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This page sets out the method for bounds in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Bounds topic page.
When a value is 'correct to' or 'to the nearest' something, halve that unit and add or subtract it.
When you must find the bounds of a calculation, ask 'what makes the answer as big as possible?' for the upper bound.
A value to the nearest centimetre can be up to 0.5 cm below the reading.
The lower bound of 7 cm to the nearest cm is 6.5 cm.
The upper bound of 7 cm to the nearest cm is 7.5 cm.
For the greatest area, multiply the two upper bounds.
For the least area, multiply the two lower bounds.
Upper area is not upper length times lower width.
Find the size of the last unit the number was rounded to, then halve it. Nearest whole number: the unit is 1, so the half-unit is 0.5. One decimal place: the unit is 0.1, so the half-unit is 0.05. Nearest 10: the unit is 10, so the half-unit is 5. A length of 8 cm to the nearest centimetre runs from 7.5 cm to 8.5 cm. A mass of 80 g to the nearest 10 g runs from 75 g to 85 g.
Write the interval with a closed lower end and an open upper end. For 6.3 to 1 decimal place, 6.25 ≤ x < 6.35. The value 6.25 rounds to 6.3, so it is included. The value 6.35 rounds to 6.4, so it is not included. In a later calculation you still use 6.35 as the upper boundary.
Match the half-unit to the accuracy in the question. A number given as 6.30 to 2 decimal places has a half-unit of 0.005, not 0.05. The extra zero is information. 6.3 to 1 decimal place and 6.30 to 2 decimal places are different intervals.
Method: 1. Name the rounding unit, then halve it to get the error. 2. Subtract that half-unit for the lower bound and add it for the upper bound. 3. Write the interval with the lower bound included and the upper bound excluded.
Both numbers push a sum the same way. The greatest sum is upper plus upper. The least sum is lower plus lower. If A = 8.4 and B = 3.2, both to 1 decimal place, then A lies from 8.35 to 8.45 and B lies from 3.15 to 3.25. The upper sum is 8.45 + 3.25 = 11.7. The lower sum is 8.35 + 3.15 = 11.5.
A difference is crossed. To make A - B as large as possible, make A large and B small. The upper bound is 8.45 - 3.15 = 5.3. To make the difference as small as possible, make A small and B large. The lower bound is 8.35 - 3.25 = 5.1. Using upper minus upper gives 8.45 - 3.25 = 5.2, which is only the difference of the rounded values' upper ends. It is not the upper bound of the difference.
Say the sentence before you calculate. 'Largest minus smallest' for the upper difference. 'Smallest minus largest' for the lower difference. Label which measurement is being subtracted so the two bounds are not swapped.
Method: 1. For a sum, add the two upper bounds and add the two lower bounds. 2. For a difference, use upper minus lower, and lower minus upper. 3. Keep the bounds in the same order as the original calculation.
For positive measurements, the greatest product uses both upper bounds and the least product uses both lower bounds. A rectangle is 8 cm by 3 cm, both to the nearest centimetre. The sides run from 7.5 to 8.5 and from 2.5 to 3.5. Upper area: 8.5 × 3.5 = 29.75 cm^2. Lower area: 7.5 × 2.5 = 18.75 cm^2. The rounded area 8 × 3 = 24 sits between them. It is not a bound.
Division crosses, for the same reason as subtraction. Speed is distance divided by time. The greatest speed uses the greatest distance and the smallest time. The least speed uses the smallest distance and the greatest time. If the distance is 80 km to the nearest 10 km, it runs from 75 to 85. If the time is 4 hours to the nearest hour, it runs from 3.5 to 4.5. Upper speed: 85 ÷ 3.5 = 170/7 km/h. Lower speed: 75 ÷ 4.5 = 50/3 km/h.
Do not divide upper by upper. That pairs a long journey with a long time and gives a middling speed. Write the four bounds first, then pick. For a quotient P/Q of positive values, upper is upper P over lower Q, and lower is lower P over upper Q. Keep the result exact. 170/7 and 50/3 are exact. A decimal is only a later rounding of those bounds.
Method: 1. For a product of positive lengths, multiply upper by upper and lower by lower. 2. For a quotient, divide upper by lower and lower by upper. 3. Do not round the bounds before you multiply or divide.
Sometimes a question asks for the result to a suitable degree of accuracy. Work out the lower bound and the upper bound first. Then round each of them to 1 decimal place, to 2 decimal places, and to the nearest whole number. If both bounds round to the same figure, that figure is safe. If they round to different figures, that place is too fine.
Suppose a calculation has lower bound 4.162 and upper bound 4.168. To 1 decimal place, 4.162 rounds to 4.2 and 4.168 rounds to 4.2. They agree. To 2 decimal places, 4.162 rounds to 4.16 and 4.168 rounds to 4.17. They do not agree. A suitable degree of accuracy is 1 decimal place, and the value to quote is 4.2. Quoting 4.16 pretends the bounds are tighter than they are.
The speed bounds 50/3 ≈ 16.7 and 170/7 ≈ 24.3 do not agree to the nearest whole number. They do both round to 20 to the nearest 10. That is a wide statement, and it is what those measurements will support. If the bounds are far apart, give the interval rather than a tight decimal.
Method: 1. Work out the lower result and the upper result separately. 2. See which digits those two results share. 3. Give the answer only as far as that shared accuracy.
An error interval uses inequalities: lower bound ≤ value < upper bound. The bounds sit half a unit of accuracy either side of the stated value. A mass of 3.6 kg to 1 decimal place has a unit of 0.1, so half a unit is 0.05 and the interval is 3.55 ≤ m < 3.65. A population of 28 000 to the nearest thousand has half a unit of 500, so 27 500 ≤ p < 28 500. The lower limit uses ≤ and the upper limit uses <, because a value exactly on the upper bound would round up to the next number.
Other roundings need care. To the nearest 5 the half unit is 2.5, so 120 becomes 117.5 ≤ x < 122.5. To the nearest half-metre the half unit is 0.25. For significant figures, find the place value of the last figure: 4.2 to 2 significant figures has a unit of 0.1, so 4.15 ≤ x < 4.25, but 420 to 2 significant figures has a unit of 10, so 415 ≤ x < 425. Always change units first, for example grams to kilograms, before you find the bounds.
Method: 1. Find the unit of accuracy, such as nearest 10, 1 decimal place or 2 significant figures. 2. Halve the unit. 3. Subtract the half unit for the lower bound and add it for the upper bound. 4. Write lower ≤ x < upper, with the correct inequality signs.
Tip: The upper bound is written with a strict inequality, even though it is still called the upper bound.
Some questions ask whether a limit will definitely be respected, such as the maximum load of a lift or a van. The way to answer is to find the worst case. For a limit that must not be exceeded, find the upper bound of the total. If even the upper bound is below the limit, then the load is definitely safe. If the upper bound is above the limit, you cannot be sure, so the answer is no.
Work piece by piece. For n identical items with a stated mass to some accuracy, the upper bound of the total is n times the upper bound of one item, not the upper bound of a rounded total. Add the totals for the different groups, then compare with the limit. Finish with a sentence that uses the numbers, such as "No, because the total could be 456 kg, which is more than 450 kg". The last mark is for that reasoned conclusion.
Method: 1. Decide whether the limit is a maximum, so you need the upper bound of the total. 2. Find the upper bound of each item and multiply by the number of items. 3. Add the group totals. 4. Compare with the limit and write a sentence using the numbers.
In a formula, decide for each letter whether to use its upper or lower bound by asking whether a bigger value makes the result bigger or smaller. For (p + q)/(r - s), the numerator is largest when p and q are both at their upper bounds. The denominator is smallest when r is at its lower bound and s is at its upper bound, because taking away a larger s leaves less. So the upper bound of the whole fraction uses the upper bounds of p and q, the lower bound of r and the upper bound of s.
Powers and roots follow the same rule for positive numbers. The upper bound of a square comes from the upper bound of the number, and the upper bound of a square root comes from the upper bound of the number inside. For v = u + at with all positive values, every letter uses its upper bound to make the upper bound of v, and its lower bound to make the lower bound of v. Write the bounds of each letter in a small table first, so that the substitution is quick and accurate.
Method: 1. Write the lower and upper bound of every letter in a table. 2. For each letter, decide whether a bigger value makes the result bigger or smaller. 3. Substitute the matching bounds for the upper bound of the result, and the opposite ones for the lower bound. 4. Round only the final answer, to the accuracy asked for.
Questions that say "By considering bounds, work out the value to a suitable degree of accuracy" follow a plan. Work out the upper bound and the lower bound of the answer, keeping at least four significant figures. Then round both bounds to the same number of significant figures, starting with 1 and increasing it until the two rounded values stop agreeing. The answer is the rounding at the greatest number of significant figures where both bounds still agree.
For example, an upper bound of 6.014 and a lower bound of 5.986 both round to 6.0 to 2 significant figures. To 3 significant figures they are 6.01 and 5.99, which differ. So the speed is 6.0 to 2 significant figures. You must write the reason: "both bounds round to 6.0 to 2 significant figures". Without that sentence the last mark is lost. Never quote more figures than the bounds support.
Method: 1. Work out the upper bound and the lower bound, keeping at least four figures. 2. Round both to 1, 2, 3 significant figures in turn. 3. Find the greatest number of significant figures where the two roundings are equal. 4. Write the answer with that accuracy and a sentence giving the reason.
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