1. Upper & Lower Bounds
What could be the highest this number could be if it
has already been rounded to the nearest 10? 60 90 80 70
74 would be rounded down to 70
but 75 would be rounded up to 80
Therefore the highest the number
could be before rounding is 74
What could be the lowest this number could be if it
has already been rounded to the nearest 10? 60 90 80 70
65 would be rounded up to 70
but 64 would be rounded down to 60
Therefore the lowest the number
could be before rounding is 65

2. Upper & Lower Bounds
This works well when counting in whole number denominations
such as people or chairs or tables.
When working with continuous denominations such as length,
weight or height where we can have any value in a range we
make an assumption.
A journey of 26 miles to the nearest mile could be as long as
…. miles or as short as 25.5 miles.
It could not be 26.5 miles, as this would round to 27 miles.
However, …. is virtually the same as 26.5
So we overcome this difficulty by saying that
26.5 is the upper bound
25.5 is the lower bound

3. It is therefore valid to give the upper bound without using
Upper & Lower Bounds
A mathematical peculiarity:
or 0.9
Let x = …. 1
or 9.9
multiply by 10
10x = …. 2
subtract from 1 2
Notice how the 0.9 disappears
9x = 9
divide by 9
x = 1
It is therefore valid to give the upper bound without using
recurring decimals
e.g. the upper and lower bounds of 7.1 (given to the nearest 1 d.p.) are
7.15
7.05

4. 1. Each of these quantities is rounded to the nearest whole number
Upper & Lower Bounds
Now try these
1. Each of these quantities is rounded to the nearest whole number
of units. Write down the minimum and maximum possible size of
each quantity.
26 g b) 4 cm c) 225 m
d) 13 litres e) 33 kg f) £249
26.5 g
25.5 g
4.5 cm
3.5 cm
225.5 m
224.5 m
12.5 g
13.5 g
33.5 kg
32.5 kg
£249.49
£248.50
Note: pence can only
be whole numbers
3. A packet weighs 2 kg, correct to the nearest 100 g.
What is the maximum possible weight?
2.050 kg
5. The weight of a toffee is 5 g correct to the nearest half gram.
What is the minimum possible weight of one toffee?
4.75 g

5. Upper & Lower Bounds
Now try these:
Write down the upper and lower bounds of each of these
values given to the accuracy stated.
8 m (1s.f.) g (1 s.f.)
26 Kg (2.s.f) m (1 d.p.)
25 min (2 s.f.) l (3 s.f.)
2.40 m (2 d.p.) weeks (2 s.f.)
0.2 kg (1 d.p.) g (2 s.f.)
0.06 s (2 d.p.) g (2 s.f.)
8.5
7.5
350
250
26.5
25.5
0.75
0.65
25.5
24.5
366.5
365.5
2.405
2.395
175
165
0.25
0.15
85.5
84.5
0.065
0.055
215
205

6. Maximum & Minimum Values
A rectangle’s sides are measured to the nearest centimetre.
26 cm
14 cm
Find the largest and smallest possible areas of this rectangle
Upper and lower bounds of 26 are:
26.5 cm and 25.5 cm
Upper and lower bounds of 14 are:
14.5 cm and 13.5 cm
The largest possible area = 26.5 x 14.5
= cm2
The smallest possible area = 25.5 x 13.5
= cm2

7. Maximum & Minimum Values
The table gives some information about the three counties in
East Anglia
Each land area is given to the nearest 100 km2 and each
Population is given to the nearest 1000 people
Population density =
Population
Land Area
Calculate the lower bound or the population density for the whole
of East Anglia.
3450
669500
5450
758500
3850
661500
12750
12750
smallest value
largest value
gives the lower bound
= 163.9