KS3 Maths · Year 7 · Chapter 8: Number · Lesson 1

Powers of 10

Year 7About 60 minutesNo calculator50 marks

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Quick questions to start.

A1
Needed today
Write $1 0^{4}$ as an ordinary number.
[1 mark]
Answer
A2
Needed today
Work out $3.6 \times 100$ .
[1 mark]
Answer
A3
Last lesson
A plumber charges £40 to call plus £25 for each hour. Work out the charge for a $3$ -hour job.
[1 mark]
Answer
A4
From chapter 4
Work out $15%$ of £60.
[1 mark]
Answer

Example, then your turn

Year 7

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

Example 1

Work out

(a) $6.284 \times 1 0^{3}$

(b) $517 \div 1 0^{2}$ .

Hint

1 0^{3} = 1000

. Multiplying by

1000

moves every digit

3

places to the left; dividing by

100

moves every digit

2

places to the right.

B1
Work out
(a) $0.0473 \times 1 0^{2}$
(b) $89.1 \div 1 0^{3}$ .
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 2

Work out

(a) $3.92 \times 1 0^{- 2}$

(b) $45 \times 1 0^{- 3}$ .

Hint

1 0^{- 2} = 0.01

, so multiplying by

1 0^{- 2}

is the same as dividing by

100

B2
Work out
(a) $7.15 \times 1 0^{- 1}$
(b) $268 \times 1 0^{- 3}$ .
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 3

Work out

(a) $5.18 \div 1 0^{- 2}$

(b) $0.063 \div 1 0^{- 3}$ .

Hint Dividing by

1 0^{- 2}

undoes multiplying by

1 0^{- 2}

, so it is the same as multiplying by

100

. The answer is bigger.

B3
Work out
(a) $2.47 \div 1 0^{- 1}$
(b) $0.0915 \div 1 0^{- 4}$ .
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 4

(a) Write $0.0001$ as a power of $10$ .

(b) Write $4800$ in the form $48 \times 1 0^{n}$ .

Hint Count the places the digits move. Moving to the left is a positive power; moving to the right is a negative power.

B4
(a) Write $0.00001$ as a power of $10$ .
(b) Write $930000$ in the form $93 \times 1 0^{n}$ .
(c) Write $0.071$ in the form $71 \times 1 0^{n}$ .
[3 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$ (c) $\dots\dots$

Example 5

One grain of sand has a mass of $7 \times 1 0^{- 4}$ grams. Work out the mass of $2000000$ grains of sand. Give your answer in kilograms.

Hint Write

7 \times 1 0^{- 4}

as a decimal first. Then multiply, and change grams to kilograms at the end.

B5
One sheet of card is $3 \times 1 0^{- 1}$ mm thick. A stack has $240$ sheets of this card. Work out the height of the stack in centimetres.
[3 marks]
Answer

Practice

Year 7

C1
Find the missing power:
(a) $6.3 \times 1 0^{□} = 0.0063$
(b) $2.4 \div 1 0^{□} = 2400$ .
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$
C2
Write these numbers in order of size. Start with the smallest.
$5 \times 1 0^{- 2}$ , $0.6 \div 10$ , $48 \times 1 0^{- 3}$ , $0.0049 \times 10$
[2 marks]
Answer

C3
Write each answer in index form.
(a) $1 \div 10000000$
(b) $6 \div 1000$
(c) $3 \div 10000$
[3 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$ (c) $\dots\dots$
C4
$1$ millimetre $= 1 0^{- 3}$ metres.
(a) Write $2650$ mm in metres.
(b) Write $0.38$ m in millimetres.
[2 marks]
Answer(a) $\dots\dots$ m (b) $\dots\dots$ mm
C5
Work out $0.36 \times 1 0^{4} \div 1 0^{- 2}$ .
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
Answer each part.
(a)
Write $1 0^{- 2}$ as a decimal.
[1 mark]
Answer
(b)
Work out $2.85 \times 1 0^{3}$ .
[1 mark]
Answer
(c)
Work out $64 \times 1 0^{- 3}$ .
[1 mark]
Answer
Total for D1: 3 marks

D2
Jamal says, "Dividing by $1 0^{- 2}$ makes a number smaller, so $3.5 \div 1 0^{- 2} = 0.035$ ."
Explain why Jamal is wrong. Work out the correct answer.
[2 marks]
Answer
D3
A bacterium is $2 \times 1 0^{- 3}$ mm long.
(a)
Write $2 \times 1 0^{- 3}$ as an ordinary number.
[1 mark]
Answer
(b)
How many of these bacteria, placed end to end, make a line $1$ cm long? You must show your working.
[2 marks]
Answer
(c)
A sample holds $3000000$ of these bacteria. They are placed end to end. Work out the length of the line in metres.
[2 marks]
Answer
Total for D3: 5 marks
D4
A grain of rice has a mass of $0.025$ grams.
(a)
Write $0.025$ in the form $25 \times 1 0^{n}$ .
[1 mark]
Answer
(b)
A bag holds $48000$ grains of rice. Work out the mass of the rice in the bag. Give your answer in kilograms.
[3 marks]
Answer
Total for D4: 4 marks
D5
Work out the value of $n$ and the value of $m$ .
(a)
$4.6 \times 1 0^{n} = 0.046$
[1 mark]
Answer $n = \dots\dots$
(b)
$0.83 \div 1 0^{m} = 830$
[1 mark]
Answer $m = \dots\dots$
Total for D5: 2 marks
D6
Ruth writes these four numbers. $0.7 \times 1 0^{- 1} 70 \times 1 0^{- 3} 0.007 \times 1 0^{2} 7 \div 1 0^{3}$
(a)
Two of the numbers are equal. Write down these two numbers.
[2 marks]
Answer
(b)
Which number is the largest?
[1 mark]
Answer
Total for D6: 3 marks

Extension

Stretch

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

E1
A dripping tap loses one drop every half a second. One drop has a volume of $5 \times 1 0^{- 2}$ cm³. $1$ litre $= 1000$ cm³. How many litres of water does the tap lose in one day? You must show your working.
[4 marks]
Hint 1 · What to try
Work out how many drops fall in one day. There are $24$ hours in a day, $60$ minutes in an hour and $60$ seconds in a minute.
Hint 2 · The first line
One day is $24 \times 60 \times 60 = 86400$ seconds, so $2 \times 86400 = 172800$ drops fall.
Hint 3 · The full method
$172800 \times 0.05 = 8640$ cm³, and $8640 \div 1000 = 8.64$ litres.
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 write a number in index form, such as $59 \times 1 0^{- 4}$ .
I can use powers of $10$ to change units and solve problems.

Answers and mark scheme

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