KS3 Maths · Year 8 · Chapter 9: Decimal numbers · Lesson 1

Powers of 10

Year 8About 60 minutesNo calculator48 marks

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Date

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Quick questions to start.

A1
Needed today
Work out $0.47 \times 1000$ .
[1 mark]
Answer
A2
Needed today
Work out $36 \div 10000$ .
[1 mark]
Answer
A3
Last lesson
Factorise $x^{2} - 81$ .
[1 mark]
Answer
A4
From chapter 7
Work out $\frac{3}{4} - \frac{2}{5}$ .
[1 mark]
Answer

Example, then your turn

Year 8

Your teacher works through each example with you. Then try the one beside it.

Example 1

Work out

(a) $0.0384 \times 1 0^{4}$

(b) $52.6 \div 1 0^{3}$ .

Hint Multiplying by

1 0^{4}

moves every digit

4

places to the left. Dividing by

1 0^{3}

moves every digit

3

places to the right.

B1
Work out
(a) $0.6125 \times 1 0^{3}$
(b) $7.9 \div 1 0^{4}$ .
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 2

Work out

(a) $4.6 \times 1 0^{- 2}$

(b) $0.38 \times 1 0^{- 3}$

Hint

1 0^{- 2} = 0.01

, so multiplying by

1 0^{- 2}

is the same as dividing by

100

. Dividing by

1 0^{- 2}

undoes that: it is the same as multiplying by

100

B2
Work out
(a) $83.2 \times 1 0^{- 1}$
(b) $0.57 \times 1 0^{- 4}$
(c) $0.042 \div 1 0^{- 3}$ .
[3 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$ (c) $\dots\dots$

Example 3

$1$ micrometre $= 1 0^{- 6}$ metres and $1$ gigawatt $= 1 0^{9}$ watts.

(a) Write $0.000045$ metres in micrometres.

(b) Write $3.2$ gigawatts in watts.

Hint To change metres into micrometres, divide by

1 0^{- 6}

: that is the same as multiplying by

1000000

B3
$1$ nanosecond $= 1 0^{- 9}$ seconds and $1$ megabyte $= 1 0^{6}$ bytes.
(a) Write $0.00000008$ seconds in nanoseconds.
(b) Write $2.7$ megabytes in bytes.
[2 marks]
Answer(a) $\dots\dots$ ns (b) $\dots\dots$ bytes

Example 4

$1$ kilometre $= 1 0^{3}$ metres and $1$ micrometre $= 1 0^{- 6}$ metres. How many micrometres are there in $1$ kilometre? Give your answer as a power of $10$ .

Hint Divide the size of the big unit by the size of the small unit. When you divide powers of

10

, subtract the indices.

B4
$1$ centimetre $= 1 0^{- 2}$ metres and $1$ nanometre $= 1 0^{- 9}$ metres. How many nanometres are there in $1$ centimetre? Give your answer as a power of $10$ .
[2 marks]
Answer $1 0^{\dots}$

Example 5

An island has an area of $2600$ km². It has a population of $120000$ people.

(a) Work out the average land area for each person, in km². Give your answer correct to $2$ significant figures.

(b) $1$ km $^{2} = 1 0^{6}$ m². Write the area for each person in m², correct to $2$ significant figures.

Hint Area for each person

=

total area

\div

number of people. Round only at the end.

B5
A country has an area of $5100$ km². It has a population of $900000$ people.
(a) Work out the average land area for each person, in km². Give your answer correct to $2$ significant figures.
(b) $1$ km $^{2} = 1 0^{6}$ m². Write the area for each person in m², correct to $2$ significant figures.
[4 marks]
Answer(a) $\dots\dots$ km² (b) $\dots\dots$ m²

Practice

Year 8

C1
Find the missing power.
(a) $0.27 \times 1 0^{□} = 2700$
(b) $615 \times 1 0^{□} = 0.615$
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$
C2
Work out
(a) $46.8 \times 0.01$
(b) $5.3 \times 0.001$
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

C3
Write each number as a decimal.
(a) $1 0^{- 5}$
(b) $2.06 \times 1 0^{- 3}$
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$
C4
Write these numbers in order of size. Start with the smallest.
$0.4 \times 1 0^{- 1}$ , $35 \times 1 0^{- 3}$ , $0.0038 \times 10$ , $3.6 \times 1 0^{- 2}$
[2 marks]
Answer
C5
Two of these calculations give $0.064$ . The other two give $640$ . Sort them into the two pairs.
P: $6.4 \times 1 0^{- 2}$ Q: $6.4 \div 1 0^{2}$
R: $6.4 \div 1 0^{- 2}$ S: $6.4 \times 100$
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
Answer each part.
(a)
Write down the value of $1 0^{0}$ .
[1 mark]
Answer
(b)
Work out $0.703 \times 1 0^{3}$ .
[1 mark]
Answer
(c)
Work out $18 \times 1 0^{- 2}$ .
[1 mark]
Answer
Total for D1: 3 marks

D2
Ben says, "To work out $0.62 \times 1 0^{- 3}$ you move the decimal point $3$ places to the right, so the answer is $620$ ."
Explain why Ben is wrong. Work out the correct answer.
[2 marks]
D3
A grain of pollen is $2.5 \times 1 0^{- 5}$ metres wide.
(a)
Write $2.5 \times 1 0^{- 5}$ as an ordinary number.
[1 mark]
Answer
(b)
Grains of pollen are laid in a straight line, touching. The line is $1$ cm long. Work out how many grains of pollen are in the line.
[2 marks]
Answer
(c)
$1$ micrometre $= 1 0^{- 6}$ metres. Write the width of one grain in micrometres.
[1 mark]
Answer
Total for D3: 4 marks
D4
Island P has a population of $3.6 \times 1 0^{5}$ and an area of $1.2 \times 1 0^{3}$ km².
Island Q has a population of $84000$ and an area of $400$ km².
(a)
Work out the average number of people for each km² on island P.
[2 marks]
Answer
(b)
Work out the average number of people for each km² on island Q.
[1 mark]
Answer
(c)
Which island is more crowded? Give a reason for your answer.
[1 mark]
Answer
(d)
$1$ km $^{2} = 1 0^{6}$ m². Work out the average land area for each person on island Q, in m². Give your answer correct to $2$ significant figures.
[2 marks]
Answer
Total for D4: 6 marks
D5
A memory card holds $64$ gigabytes. A photo uses $4$ megabytes.
$1$ gigabyte $= 1 0^{9}$ bytes and $1$ megabyte $= 1 0^{6}$ bytes.
Work out the number of photos that the memory card can hold. You must show your working.
[3 marks]
Answer

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Light travels $3 \times 1 0^{8}$ metres in one second. A computer chip carries out one step every $0.4$ nanoseconds ( $1$ nanosecond $= 1 0^{- 9}$ seconds). Work out how far light travels during one step. Give your answer in centimetres.
[3 marks]
Hint 1 · What to try
Write the time for one step in seconds, then use distance $=$ speed $\times$ time.
Hint 2 · The first line
$0.4$ nanoseconds $= 0.4 \times 1 0^{- 9} = 4 \times 1 0^{- 10}$ seconds.
Hint 3 · The full method
$3 \times 1 0^{8} \times 4 \times 1 0^{- 10} = 12 \times 1 0^{- 2} = 0.12$ m, and $0.12$ m $= 12$ cm.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can multiply and divide by positive and negative powers of $10$ .
I can change units that use prefixes such as micro, nano, mega and giga.
I can work out an amount for each person and round it to $2$ significant figures.

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