KS3 Maths · Year 8 · Chapter 9: Decimal numbers · Lesson 5

Upper and lower bounds

Year 8About 70 minutesCalculator allowed50 marks

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Date

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Quick questions to start.

A1
Needed today
Round $2650$ to the nearest hundred.
[1 mark]
Answer
A2
Last lesson
Work out $(8 \times 1 0^{6}) \div (2 \times 1 0^{- 3})$ . Give your answer in standard form.
[1 mark]
Answer
A3
From chapter 8
Factorise $x^{2} + 7 x + 12$ .
[1 mark]
Answer
A4
Year 7
Round $0.0538$ to $1$ significant figure.
[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

Write down the lower bound and the upper bound of each measurement.

(a) $70$ cm, to the nearest $10$ cm

(b) $3.6$ kg, to $1$ decimal place

Hint Go half a unit each way. Half of

10

cm is

5

cm; half of

0.1

kg is

0.05

kg.

B1
Write down the lower bound and the upper bound of each measurement.
(a) $400$ m, to the nearest $100$ m
(b) $12.8$ seconds, to $1$ decimal place
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 2

Write the error interval for each length $L$ .

(a) $L = 6000$ m, to $1$ significant figure

(b) $L = 0.47$ m, to $2$ significant figures

Hint The last significant figure tells you the unit:

6000

1

s.f. is to the nearest

1000

. Use

\leq

at the bottom and

<

at the top.

B2
Write the error interval for each mass $m$ .
(a) $m = 900$ g, to $1$ significant figure
(b) $m = 0.082$ kg, to $2$ significant figures
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 3

Each length of this rectangle is given to the nearest centimetre. Work out the least and the greatest possible perimeter.

Not drawn accurately

Hint Use the lower bounds for the least perimeter and the upper bounds for the greatest.

B3
Each side of this triangle is given to the nearest centimetre. Work out the least and the greatest possible perimeter.
[3 marks]
Not drawn accurately
Answerleast $\dots\dots$ greatest $\dots\dots$

Example 4

Each length of this rectangle is given to the nearest metre. Work out the least and the greatest possible area.

Not drawn accurately

Hint Multiply the two lower bounds, then the two upper bounds.

B4
This square has sides of $6.4$ cm, to $1$ decimal place. Work out the least and the greatest possible area.
[3 marks]
Not drawn accurately
Answerleast $\dots\dots$ greatest $\dots\dots$

Example 5

Ella runs $100$ m, to the nearest metre, in $12$ seconds, to the nearest second. Work out her greatest and least possible average speed. Give your answers to $3$ significant figures.

Hint Speed

=

distance

\div

time. A big answer needs a big distance and a small time.

B5
Kai cycles $36$ km, to the nearest km, in $1.5$ hours, to the nearest $0.1$ hour. Work out his greatest and least possible average speed. Give your answers to $1$ decimal place.
[4 marks]
Answergreatest $\dots\dots$ least $\dots\dots$

Practice

Year 8

C1
The number of people at a concert is $2700$ , to the nearest $100$ . Write down the least and the greatest possible number of people.
[2 marks]
Answerleast $\dots\dots$ greatest $\dots\dots$
C2
Two pieces of wood are $15$ cm and $8$ cm long, each to the nearest centimetre. Work out the greatest possible difference between their lengths.
[2 marks]
Answer

C3
$x = 7.2$ and $y = 3.5$ , both correct to $1$ decimal place. Work out the least possible value of $x \times y$ .
[2 marks]
Answer
C4
A shelf is $80$ cm long, to the nearest cm. Twenty books are each $4.0$ cm thick, to the nearest mm. Can you be sure that all $20$ books will fit on the shelf? Give a reason.
[2 marks]

Exam-style questions

GCSE Foundation

D1
Answer each part.
(a)
A length is $540$ mm, to the nearest $10$ mm. Write down its lower bound.
[1 mark]
Answer
(b)
A mass is $8.7$ kg, to $1$ decimal place. Write down its upper bound.
[1 mark]
Answer
(c)
A time is $t = 30$ seconds, to the nearest $5$ seconds. Write down the error interval for $t$ .
[2 marks]
Answer
Total for D1: 4 marks

D2
A rectangle is $5$ cm long and $3$ cm wide, both to the nearest centimetre. Sam says, "The greatest possible area is $6 \times 4 = 24$ cm²."
Explain what Sam has done wrong. Work out the greatest possible area of the rectangle.
[3 marks]
Not drawn accurately
D3
Ben lays $30$ tiles in a row along a wall. Each tile is $20$ cm long, to the nearest centimetre. The wall is $6$ m long, to the nearest $10$ cm. Can Ben be sure that the row of tiles covers the whole length of the wall? You must show your working.
[4 marks]
D4
A bag holds $25$ kg of sand, to the nearest kg. Some of the sand is used. The bag now holds $21.6$ kg of sand, to $1$ decimal place.
(a)
Write down the upper bound of $25$ kg.
[1 mark]
Answer
(b)
Write down the lower bound of $21.6$ kg.
[1 mark]
Answer
(c)
Work out the greatest possible mass of sand that was used.
[2 marks]
Answer
(d)
Work out the least possible mass of sand that was used.
[1 mark]
Answer
Total for D4: 5 marks
D5
The base and the height of this triangle are each given to the nearest centimetre. Work out the least and the greatest possible area of the triangle.
[4 marks]
Not drawn accurately
Answerleast $\dots\dots$ greatest $\dots\dots$

Extension

Stretch

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

E1
This rectangle has an area of $48$ cm², to the nearest cm². Its length is $8$ cm, to the nearest cm. Work out the greatest and the least possible width. Give your answers to $3$ significant figures.
[4 marks]
Not drawn accurately
Hint 1 · What to try
Width $=$ area $\div$ length. Write down the bounds of the area and of the length.
Hint 2 · The first line
The area is between $47.5$ and $48.5$ ; the length is between $7.5$ and $8.5$ .
Hint 3 · The full method
Greatest width $= 48.5 \div 7.5$ . Least width $= 47.5 \div 8.5$ .
Answergreatest $\dots\dots$ least $\dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can write down the upper and lower bounds of a rounded measurement.
I can write an error interval such as $5500 \leq L < 6500$ .
I can choose which bounds give the least and greatest perimeter, area, speed or difference.

Answers and mark scheme

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