### GCSE Maths

Numbers
Algebra
Geometry and Measures
Probability
Statistics

# Greatest and Least Possible Values

When we measure a length of an object, such as 4 cm (correct to the nearest centimetre), the actual length falls within a range. For 4 cm, this range is 4.0 ≤ 4 cm < 4.5, with 4.0 being the least possible value and 4.5 being the greatest possible value.

It’s important to remember

• > means greater than
• < means less than
• ≥ means greater than or equal to
• ≤ means less than or equal to

On a number line, the interval a < x < b represents all values of x between a and b, but not including a and b themselves.

The interval a ≤ x ≤ b represents all values of x between a and b, including a and b. On a number line, this is shown as:

It’s also important to remember to know the following intervals and how they look on a number line:

Let’s look at some examples.

## Examples

Example 1:

To 1 decimal place, the length of a swimming pool measures 15.4 m and the width measures 8.7 m. Calculate the perimeter of the swimming pool, giving the smallest and greatest possible value.

Length: 15.35 ≤ 15.4 < 15.45

Width: 8.65 ≤ 8.7 < 8.75

Perimeter = 2 × (length + width)

Smallest possible perimeter: 2 × (15.35 + 8.65) = 48.0 m

Greatest possible perimeter: 2 × (15.45 + 8.75) = 48.4 m

The smallest possible perimeter of the swimming pool is 48.0 m, and the greatest possible perimeter is 48.4 m.

Example 2:

Given that 2.50 ≤ x < 3.75 and 1.25 ≤ y < 2.50, find the maximum and minimum values of:

1. x/y

2. x – y

3. x² – y²

4. y/x

1. x/y

• min(x/y) = 2.50 / 2.50 = 1

• max(x/y) = 3.75 / 1.25 = 3

2. x – y

• min(x – y) = 2.50 – 2.50 = 0

• max(x – y) = 3.75 – 1.25 = 2.50

3. x² – y²

• min(x² – y²) = 2.50² – 2.50² = 0

• max(x² – y²) = 3.75² – 1.25² = 14.0625 – 1.5625 = 12.50

4. y/x

• min(y/x) = 1.25 / 3.75 = 1/3

• max(y/x) = 2.50 / 2.50 = 1

Example 3:

A rectangle has a length of 74 mm and a width of 47 mm, each correct to the nearest millimetre. Calculate the lower and upper bounds for:

a) the length

b) the width

c) the perimeter of the rectangle

d) the area of the rectangle

Length: 73.5 ≤ length < 74.5

Width: 46.5 ≤ width < 47.5

a) Lower bound for the length = 73.5 mm Upper bound for the length = 74.5 mm

b) Lower bound for the width = 46.5 mm Upper bound for the width = 47.5 mm

c) Lower bound for the perimeter: 2 × (73.5 + 46.5) = 240 mm Upper bound for the perimeter: 2 × (74.5 + 47.5) = 244 mm

d) Lower bound for the area: 73.5 × 46.5 = 3417.25 mm² Upper bound for the area: 74.5 × 47.5 = 3538.75 mm²

Example 4:

A square has a side length measured as 9.5 cm, correct to the nearest 0.1 cm. Calculate the least and greatest possible values of the square’s perimeter and area.

Side length: 9.45 cm < side length ≤ 9.55 cm

Least possible value of the perimeter:

4 × side length = 4 × 9.45 cm = 37.8 cm

Greatest possible value of the perimeter:

4 × side length = 4 × 9.55 cm = 38.2 cm

Least possible value of the area:

side length² = (9.45 cm)² = 89.3025 cm²

Greatest possible value of the area:

side length² = (9.55 cm)² = 91.2025 cm²

So, the square’s perimeter ranges from 37.8 cm to 38.2 cm, and its area ranges from 89.3025 cm² to 91.2025 cm².