Comparing/calculating very large and small quantities | Year 9

01 / LEARN

The idea, step by step

Compare the scale first

For positive values in standard scientific notation, compare the exponents first. If they match, compare the coefficients. For example, $8.7 \times 10^{4} < 1.2 \times 10^{5}$ , even though 8.7 is greater than 1.2.

To multiply, multiply coefficients and add exponents. To divide, divide coefficients and subtract exponents. Rewrite the result if its coefficient is outside $1 \leq a < 10$ .

Addition needs a shared unit of scale

Powers of ten act like units. Before adding $4.2 \times 10^{5}$ and $7 \times 10^{4}$ , rewrite the second as $0.7 \times 10^{5}$ . Then the coefficients can be added to give $4.9 \times 10^{5}$ .

Unlike multiplication, addition does not use the sum of exponents. Keep track of the question units and use an estimate to check the final scale. A ratio of two quantities in the same units is dimensionless.

A common trap

Adding exponents when adding two numbers in scientific notation.

Instead: Align the powers before adding coefficients. Adding exponents is a multiplication law.

02 / WORKED EXAMPLES

See the thinking

Example 1

Multiply and normalise

Calculate $(6 \times 10^{4}) (5 \times 10^{- 2})$ .

Multiply coefficients: $6 \times 5 = 30$ .
Add exponents: $4 + (- 2) = 2$ , giving $30 \times 10^{2}$ .
Normalise: $30 \times 10^{2} = 3 \times 10^{3}$ .

3 \times 10^{3}

Example 2

Subtract matching scales

Calculate $8.1 \times 10^{6} - 6 \times 10^{5}$ .

Rewrite $6 \times 10^{5} = 0.6 \times 10^{6}$ .
Subtract coefficients: $8.1 - 0.6 = 7.5$ .
The result is $7.5 \times 10^{6}$ .

7.5 \times 10^{6}

Check your understanding

Which is larger: $9.1 \times 10^{- 4}$ or $2.3 \times 10^{- 3}$ ?

Show answer

$2.3 \times 10^{- 3}$

The second equals $23 \times 10^{- 4}$ , which is larger than $9.1 \times 10^{- 4}$ .

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Order from smallest to largest: $3.2 \times 10^{5}$ , $7.4 \times 10^{4}$ , $2.9 \times 10^{5}$ .
2 marks
Calculate $(4 \times 10^{3}) (2 \times 10^{5})$ .
2 marks
Calculate $\frac{9 \times 10^{7}}{3 \times 10^{2}}$ .
2 marks
Calculate $2.4 \times 10^{6} + 5 \times 10^{5}$ .
2 marks
Calculate $(8 \times 10^{- 3}) (5 \times 10^{- 4})$ .
2 marks
A file of $6 \times 10^{8}$ bytes is sent at $2 \times 10^{6}$ bytes per second. Find the time, assuming a constant rate.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Order from smallest to largest: $3.2 \times 10^{5}$ , $7.4 \times 10^{4}$ , $2.9 \times 10^{5}$ .
2 marks
The $10^{4}$ quantity is smaller than both $10^{5}$ quantities.
For matching powers compare 2.9 with 3.2.
$7.4 \times 10^{4}, 2.9 \times 10^{5}, 3.2 \times 10^{5}$
Calculate $(4 \times 10^{3}) (2 \times 10^{5})$ .
2 marks
$4 \times 2 = 8$ and $3 + 5 = 8$ .
$8 \times 10^{8}$
Calculate $\frac{9 \times 10^{7}}{3 \times 10^{2}}$ .
2 marks
$9 \div 3 = 3$ and $7 - 2 = 5$ .
$3 \times 10^{5}$
Calculate $2.4 \times 10^{6} + 5 \times 10^{5}$ .
2 marks
$5 \times 10^{5} = 0.5 \times 10^{6}$ .
$(2.4 + 0.5) \times 10^{6} = 2.9 \times 10^{6}$ .
$2.9 \times 10^{6}$
Calculate $(8 \times 10^{- 3}) (5 \times 10^{- 4})$ .
2 marks
$8 \times 5 = 40$ and $- 3 + (- 4) = - 7$ .
$40 \times 10^{- 7} = 4 \times 10^{- 6}$ .
$4 \times 10^{- 6}$
A file of $6 \times 10^{8}$ bytes is sent at $2 \times 10^{6}$ bytes per second. Find the time, assuming a constant rate.
3 marks
$t = \frac{6 \times 10^{8}}{2 \times 10^{6}} = 3 \times 10^{2}$ .
$3 \times 10^{2} = 300$ seconds.
$300$ seconds

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum

Year 9 numerical integer exponent laws and their extension to variables; scientific notation and practical measurement applications. Excludes fractional-index and full surd algebra taught in Year 10.

Official curriculum source ↗

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