Worked examples
The question
Question: A rectangle is 8 cm by 5 cm, both to the nearest cm.
Find the upper bound of its area.
1 Bounds of each number
The length is 8 cm, to the nearest cm.
Half a step is 0.5.
So it is from 7.5 up to 8.5.
7.5 ≤ length < 8.5
8 ± 0.5
The width is 5 cm, to the nearest cm.
Half a step is 0.5.
So it is from 4.5 up to 5.5.
4.5 ≤ width < 5.5
5 ± 0.5
2 Choose the bounds
To make the area as big as it can be, both numbers must be as big as they can be.
So we use 8.5 and 5.5.
upper bound = 8.5 × 5.5
biggest: biggest × biggest
Careful!
We do not change the answer by half a step.
We work it out again with the biggest numbers.
Careful: not 40 + 0.5 ✗
3 Work it out
That is 46.75 cm².
upper bound = 46.75 cm²
To multiply decimals: first 85 times 55, as whole numbers.
That is 4675.
Then count the decimal places: 2.
85 × 55 = 4675
2 decimal places: 46.75
So the upper bound of the area is 46.75 cm².
4 Check
Let's check.
With the rounded numbers we get 40.
with the rounded numbers: 8 × 5 = 40
The upper bound, 46.75, is more than that, as it should be.
So the answer is right!
46.75 is more than 40
Let's look back.
First: bounds of each number.
Then: choose the bounds.
Then: work it out.
Last: check.
Answer 2 decimal places: 46.75
▶ Watch it solved
The question
Question: A runner goes 100 m (to the nearest m) in 12 s (to the nearest s).
Find the lower bound of the speed.
1 Bounds of each number
The distance is 100 m, to the nearest m.
Half a step is 0.5.
So it is from 99.5 up to 100.5.
99.5 ≤ distance < 100.5
100 ± 0.5
The time is 12 s, to the nearest s.
Half a step is 0.5.
So it is from 11.5 up to 12.5.
11.5 ≤ time < 12.5
12 ± 0.5
2 Choose the bounds
Dividing by a bigger number gives a smaller answer.
So the smallest speed is the smallest distance divided by the biggest time.
So we use 99.5 and 12.5.
lower bound = 99.512.5
smallest: smallest ÷ biggest
Careful!
A smaller bottom makes the answer bigger.
So the bottom goes the other way.
Careful: not lower ÷ lower: 99.5 ÷ 11.5 ✗
3 Work it out
That is 7.96 m/s.
lower bound = 7.96 m/s
So the lower bound of the speed is 7.96 m/s.
4 Check
Let's check.
With the rounded numbers we get 8.33.
with the rounded numbers: 100 ÷ 12 ≈ 8.33
The lower bound, 7.96, is less than that, as it should be.
So the answer is right!
7.96 is less than 8.33
Let's look back.
First: bounds of each number.
Then: choose the bounds.
Then: work it out.
Last: check.
Answer lower bound = 7.96 m/s
▶ Watch it solved
The question
Question: x = 36 and y = 8, both to the nearest whole number.
Find the upper bound of x ÷ y.
1 Bounds of each number
The number x is 36, to the nearest whole number.
Half a step is 0.5.
So it is from 35.5 up to 36.5.
35.5 ≤ x < 36.5
36 ± 0.5
The number y is 8, to the nearest whole number.
Half a step is 0.5.
So it is from 7.5 up to 8.5.
7.5 ≤ y < 8.5
8 ± 0.5
2 Choose the bounds
Dividing by a bigger number gives a smaller answer.
So the biggest answer is the biggest x divided by the smallest y.
So we use 36.5 and 7.5.
upper bound = 36.57.5
biggest: biggest ÷ smallest
Careful!
A bigger bottom makes the answer smaller.
So the bottom goes the other way.
Careful: not upper ÷ upper: 36.5 ÷ 8.5 ✗
3 Work it out
With a calculator: about 4.87, to 3 significant figures.
upper bound ≈ 4.87
So the upper bound of the answer is 4.87.
4 Check
Let's check.
With the rounded numbers we get 4.5.
with the rounded numbers: 36 ÷ 8 = 4.5
The upper bound, 4.87, is more than that, as it should be.
So the answer is right!
4.87 is more than 4.5
Let's look back.
First: bounds of each number.
Then: choose the bounds.
Then: work it out.
Last: check.
Answer upper bound ≈ 4.87
▶ Watch it solved
