A rounded measurement stands for a whole range of true values. To recover the interval, work out the rounding unit, halve it, then subtract and add that half-unit. This appears in bounds questions on length, mass, time and capacity.
It builds on choosing sensible significant figures and prepares you for the calculation lessons in precision bounds and measurement.
What does a rounded value really mean?
Suppose a length is 7.3 cm to 1 decimal place. The true length could be 7.26, 7.3 or 7.34, because all of these round to 7.3.
The smallest true length that still rounds to 7.3 is 7.25. The largest values approach 7.35, but 7.35 itself rounds up to 7.4.
How do you find the interval?
- Find the rounding unit. To the nearest 0.1 the unit is 0.1. To the nearest 10 it is 10. To 2 significant figures, look at where the second figure sits.
- Halve the unit. The half-unit is the largest gap between the true value and the stated value.
- Subtract it for the lower bound and add it for the upper bound.
- Write the interval as lower bound ≤ x < upper bound, with units.
Worked example
A plank is measured as 140 cm to the nearest 5 cm. Find the interval for its true length, h.
Step 1, unit: the rounding unit is 5 cm.
Step 2, half-unit: 5 ÷ 2 = 2.5 cm.
Step 3, bounds: lower bound = 140 − 2.5 = 137.5 cm. Upper bound = 140 + 2.5 = 142.5 cm.
Step 4, interval: 137.5 ≤ h < 142.5 cm.
The same method handles significant figures. For 4500 to 2 significant figures, the second figure is the 5 in the hundreds place, so the unit is 100 and the half-unit is 50. The interval is 4450 ≤ x < 4550.
The mistake to watch for
A common slip is to add and subtract the whole unit instead of half of it.
Mistaken answer: 7.3 to 1 decimal place gives 7.2 ≤ L < 7.4.
The student used 0.1 instead of 0.05, so the interval is twice as wide as it should be.
The correct interval is 7.25 ≤ L < 7.35. A quick check is to test the ends: 7.2 would round to 7.2, not 7.3, so it cannot belong to the interval for 7.3.
Another slip is writing the upper bound as 7.34999 or as 7.349. Give 7.35 as the upper bound and write the strict inequality.
Check yourself
Try these, then open each answer.
1. A mass is 12.8 kg to 1 decimal place. Write the interval for the true mass, m.
Show answer
The unit is 0.1, so the half-unit is 0.05. Lower bound 12.8 − 0.05 = 12.75. Upper bound 12.8 + 0.05 = 12.85.
12.75 ≤ m < 12.85 kg
2. A crowd is 650 to the nearest 10. Write the interval for the true number, n.
Show answer
The unit is 10, so the half-unit is 5. Lower bound 645. Upper bound 655.
645 ≤ n < 655
3. A value is 0.046 to 2 significant figures. Write the interval for x.
Show answer
The second figure, 6, is in the thousandths place, so the unit is 0.001 and the half-unit is 0.0005. Lower bound 0.0455. Upper bound 0.0465.
0.0455 ≤ x < 0.0465
Where this leads next
Once you can build an interval, combine two of them in bound a sum and a difference. The bounds and rounding explainer draws the interval for any rounded value and lets you check your own.
Some students can follow the half-unit rule but mix up the two ends or the inequality signs. A teacher in online one-to-one Mathematics tuition can go through your written working and show where that happens.