To find the upper bound of A minus B, which bound do you use for B?

The lower bound of B. A minus B is biggest when B is as small as possible, because what you subtract works against the result. So A takes its upper bound but B takes its lower bound. For A = 50 and B = 20 to the nearest 10, the upper bound of A minus B is 55 minus 15 = 40, not the all-upper 55 minus 25 = 30.

How do you find the upper bound of a division like a over b?

Divide by the lower bound of b. Dividing by a smaller number gives a bigger result, so the divisor works against you, just like a subtracted quantity. For a = 4.72 and b = 158 to 3 significant figures, the upper bound of a over b is 4.725 over 157.5 = 0.03, not the all-upper 4.725 over 158.5 = 0.0298.

Is a value recorded as 8 after truncation the same interval as after rounding?

No. A value rounded to 8 to the nearest whole number lies in 7.5 to just below 8.5, half a unit each side. A value truncated to 8, that is rounded down, lies in 8 to just below 9, because everything from 8 up to just below 9 truncates down to 8. Truncation does not use the half-unit rule, and the stated figure is the lower bound.

GCSE Maths Higher · AQA · Bounds & error intervals

Bounds of a calculation: subtraction and division flip the bound

To find the upper bound of a calculation, the instinct is to push every quantity to its upper bound. That works for adding and multiplying, but it fails for subtraction and division. $A - B$ is biggest when $B$ is smallest, and $A / B$ is biggest when $B$ is smallest. A quantity you subtract or divide by works against the result, so it takes the opposite bound.

The all-upper habit is one of the most reliable grade 7 to 9 mark-losers in bounds questions, because it gives an answer that looks plausible but is on the wrong side. The fix is to ask, term by term, which way each quantity pushes the answer, then pick the bound that drives the result the way you want.

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How to spot it in your own work

You took the upper bound of every quantity to find an upper bound, even the one being subtracted.
For $A - B$ you used the upper bound of $B$ , when a smaller $B$ would have made the result bigger.
For $A / B$ you divided by the upper bound of $B$ , when dividing by the lower bound gives a larger answer.
You applied the half-unit rounding interval to a value that was truncated.

An exam question that triggers it

Here is the structure of JUN23 Paper 2 Question 24:

$a = 65$ to the nearest integer, and $b = 30$ to 1 significant figure.

Work out the upper bound of $2a^2 - b^2$ .

The misconception pushes everything up, $2(65.5)^2 - 35^2 = 8580.5 - 1225 = 7355.5$ . The fix takes $a$ at its upper bound but the subtracted $b$ at its lower bound: $2(65.5)^2 - 25^2 = 8580.5 - 625 = 7955.5$ .

Why students fall for this

The first bounds questions students meet are sums and products, where every quantity really does take the same bound for an upper bound. The rule "upper bound means take all the upper bounds" gets overlearned before subtraction and division ever appear.

The flip is counter-intuitive. It feels as though a bigger upper bound on every input should give a bigger answer, when in fact a quantity that is subtracted or divided by pulls the answer the other way, so its bigger value makes the result smaller.

Truncation hides in the same family. A value recorded as 8 is assumed to sit in the middle of a half-unit interval, when a truncated value sits at the bottom of its interval instead, so the wrong endpoints get used as bounds.

The fix: Identify the operation, flip the subtracted or divided-by term

Step 1: identify the operation on each quantity. Is it added to, multiplied by, subtracted, or divided by?

Step 2: for + and x, push to the same bound. For an upper bound take all upper bounds; for a lower bound take all lower bounds.

Step 3: for - and /, flip the second quantity. A subtracted or divided-by quantity works against the result, so it takes the opposite bound. $A - B$ upper uses $B$ lower; $A / B$ upper uses $B$ lower.

Step 4: sense-check the direction. An upper bound must be the largest the calculation can reach. If swapping a bound would make it bigger, you chose the wrong one.

Worked example

JUN23·2·24 structure: $a = 65$ to the nearest integer, $b = 30$ to 1 s.f. Find the upper bound of $2a^2 - b^2$ .

Intervals. $a \in [64.5, 65.5)$ and $b \in [25, 35)$ .
Flip the subtracted term. Take $a$ high but $b$ low, since $b^2$ is subtracted.
$2(65.5)^2 - 25^2 = 8580.5 - 625 = 7955.5$
Trap: all upper, $8580.5 - 35^2 = 7355.5$ .

NOV21·3·28 structure: $a = 4.72$ and $b = 158$ , both to 3 s.f. Find the upper bound of $a / b$ .

Intervals. $a \in [4.715, 4.725)$ and $b \in [157.5, 158.5)$ .
Divide by the lower bound of b. $4.725 / 157.5 = 0.03$ . Trap: divide by the upper b, $4.725 / 158.5 = 0.0298$ .

JUN19·2·17 structure: $m = (p - 2b)/2$ , $p = 68.3$ and $b = 8.7$ , both to 1 d.p. Find the lower bound of $m$ .

Minimise the numerator. Take $p$ low but $b$ high, since $2b$ is subtracted. $p \in [68.25, 68.35)$ , $b \in [8.65, 8.75)$ .
Compute.
$m = \frac{68.25 - 2 \times 8.75}{2} = \frac{50.75}{2} = 25.375$
Trap: all lower, $(68.25 - 2 \times 8.65)/2 = 25.475$ .

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Common questions

For the upper bound of $A - B$ , which bound of B do you use?: The lower bound. $A - B$ is biggest when $B$ is smallest, because what you subtract works against the result. So $A$ takes its upper bound and $B$ takes its lower bound.
How do you find the upper bound of $a / b$ ?: Divide by the lower bound of $b$ . Dividing by a smaller number gives a bigger result, so $4.725 / 157.5 = 0.03$ , not the all-upper $4.725 / 158.5 = 0.0298$ .
Is a truncated 8 the same interval as a rounded 8?: No. Rounded, $7.5 \le v < 8.5$ , half a unit each side. Truncated, $8 \le w < 9$ , because everything from 8 up to just below 9 chops down to 8. Truncation does not use the half-unit rule.

Related misconceptions

Worst-case bound in a decisionTo prove an at-least or safely-under-the-limit claim, test the single worst case using the bounds: all lower bounds to prove at least, all upper bounds to prove it stays under a limit.
Error intervals and boundsWriting the half-unit error interval for a rounded value, the foundation skill that this calculation-bounds work builds on.

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