Units of Measurement 2
Upper and Lower Bounds

When measurements are made, they can only be obtained to a limited degree of accuracy. For example if the length of a line is given as 11 mm, this means that it is 11 mm to the nearest mm. In fact, if l is the length then it lies in the range

10.5 ≤ l < 11.5

Worked Examples

1

For each length below state the range of values within which the length must be.

If l represents the given length then:
(a)
18 cm Show me 17.5 ≤ l < 18.5
(b)
12.7 cm Show me 12.65 ≤ l < 12.75
(c)
11.06 m Show me 11.055 ≤ l < 11.065

Note

The lower bound is the smallest number which will round to the given number, while the upper bound is the smallest number which will not round to the given number.

2

If the length of the sides of the rectangle are given to the nearest cm, find:

First note that the length, l, of the rectangle is 8 cm to the nearest cm, so 7.5 ≤ l < 8.5.
Similarly the width, w, of the rectangle lies in the range 3.5 ≤ w < 4.5.
(a)

the minimum possible perimeter

Show me

To find the minimum possible perimeter, use the minimum possible length,

Minimum perimeter = 7.5 + 3.5 + 7.5 + 3.5
= 22 cm
(b)

the range of possible areas.

Show me

The range of values for the area can be calculated using the maximum and minimum lengths of each side.

Minimum area = 3.5 × 7.5
= 26.25.
Maximum area = 4.5 × 8.5
= 38.25

So

26.25 ≤ area < 38.25

Exercises

For each quantity below, state the range of values within which it must lie.

(a)
x = 4.7 ≤ x <
(b)
l = 42 ≤ l <
(c)
A = 15.62 ≤ A <
(d)
d = 16.2 ≤ d <
(e)
r = 11.68 ≤ r <
(f)
m = 14.24 ≤ m <
(g)
w = 218 ≤ w <
(h)
l = 15.20 ≤ l <
(i)
w = 5.00 ≤ w <
(j)
v = 20.0 ≤ v <
(k)
A = 18.09 ≤ A <
(l)
r = 31.451 ≤ r <

The lengths of the rectangle are given to the nearest cm.

(a)

State the range of values within which the length, l, and the width, w, must lie.

≤ l <

≤ w <

(b)

Find the range of values within which,

(i) the perimeter

(ii) the area

of the rectangle must lie.

(i) ≤ perimeter <

(ii) ≤ area <

The radius of a circle is 12 mm to the nearest mm.

(a)

State the range of possible values for the radius.

≤ radius <
(b)

Find the range of possible values for the:

Give your answer rounded to 1 d.p. where necessary

(i)
diameter ≤ diameter <
(ii)
circumference ≤ circumference <
(iii)
area. ≤ area <

The quantities x and y are given to 1 significant figure as x = 20 and y = 40 .
Find the minimum possible value of each expression below.

Give your answer rounded to 2 d.p. where needed.

(a)
x + y
(b)
y − x
(c)
xy
(d)
(e)

A student runs 100 m in 12.8 seconds. Rounding answers to 3 decimal places where necessary, find the possible speeds of the student in m/sec if:

(a)

the distance is to the nearest metre and the time is to the nearest 0.1 s,

to
(b)

the distance is to the nearest cm and the time is to the nearest 0.1 s.

to
(a)

The mass of a plant on kitchen scales is 2.474 kg. What is the possible mass of the plant?

≤ mass <
(b)

The mass of another plant on scientific scales is 1.6280 kg. What are the upper and lower bounds of the mass of the plant?

,

Twenty computer monitors are packed in a single container. Each monitor weighs 12.6 kg, to the nearest 0.1 kg.

(a)

Calculate the lower boundary for the total weight of the monitors.

kg
(b)

Calculate the difference between the upper and lower boundaries for the total weight of the monitors.

kg

Jafar has a piece of wood that has a length of 30 cm, correct to the nearest centimetre.

(a)

Write down the minimum length of the piece of wood.

cm

Fatima has a different piece of wood that has a length of 18.4 cm, correct to the nearest millimetre.

(b)

Write down the maximum and minimum lengths between which the length of the piece of wood must lie.

≤ length <

A full jar of coffee weighs 750 g. The empty jar weighs 545 g. Both weights are accurate to the nearest 5 g.

Calculate, to the nearest whole gram, the maximum and minimum possible values of the weight of coffee in the jar.

g, g

Four shelf units are to be fitted along a library wall.

The wall is 9 m long.

All measurements are to the nearest cm.

What is the maximum length of the fourth shelf unit?

m cm