Step by step
The question
Question: the time is 45 s, to the nearest s.
Find the upper and lower bounds.
Time: 45 s, to the nearest s
The real time was rounded.
We want the smallest and the biggest it could have been.
1 Find the step
It was rounded to the nearest s.
So the step is 1 s.
step = 1 s
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 45.
Half of 1 is 0.5.
half a step = 1 ÷ 2 = 0.5 s
3 Find the bounds
The lower bound is 45 minus 0.5: 44.5.
lower bound = 45 − 0.5 = 44.5 s
Careful!
We go only half a step down and half a step up.
44 would round to 44, not to 45.
Careful: not 44 to 46 ✗
The upper bound is 45 plus 0.5: 45.5.
upper bound = 45 + 0.5 = 45.5 s
4 Show it on a number line
So 44.5 ≤ time < 45.5.
This is called the error interval.
44.5 ≤ time < 45.5
5 Check
Let's check.
44.5, rounded to the nearest s, is 45.
44.5 rounds to 45
And 45.49, just under the upper bound, also rounds to 45.
So the bounds are right!
45.49 rounds to 45
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
44.5 ≤ time < 45.5