the zero after the point. It shows the answer has three significant figures.
(i) A bacterium is \(2.4 \times 10^{-6}\) m long. How many of them, end to end, would stretch \(3.5\) cm?
(ii) \(\dfrac{5.2 \times 10^{-3}}{7 \times 10^{2}}\)
(i) Use the same unit: \(3.5\) cm \(= 0.035\) m. Then divide:
| \(\text{number}\) | \(=\) | \(\dfrac{0.035}{2.4 \times 10^{-6}}\) |
| \(\) | \(=\) | \(14\,583.33\ldots\) |
| \(\) | \(=\) | \(1.4583\ldots \times 10^{4}\) |
| \(\) | \(\approx\) | \(\mathbf{1.46 \times 10^{4}}\) |
(ii) Use brackets on the calculator for the whole bottom line:
| \(\dfrac{5.2 \times 10^{-3}}{7 \times 10^{2}}\) | \(=\) | \(0.000\,007\,428\,57\ldots\) |
| \(\) | \(=\) | \(7.42857\ldots \times 10^{-6}\) |
| \(\) | \(\approx\) | \(\mathbf{7.43 \times 10^{-6}}\) |
Round once, at the end, from the full calculator value.
Common pitfalls
Frequently asked questions
How do you multiply decimals?
Ignore the decimal points and multiply the digits. Then count the decimal places in both numbers and put that many in the answer. For example, \(0.7 \times 0.04\): \(7 \times 4 = 28\), with \(3\) places, so the answer is \(0.028\).
How do you divide by a decimal?
Multiply both numbers by \(10\), \(100\) or \(1000\) until the number you are dividing by is a whole number, then divide as usual. For example, \(5.6 \div 0.8 = 56 \div 8 = 7\).
What are significant figures?
Significant figures are the digits that carry meaning in a number, counted from the first non-zero digit. \(0.0045\) has \(2\): the \(4\) and the \(5\).
Do zeros count as significant figures?
Leading zeros never count. Zeros between other digits always count. Zeros at the end count after a decimal point, and are usually placeholders in a whole number.
What is the difference between decimal places and significant figures?
Decimal places count the digits after the point. Significant figures count from the first non-zero digit. \(0.00412\) has \(5\) decimal places but \(3\) significant figures.
How do you give an answer in scientific notation to 3 significant figures?
Write the full answer in scientific notation, then round the front number to \(3\) digits: \(2361.96 = 2.36196 \times 10^3 \approx 2.36 \times 10^3\).