Step by step
The question
Question: the length is 8 cm, to the nearest cm.
Find the upper and lower bounds.
Length: 8 cm, to the nearest cm
The real length was rounded.
We want the smallest and the biggest it could have been.
1 Find the step
It was rounded to the nearest cm.
So the step is 1 cm.
step = 1 cm
upper bound = the biggest it could be
lower bound = the smallest it could be
2 Half a step
Anything less than half a step away rounds to 8.
Half of 1 is 0.5.
half a step = 1 ÷ 2 = 0.5 cm
3 Find the bounds
The lower bound is 8 minus 0.5: 7.5.
lower bound = 8 − 0.5 = 7.5 cm
Careful!
We go only half a step down and half a step up.
7 would round to 7, not to 8.
Careful: not 7 to 9 ✗
The upper bound is 8 plus 0.5: 8.5.
upper bound = 8 + 0.5 = 8.5 cm
4 Show it on a number line
So 7.5 ≤ length < 8.5.
This is called the error interval.
7.5 ≤ length < 8.5
5 Check
Let's check.
7.5, rounded to the nearest cm, is 8.
7.5 rounds to 8
And 8.49, just under the upper bound, also rounds to 8.
So the bounds are right!
8.49 rounds to 8
Let's look back.
First: find the step.
Then: half a step.
Then: find the bounds.
Then: show it on a number line.
Last: check.
Answer
7.5 ≤ length < 8.5