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.

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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.

A straight horizontal line segment with two endpoints labelled 'a' and 'b'. An arrowhead on the right side indicates the direction from 'a' to 'b'.

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:

A straight horizontal line segment with two darkened endpoints labelled 'a' and 'b'. An arrowhead on the right side indicates the direction from 'a' to 'b'.

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

A series of horizontal line segments representing inequalities on a number line. The first line has an open circle at point 'a' indicating 'x > a' with the note '(a not included)'. The second line has a filled circle at point 'a' representing 'x ≥ a' with the note '(a included)'. The third line has an open circle at point 'a' and an arrow pointing left denoting 'x < a'. The fourth line features a filled circle at point 'a' and an arrow pointing right, illustrating 'x ≤ a'. The last line shows an open circle at point 'a' and a filled circle at point 'b' for the inequality 'a < x ≤ b' with a note stating '(Here, a is not included but b is included)'.

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.

Solution

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

Solution

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

Solution

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.

Solution

• 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.

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