Sec 1 · G2 (N(A))

Approximation and estimation

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1. Rounding

A number line from 3.4 to 3.5 with the halfway mark 3.45 dashed. 3.47 is past halfway, so it rounds up to 3.5.

Round to	Example	Answer
nearest $10$	$3846$	$3850$
nearest whole number	$27.6$	$28$
$1$ decimal place	$3.47$	$3.5$
$2$ decimal places	$0.08362$	$0.08$
$3$ significant figures	$0.0045082$	$0.00451$
$2$ significant figures	$63\,750$	$64\,000$

📖 Worked example

Use a calculator to evaluate $\dfrac{70}{9}$ , giving your answer correct to 2 decimal places.

Step 1 of 2

Work it out on the calculator: write down at least 4 decimal places

\dfrac{70}{9} = 7.77777\ldots

📖 Worked example

Use a calculator to evaluate $\dfrac{\sqrt[3]{100} + 3.4^{2}}{12.6 - 1.9}$ , giving your answer correct to 3 significant figures.

Step 1 of 4

Work out the numerator on its own (keep at least 4 decimal places)

\sqrt[3]{100} + 3.4^{2} = 16.2015\ldots

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2. Range of possible values

Given	Half a unit	Range
$h = 42$ m, to the nearest m	$0.5$ m	$41.5 \leq h < 42.5$
$d = 3.7$ km, to $1$ decimal place	$0.05$ km	$3.65 \leq d < 3.75$

📖 Worked example

The height of an HDB block is $42$ m, correct to the nearest metre. Given that the actual height of the block is $h$ m, write down the range of possible values of $h$ .

Step 1 of 2

The actual value can be up to half a rounding unit either side

Half of

1

= 0.5

📖 Worked example

The population of a town is $62\,500$ , correct to 3 significant figures. Find the smallest and the largest possible population.

Step 1 of 4

Which place is the last significant figure in? (count 3 figures from the left)

62\,500

to 3 significant figures is rounded to the nearest

100

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3. Significant figures

Zeros	Significant?	Example (s.f.)
between non-zero digits	yes	$3\mathbf{0}5$ has $3$
at the front	no	$0.00\mathbf{4}\mathbf{7}$ has $2$
at the end, after a decimal point	yes	$2.5\mathbf{0}$ has $3$
at the end of a whole number	may or may not be	$4500$ has $2$ , $3$ or $4$

📖 Worked example

Write $0.02995$ correct to 3 significant figures.

Step 1 of 3

Count 3 significant figures from the first non-zero digit, then look at the next digit

The 4th significant figure is

5

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4. Estimation

Round each number to 1 s.f., then work it out

\displaystyle \dfrac{48.7 \times 3.12}{9.8}

$\dfrac{50 \times 3}{10}$ , which is $15$
the calculator gives $15.5$ (3 s.f.): close, so it is sensible

Square root: use the nearest perfect square

\displaystyle \sqrt{63.5}

$\sqrt{64}$ , which is $8$

📖 Worked example

By writing each number correct to 1 significant figure, estimate the value of $\dfrac{812 \times 2.07}{38.6}$ .

Step 1 of 2

Round each number to 1 significant figure

\dfrac{812 \times 2.07}{38.6} \approx \dfrac{800 \times 2}{40}

📖 Worked example

Nur Aisyah estimates the value of $\dfrac{8.96}{4.03}$ by writing each number correct to 1 significant figure. Without finding the exact value, explain whether the estimate is greater than or less than the exact value.

Step 1 of 2

Round each number to 1 significant figure: did it go up or down?

8.96 \rightarrow 9

is rounded up;

4.03 \rightarrow 4

is rounded down.

📖 Worked example

By rounding each number to the nearest whole number, estimate the value of $\dfrac{7.92 \times 3.15}{3.23}$ . Show your working clearly.

Step 1 of 2

Round each number to the nearest whole number (look at the first decimal place)

\dfrac{7.92 \times 3.15}{3.23} \approx \dfrac{8 \times 3}{3}

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Sec 1 · G2 (N(A))

Approximation and estimation

Practise approximation and estimation with step-by-step help. Free account.

Start free

1. Rounding

Round to	Example	Answer
nearest $10$	$3846$	$3850$
nearest whole number	$27.6$	$28$
$1$ decimal place	$3.47$	$3.5$
$2$ decimal places	$0.08362$	$0.08$
$3$ significant figures	$0.0045082$	$0.00451$
$2$ significant figures	$63\,750$	$64\,000$

📖 Worked example

Use a calculator to evaluate $\dfrac{70}{9}$ , giving your answer correct to 2 decimal places.

Step 1 of 2

Work it out on the calculator: write down at least 4 decimal places

\dfrac{70}{9} = 7.77777\ldots

📖 Worked example

Use a calculator to evaluate $\dfrac{\sqrt[3]{100} + 3.4^{2}}{12.6 - 1.9}$ , giving your answer correct to 3 significant figures.

Step 1 of 4

Work out the numerator on its own (keep at least 4 decimal places)

\sqrt[3]{100} + 3.4^{2} = 16.2015\ldots

Practise this →

2. Range of possible values

Given	Half a unit	Range
$h = 42$ m, to the nearest m	$0.5$ m	$41.5 \leq h < 42.5$
$d = 3.7$ km, to $1$ decimal place	$0.05$ km	$3.65 \leq d < 3.75$

📖 Worked example

The height of an HDB block is $42$ m, correct to the nearest metre. Given that the actual height of the block is $h$ m, write down the range of possible values of $h$ .

Step 1 of 2

The actual value can be up to half a rounding unit either side

Half of

1

= 0.5

📖 Worked example

The population of a town is $62\,500$ , correct to 3 significant figures. Find the smallest and the largest possible population.

Step 1 of 4

Which place is the last significant figure in? (count 3 figures from the left)

62\,500

to 3 significant figures is rounded to the nearest

100

Practise this →

3. Significant figures

Zeros	Significant?	Example (s.f.)
between non-zero digits	yes	$3\mathbf{0}5$ has $3$
at the front	no	$0.00\mathbf{4}\mathbf{7}$ has $2$
at the end, after a decimal point	yes	$2.5\mathbf{0}$ has $3$
at the end of a whole number	may or may not be	$4500$ has $2$ , $3$ or $4$

📖 Worked example

Write $0.02995$ correct to 3 significant figures.

Step 1 of 3

Count 3 significant figures from the first non-zero digit, then look at the next digit

The 4th significant figure is

5

Practise this →

4. Estimation

Round each number to 1 s.f., then work it out

\displaystyle \dfrac{48.7 \times 3.12}{9.8}

$\dfrac{50 \times 3}{10}$ , which is $15$
the calculator gives $15.5$ (3 s.f.): close, so it is sensible

Square root: use the nearest perfect square

\displaystyle \sqrt{63.5}

$\sqrt{64}$ , which is $8$

📖 Worked example

By writing each number correct to 1 significant figure, estimate the value of $\dfrac{812 \times 2.07}{38.6}$ .

Step 1 of 2

Round each number to 1 significant figure

\dfrac{812 \times 2.07}{38.6} \approx \dfrac{800 \times 2}{40}

📖 Worked example

Step 1 of 2

Round each number to 1 significant figure: did it go up or down?

8.96 \rightarrow 9

is rounded up;

4.03 \rightarrow 4

is rounded down.

📖 Worked example

By rounding each number to the nearest whole number, estimate the value of $\dfrac{7.92 \times 3.15}{3.23}$ . Show your working clearly.

Step 1 of 2

Round each number to the nearest whole number (look at the first decimal place)

\dfrac{7.92 \times 3.15}{3.23} \approx \dfrac{8 \times 3}{3}

Practise this →

Practise approximation and estimation with step-by-step help

Endless questions from basic to exam level. Get one wrong and see where it went wrong. Free account.

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