When you add or subtract rounded measurements, each one has an interval, and the answer has an interval too. For a sum, add the lower bounds and add the upper bounds. For a difference, pair the lower bound of the first with the upper bound of the second. This comes up in perimeter, total mass, distance between two points and time gap questions.
It needs the intervals from recovering an interval from a rounded measurement, so build those first.
Why does a sum use like with like?
The smallest possible total happens when both parts are at their smallest. The largest total happens when both are at their largest.
So if a has bounds aL and aU, and b has bounds bL and bU:
- lower bound of a + b = aL + bL
- upper bound of a + b = aU + bU
Why does a difference cross over?
To make a − b as small as possible, take a as small as it can be and subtract as much as possible. To make it as large as possible, take a as large as it can be and subtract as little as possible.
- lower bound of a − b = aL − bU
- upper bound of a − b = aU − bL
Worked example
Two lengths are A = 8.4 cm and B = 5.7 cm, each to 1 decimal place. Find the bounds of A + B and A − B.
Step 1, intervals: 8.35 ≤ A < 8.45 and 5.65 ≤ B < 5.75.
Step 2, sum: lower bound = 8.35 + 5.65 = 14.00. Upper bound = 8.45 + 5.75 = 14.20. So 14.00 ≤ A + B < 14.20 cm.
Step 3, difference: lower bound = 8.35 − 5.75 = 2.60. Upper bound = 8.45 − 5.65 = 2.80. So 2.60 ≤ A − B < 2.80 cm.
Check the nominal value: 8.4 − 5.7 = 2.7, which sits in the middle of 2.60 and 2.80. That is what you should expect from a correct interval.
The mistake to watch for
A common slip is pairing like with like in a subtraction.
Mistaken working: lower bound of A − B = 8.35 − 5.65 = 2.70.
The student subtracted the two lower bounds. That value is just the nominal answer, so it is not a lower bound at all.
The correct pairing subtracts the largest possible B: 8.35 − 5.75 = 2.60. A quick test is to ask “could the answer be smaller than my lower bound?” If yes, you have not used the extreme values.
Write the interval for each quantity first, in a small table, and label which value you use in each calculation.
Check yourself
Try these, then open each answer.
1. Two planks are 120 cm and 85 cm, each to the nearest centimetre. Find the bounds of their total length.
Show answer
Intervals: 119.5 ≤ first < 120.5 and 84.5 ≤ second < 85.5. Lower bound 119.5 + 84.5 = 204. Upper bound 120.5 + 85.5 = 206.
204 ≤ total < 206 cm
2. Using the same planks, find the bounds of the difference in their lengths (120 − 85).
Show answer
Lower bound 119.5 − 85.5 = 34. Upper bound 120.5 − 84.5 = 36.
34 ≤ difference < 36 cm
3. Two journeys take 2.4 h and 1.9 h, each to 1 decimal place. Find the bounds of how much longer the first journey is.
Show answer
Intervals: 2.35 ≤ first < 2.45 and 1.85 ≤ second < 1.95. Lower bound 2.35 − 1.95 = 0.40. Upper bound 2.45 − 1.85 = 0.60.
0.40 ≤ difference < 0.60 h
Where this leads next
Products and quotients use a different rule, which you will meet in bound a product with positive measurements. The bounds and rounding explainer shows each pairing so you can compare with your own working.
Students who understand the rule still sometimes swap the pairing under exam pressure. A teacher in online one-to-one Mathematics tuition can practise that step with you until it is automatic.