GCSE Maths: Upper and lower bounds in calculations (Higher)
Part of the GCSE Maths decimals, rounding and estimation chapter (Higher). Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Upper and lower bounds in calculations
In section 4.8 you found the lower bound and the upper bound of a single rounded measurement. In real problems we often calculate with measurements, for example finding an area from a length and a width. If both are rounded, the answer also has a range of possible values. The upper bound of the answer is the biggest value it could be (the limit at the top), and the lower bound is the smallest.
The key idea is to choose the bounds that make the answer as big (or as small) as possible. It is not always the upper bound of everything, because some measurements work against the answer. To make a − b as big as possible, take the biggest a and subtract the smallest b. To make a ÷ b as big as possible, divide the biggest a by the smallest b (dividing by a small number gives a big answer).
| Calculation | Upper bound of the answer | Lower bound of the answer |
|---|---|---|
| a + b | UB of a + UB of b | LB of a + LB of b |
| a − b | UB of a − LB of b | LB of a − UB of b |
| a × b | UB of a × UB of b | LB of a × LB of b |
| a ÷ b | UB of a ÷ LB of b | LB of a ÷ UB of b |
(UB means upper bound and LB means lower bound.) Always check the numbers by asking: does this combination really give the biggest (or smallest) answer?
Answering to an appropriate degree of accuracy
Once you have the lower bound and upper bound of the answer, you can say how accurately the answer is known. Round both bounds to 1, 2, 3… significant figures. Find the most accurate version on which they agree. For the area above, the bounds 105.5925 and 107.6925 both round to 110 to 2 significant figures, but to 3 significant figures they are 106 and 108, which are different. So we can only say the area is 110 cm² to 2 s.f.
Exact values and truncation
Some values are exact, for example “5 identical bags”. An exact value has no error, so its lower bound and upper bound are both the value itself. If a value is truncated, the lower bound is the value shown and the upper bound is one whole unit more.
Key ideas
- Sum or product: upper bound with upper bounds, lower bound with lower bounds.
- Difference a − b: UB = UBa − LBb, LB = LBa − UBb.
- Quotient a ÷ b: UB = UBa ÷ LBb, LB = LBa ÷ UBb.
- Exact values have no error: use them as they are.
- To give an answer to an appropriate accuracy, round the bounds to different numbers of significant figures and find the most accurate level on which they agree.
Key words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.
Method. Find the bounds of every measurement, then choose the combination that makes the area as big (upper bound) or as small (lower bound) as possible.
- Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.length = 12.4 cm (to 1 decimal place): unit = 0.1, half a unit = 0.05
lower bound = 12.4 − 0.05 = 12.35, upper bound = 12.4 + 0.05 = 12.45
width = 8.6 cm (to 1 decimal place): unit = 0.1, half a unit = 0.05
lower bound = 8.6 − 0.05 = 8.55, upper bound = 8.6 + 0.05 = 8.65 - Decide which bounds to use. To get the biggest possible area, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest area, do the opposite.length: a bigger length makes the area bigger, so for the upper bound of the area use the upper bound of length, and for the lower bound of the area use the lower bound.
width: a bigger width makes the area bigger, so for the upper bound of the area use the upper bound of width, and for the lower bound of the area use the lower bound. - Calculate the upper bound of the area. Put the bounds you chose into the calculation. Use a calculator if the numbers are not easy.12.45 × 8.65 = 107.6925 cm²
- Calculate the lower bound of the area. Put the other bounds into the calculation.12.35 × 8.55 = 105.5925 cm²
- Compare the two bounds rounded to 1, 2, 3… significant figures. If the lowest and highest possible values both round to the same number, then we can be sure of that number to that accuracy.to 1 s.f.: lower bound 100, upper bound 100 (the same)
to 2 s.f.: lower bound 110, upper bound 110 (the same)
to 3 s.f.: lower bound 106, upper bound 108 (different) - Write the answer to the most accurate level on which they agree. Both bounds round to 110 to 2 s.f., but to 3 s.f. they are different. So 110 is the answer we can be certain of.110 cm² (to 2 s.f.)
Check. Check the bounds make sense: the area found from the original values is 106.64 cm², which lies between the lower bound and the upper bound. ✓
Watch out. Use the upper bound of every measurement for the upper bound of the answer, and the lower bound of every measurement for the lower bound. Do not mix them in a sum or product.
Key words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.
Method. Find the bounds of every measurement, then choose the combination that makes the difference as big (upper bound) or as small (lower bound) as possible.
- Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.a = 8.4 (to 1 decimal place): unit = 0.1, half a unit = 0.05
lower bound = 8.4 − 0.05 = 8.35, upper bound = 8.4 + 0.05 = 8.45
b = 3.7 (to 1 decimal place): unit = 0.1, half a unit = 0.05
lower bound = 3.7 − 0.05 = 3.65, upper bound = 3.7 + 0.05 = 3.75 - Decide which bounds to use. To get the biggest possible difference, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest difference, do the opposite.a: a bigger a makes the difference bigger, so for the upper bound of the difference use the upper bound of a, and for the lower bound of the difference use the lower bound.
b: a bigger b makes the difference smaller, so for the upper bound of the difference use the lower bound of b, and for the lower bound of the difference use the upper bound. - Calculate the upper bound of the difference. Put the bounds you chose into the calculation. Use a calculator if the numbers are not easy.8.45 − 3.65 = 4.8
- Calculate the lower bound of the difference. Put the other bounds into the calculation.8.35 − 3.75 = 4.6
Check. Check the bounds make sense: the difference found from the original values is 4.7, which lies between the lower bound and the upper bound. ✓
Watch out. For a subtraction or a division, the bound you need is not the same one each time: to make a − b biggest you need the upper bound of a but the lower bound of b. The same is true for dividing by b.
Key words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.
Method. Find the bounds of every measurement, then choose the combination that makes the speed as big (upper bound) or as small (lower bound) as possible.
- Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.distance = 150 km (to the nearest whole number): unit = 1, half a unit = 0.5
lower bound = 150 − 0.5 = 149.5, upper bound = 150 + 0.5 = 150.5
time = 2.4 hours (to 1 decimal place): unit = 0.1, half a unit = 0.05
lower bound = 2.4 − 0.05 = 2.35, upper bound = 2.4 + 0.05 = 2.45 - Decide which bounds to use. To get the biggest possible speed, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest speed, do the opposite.distance: a bigger distance makes the speed bigger, so for the upper bound of the speed use the upper bound of distance, and for the lower bound of the speed use the lower bound.
time: a bigger time makes the speed smaller, so for the upper bound of the speed use the lower bound of time, and for the lower bound of the speed use the upper bound. - Calculate the upper bound of the speed. Put the bounds you chose into the calculation. Use a calculator if the numbers are not easy.150.5 ÷ 2.35 = 64.04 km/h (rounded)
- Calculate the lower bound of the speed. Put the other bounds into the calculation.149.5 ÷ 2.45 = 61.02 km/h (rounded)
- Compare the two bounds rounded to 1, 2, 3… significant figures. If the lowest and highest possible values both round to the same number, then we can be sure of that number to that accuracy.to 1 s.f.: lower bound 60, upper bound 60 (the same)
to 2 s.f.: lower bound 61, upper bound 64 (different) - Write the answer to the most accurate level on which they agree. Both bounds round to 60 to 1 s.f., but to 2 s.f. they are different. So 60 is the answer we can be certain of.60 km/h (to 1 s.f.)
Check. Check the bounds make sense: the speed found from the original values is 62.5 km/h, which lies between the lower bound and the upper bound. ✓
Watch out. For a subtraction or a division, the bound you need is not the same one each time: to make a − b biggest you need the upper bound of a but the lower bound of b. The same is true for dividing by b.
Key words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.
Method. Find the bounds of every measurement, then choose the combination that makes the perimeter as big (upper bound) or as small (lower bound) as possible.
- Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.length = 7.6 cm (truncated to 1 decimal place): unit = 0.1
lower bound = 7.6 (truncating never rounds up, so the value shown is the smallest possible), upper bound = 7.6 + 0.1 = 7.7
width = 5 cm is an exact value, so it has no error: lower bound = upper bound = 5 - Decide which bounds to use. To get the biggest possible perimeter, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest perimeter, do the opposite.length: a bigger length makes the perimeter bigger, so for the upper bound of the perimeter use the upper bound of length, and for the lower bound of the perimeter use the lower bound.
- Calculate the upper bound of the perimeter. Put the bounds you chose into the calculation. Use a calculator if the numbers are not easy.2 × (7.7 + 5) = 25.4 cm
- Calculate the lower bound of the perimeter. Put the other bounds into the calculation.2 × (7.6 + 5) = 25.2 cm
Check. Check the bounds make sense: the perimeter found from the original values is 25.2 cm, which lies between the lower bound and the upper bound. ✓
Watch out. Use the upper bound of every measurement for the upper bound of the answer, and the lower bound of every measurement for the lower bound. Do not mix them in a sum or product.
Watch out
For subtraction and division you do not use the upper bounds of both numbers. For the upper bound of a − b you need the lower bound of b. Check each time: which choice makes the answer biggest?Exercise 4J Bounds in calculations
Calculator allowedKey words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.
Method. Find the bounds of every measurement, then choose the combination that makes the sum as big (upper bound) or as small (lower bound) as possible.
- Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.a = 5.6 (to 1 decimal place): unit = 0.1, half a unit = 0.05
lower bound = 5.6 − 0.05 = 5.55, upper bound = 5.6 + 0.05 = 5.65
b = 3.2 (to 1 decimal place): unit = 0.1, half a unit = 0.05
lower bound = 3.2 − 0.05 = 3.15, upper bound = 3.2 + 0.05 = 3.25 - Decide which bounds to use. To get the biggest possible sum, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest sum, do the opposite.a: a bigger a makes the sum bigger, so for the upper bound of the sum use the upper bound of a, and for the lower bound of the sum use the lower bound.
b: a bigger b makes the sum bigger, so for the upper bound of the sum use the upper bound of b, and for the lower bound of the sum use the lower bound. - Calculate the upper bound of the sum. Put the bounds you chose into the calculation. Use a calculator if the numbers are not easy.5.65 + 3.25 = 8.9
Check. Check the bounds make sense: the sum found from the original values is 8.8, which lies between the lower bound and the upper bound. ✓
Watch out. Use the upper bound of every measurement for the upper bound of the answer, and the lower bound of every measurement for the lower bound. Do not mix them in a sum or product.
Key words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.
Method. Find the bounds of every measurement, then choose the combination that makes the difference as big (upper bound) or as small (lower bound) as possible.
- Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.a = 9.5 (to 1 decimal place): unit = 0.1, half a unit = 0.05
lower bound = 9.5 − 0.05 = 9.45, upper bound = 9.5 + 0.05 = 9.55
b = 4.2 (to 1 decimal place): unit = 0.1, half a unit = 0.05
lower bound = 4.2 − 0.05 = 4.15, upper bound = 4.2 + 0.05 = 4.25 - Decide which bounds to use. To get the biggest possible difference, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest difference, do the opposite.a: a bigger a makes the difference bigger, so for the upper bound of the difference use the upper bound of a, and for the lower bound of the difference use the lower bound.
b: a bigger b makes the difference smaller, so for the upper bound of the difference use the lower bound of b, and for the lower bound of the difference use the upper bound. - Calculate the lower bound of the difference. Put the other bounds into the calculation.9.45 − 4.25 = 5.2
Check. Check the bounds make sense: the difference found from the original values is 5.3, which lies between the lower bound and the upper bound. ✓
Watch out. For a subtraction or a division, the bound you need is not the same one each time: to make a − b biggest you need the upper bound of a but the lower bound of b. The same is true for dividing by b.
Key words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.
Method. Find the bounds of every measurement, then choose the combination that makes the product as big (upper bound) or as small (lower bound) as possible.
- Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.a = 3.5 (to 1 decimal place): unit = 0.1, half a unit = 0.05
lower bound = 3.5 − 0.05 = 3.45, upper bound = 3.5 + 0.05 = 3.55
b = 2.4 (to 1 decimal place): unit = 0.1, half a unit = 0.05
lower bound = 2.4 − 0.05 = 2.35, upper bound = 2.4 + 0.05 = 2.45 - Decide which bounds to use. To get the biggest possible product, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest product, do the opposite.a: a bigger a makes the product bigger, so for the upper bound of the product use the upper bound of a, and for the lower bound of the product use the lower bound.
b: a bigger b makes the product bigger, so for the upper bound of the product use the upper bound of b, and for the lower bound of the product use the lower bound. - Calculate the upper bound of the product. Put the bounds you chose into the calculation. Use a calculator if the numbers are not easy.3.55 × 2.45 = 8.6975
Check. Check the bounds make sense: the product found from the original values is 8.4, which lies between the lower bound and the upper bound. ✓
Watch out. Use the upper bound of every measurement for the upper bound of the answer, and the lower bound of every measurement for the lower bound. Do not mix them in a sum or product.
Key words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.
Method. Find the bounds of every measurement, then choose the combination that makes the quotient as big (upper bound) or as small (lower bound) as possible.
- Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.a = 13 (to the nearest whole number): unit = 1, half a unit = 0.5
lower bound = 13 − 0.5 = 12.5, upper bound = 13 + 0.5 = 13.5
b = 3 (to the nearest whole number): unit = 1, half a unit = 0.5
lower bound = 3 − 0.5 = 2.5, upper bound = 3 + 0.5 = 3.5 - Decide which bounds to use. To get the biggest possible quotient, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest quotient, do the opposite.a: a bigger a makes the quotient bigger, so for the upper bound of the quotient use the upper bound of a, and for the lower bound of the quotient use the lower bound.
b: a bigger b makes the quotient smaller, so for the upper bound of the quotient use the lower bound of b, and for the lower bound of the quotient use the upper bound. - Calculate the upper bound of the quotient. Put the bounds you chose into the calculation. Use a calculator if the numbers are not easy.13.5 ÷ 2.5 = 5.4
Check. Check the bounds make sense: the quotient found from the original values is 4.333, which lies between the lower bound and the upper bound. ✓
Watch out. For a subtraction or a division, the bound you need is not the same one each time: to make a − b biggest you need the upper bound of a but the lower bound of b. The same is true for dividing by b.
Key words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.
Method. Find the bounds of every measurement, then choose the combination that makes the area as big (upper bound) or as small (lower bound) as possible.
- Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.length = 7.2 cm (to 1 decimal place): unit = 0.1, half a unit = 0.05
lower bound = 7.2 − 0.05 = 7.15, upper bound = 7.2 + 0.05 = 7.25
width = 4.5 cm (to 1 decimal place): unit = 0.1, half a unit = 0.05
lower bound = 4.5 − 0.05 = 4.45, upper bound = 4.5 + 0.05 = 4.55 - Decide which bounds to use. To get the biggest possible area, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest area, do the opposite.length: a bigger length makes the area bigger, so for the upper bound of the area use the upper bound of length, and for the lower bound of the area use the lower bound.
width: a bigger width makes the area bigger, so for the upper bound of the area use the upper bound of width, and for the lower bound of the area use the lower bound. - Calculate the lower bound of the area. Put the other bounds into the calculation.7.15 × 4.45 = 31.8175 cm²
Check. Check the bounds make sense: the area found from the original values is 32.4 cm², which lies between the lower bound and the upper bound. ✓
Watch out. Use the upper bound of every measurement for the upper bound of the answer, and the lower bound of every measurement for the lower bound. Do not mix them in a sum or product.
Key words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.
Method. Find the bounds of every measurement, then choose the combination that makes the perimeter as big (upper bound) or as small (lower bound) as possible.
- Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.length = 7.2 cm (to 1 decimal place): unit = 0.1, half a unit = 0.05
lower bound = 7.2 − 0.05 = 7.15, upper bound = 7.2 + 0.05 = 7.25
width = 4.5 cm (to 1 decimal place): unit = 0.1, half a unit = 0.05
lower bound = 4.5 − 0.05 = 4.45, upper bound = 4.5 + 0.05 = 4.55 - Decide which bounds to use. To get the biggest possible perimeter, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest perimeter, do the opposite.length: a bigger length makes the perimeter bigger, so for the upper bound of the perimeter use the upper bound of length, and for the lower bound of the perimeter use the lower bound.
width: a bigger width makes the perimeter bigger, so for the upper bound of the perimeter use the upper bound of width, and for the lower bound of the perimeter use the lower bound. - Calculate the upper bound of the perimeter. Put the bounds you chose into the calculation. Use a calculator if the numbers are not easy.2 × (7.25 + 4.55) = 23.6 cm
Check. Check the bounds make sense: the perimeter found from the original values is 23.4 cm, which lies between the lower bound and the upper bound. ✓
Watch out. Use the upper bound of every measurement for the upper bound of the answer, and the lower bound of every measurement for the lower bound. Do not mix them in a sum or product.
Key words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.
Method. Find the bounds of every measurement, then choose the combination that makes the total mass as big (upper bound) or as small (lower bound) as possible.
- Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.mass of box A = 40 g (to the nearest ten): unit = 10, half a unit = 5
lower bound = 40 − 5 = 35, upper bound = 40 + 5 = 45
mass of box B = 60 g (to the nearest ten): unit = 10, half a unit = 5
lower bound = 60 − 5 = 55, upper bound = 60 + 5 = 65 - Decide which bounds to use. To get the biggest possible total mass, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest total mass, do the opposite.mass of box A: a bigger mass of box A makes the total mass bigger, so for the upper bound of the total mass use the upper bound of mass of box A, and for the lower bound of the total mass use the lower bound.
mass of box B: a bigger mass of box B makes the total mass bigger, so for the upper bound of the total mass use the upper bound of mass of box B, and for the lower bound of the total mass use the lower bound. - Calculate the upper bound of the total mass. Put the bounds you chose into the calculation. Use a calculator if the numbers are not easy.45 + 65 = 110 g
- Calculate the lower bound of the total mass. Put the other bounds into the calculation.35 + 55 = 90 g
Check. Check the bounds make sense: the total mass found from the original values is 100 g, which lies between the lower bound and the upper bound. ✓
Watch out. Use the upper bound of every measurement for the upper bound of the answer, and the lower bound of every measurement for the lower bound. Do not mix them in a sum or product.
Key words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.
Method. Find the bounds of every measurement, then choose the combination that makes the density as big (upper bound) or as small (lower bound) as possible.
- Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.mass = 240 g (to the nearest ten): unit = 10, half a unit = 5
lower bound = 240 − 5 = 235, upper bound = 240 + 5 = 245
volume = 85 cm³ (to the nearest whole number): unit = 1, half a unit = 0.5
lower bound = 85 − 0.5 = 84.5, upper bound = 85 + 0.5 = 85.5 - Decide which bounds to use. To get the biggest possible density, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest density, do the opposite.mass: a bigger mass makes the density bigger, so for the upper bound of the density use the upper bound of mass, and for the lower bound of the density use the lower bound.
volume: a bigger volume makes the density smaller, so for the upper bound of the density use the lower bound of volume, and for the lower bound of the density use the upper bound. - Calculate the upper bound of the density. Put the bounds you chose into the calculation. Use a calculator if the numbers are not easy.245 ÷ 84.5 = 2.899 g/cm³ (rounded)
- Calculate the lower bound of the density. Put the other bounds into the calculation.235 ÷ 85.5 = 2.749 g/cm³ (rounded)
- Compare the two bounds rounded to 1, 2, 3… significant figures. If the lowest and highest possible values both round to the same number, then we can be sure of that number to that accuracy.to 1 s.f.: lower bound 3, upper bound 3 (the same)
to 2 s.f.: lower bound 2.7, upper bound 2.9 (different) - Write the answer to the most accurate level on which they agree. Both bounds round to 3 to 1 s.f., but to 2 s.f. they are different. So 3 is the answer we can be certain of.3 g/cm³ (to 1 s.f.)
Check. Check the bounds make sense: the density found from the original values is 2.824 g/cm³, which lies between the lower bound and the upper bound. ✓
Watch out. For a subtraction or a division, the bound you need is not the same one each time: to make a − b biggest you need the upper bound of a but the lower bound of b. The same is true for dividing by b.
Key words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.
Method. Find the bounds of every measurement, then choose the combination that makes the speed as big (upper bound) or as small (lower bound) as possible.
- Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.distance = 100 m (to the nearest whole number): unit = 1, half a unit = 0.5
lower bound = 100 − 0.5 = 99.5, upper bound = 100 + 0.5 = 100.5
time = 12.4 s (to 1 decimal place): unit = 0.1, half a unit = 0.05
lower bound = 12.4 − 0.05 = 12.35, upper bound = 12.4 + 0.05 = 12.45 - Decide which bounds to use. To get the biggest possible speed, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest speed, do the opposite.distance: a bigger distance makes the speed bigger, so for the upper bound of the speed use the upper bound of distance, and for the lower bound of the speed use the lower bound.
time: a bigger time makes the speed smaller, so for the upper bound of the speed use the lower bound of time, and for the lower bound of the speed use the upper bound. - Calculate the upper bound of the speed. Put the bounds you chose into the calculation. Use a calculator if the numbers are not easy.100.5 ÷ 12.35 = 8.138 m/s (rounded)
- Calculate the lower bound of the speed. Put the other bounds into the calculation.99.5 ÷ 12.45 = 7.992 m/s (rounded)
- Compare the two bounds rounded to 1, 2, 3… significant figures. If the lowest and highest possible values both round to the same number, then we can be sure of that number to that accuracy.to 1 s.f.: lower bound 8, upper bound 8 (the same)
to 2 s.f.: lower bound 8, upper bound 8.1 (different) - Write the answer to the most accurate level on which they agree. Both bounds round to 8 to 1 s.f., but to 2 s.f. they are different. So 8 is the answer we can be certain of.8 m/s (to 1 s.f.)
Check. Check the bounds make sense: the speed found from the original values is 8.065 m/s, which lies between the lower bound and the upper bound. ✓
Watch out. For a subtraction or a division, the bound you need is not the same one each time: to make a − b biggest you need the upper bound of a but the lower bound of b. The same is true for dividing by b.
Key words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.
Method. Find the bounds of every measurement, then choose the combination that makes the value as big (upper bound) or as small (lower bound) as possible.
- Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.x = 4.2 (to 1 decimal place): unit = 0.1, half a unit = 0.05
lower bound = 4.2 − 0.05 = 4.15, upper bound = 4.2 + 0.05 = 4.25
y = 3.1 (to 1 decimal place): unit = 0.1, half a unit = 0.05
lower bound = 3.1 − 0.05 = 3.05, upper bound = 3.1 + 0.05 = 3.15
z = 0.5 (to 1 decimal place): unit = 0.1, half a unit = 0.05
lower bound = 0.5 − 0.05 = 0.45, upper bound = 0.5 + 0.05 = 0.55 - Decide which bounds to use. To get the biggest possible value, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest value, do the opposite.x: a bigger x makes the value bigger, so for the upper bound of the value use the upper bound of x, and for the lower bound of the value use the lower bound.
y: a bigger y makes the value bigger, so for the upper bound of the value use the upper bound of y, and for the lower bound of the value use the lower bound.
z: a bigger z makes the value smaller, so for the upper bound of the value use the lower bound of z, and for the lower bound of the value use the upper bound. - Calculate the upper bound of the value. Put the bounds you chose into the calculation. Use a calculator if the numbers are not easy.(4.25 + 3.15) ÷ 0.45 = 16.44 (rounded)
- Calculate the lower bound of the value. Put the other bounds into the calculation.(4.15 + 3.05) ÷ 0.55 = 13.09 (rounded)
Check. Check the bounds make sense: the value found from the original values is 14.6, which lies between the lower bound and the upper bound. ✓
Watch out. For a subtraction or a division, the bound you need is not the same one each time: to make a − b biggest you need the upper bound of a but the lower bound of b. The same is true for dividing by b.
Method. Find the upper bound of the area and compare it with 107.
- Find the upper bounds of the length and width. Half of 0.1 is 0.05.length: 12.4 + 0.05 = 12.45
width: 8.6 + 0.05 = 8.65 - Multiply the upper bounds. The biggest area uses the biggest length and width.12.45 × 8.65 = 107.6925
- Compare with 107. 107.6925 > 107, so the area could be more than 107 cm². Values just below 12.45 and 8.65 give areas just below 107.6925, which are still more than 107.
Check. 12.44 × 8.64 = 107.4816, which is more than 107. ✓
Watch out. The upper bound itself is not reached, but you can get as near as you like.
Method. Compare 108 with the upper bound of the area.
- The biggest possible area is just under 107.6925. The upper bound of the area is 107.6925.107.6925
- Compare. 108 is bigger than the upper bound.108 > 107.6925
- Answer. No, 108 cm² is too big. It is more than the upper bound.
Check. Even with the biggest length and width the area is less than 108. ✓
Watch out. Compare with the bounds, not with the area from the values given.
Method. Round the lower bound and the upper bound to different numbers of significant figures. The answer can only be given to an accuracy at which both bounds give the same number.
- Write the two bounds. The true value is between these.lower bound = 7.31, upper bound = 7.34
- Round both to 1, 2, 3… significant figures and compare. Keep going until the two rounded values are different.1 s.f.: lower bound 7, upper bound 7 — the same
2 s.f.: lower bound 7.3, upper bound 7.3 — the same
3 s.f.: lower bound 7.31, upper bound 7.34 — different - Choose the most accurate value on which they agree. The bounds agree to 2 s.f. but not to 3 s.f. We cannot be sure of any more digits.7.3 m (to 2 s.f.)
Check. Any value between 7.31 and 7.34 rounds to 7.3 to 2 s.f. ✓
Watch out. If the bounds do not agree to 1 s.f., the answer cannot be given to any useful accuracy with this data.
Method. Round the lower bound and the upper bound to different numbers of significant figures. The answer can only be given to an accuracy at which both bounds give the same number.
- Write the two bounds. The true value is between these.lower bound = 3.846, upper bound = 3.912
- Round both to 1, 2, 3… significant figures and compare. Keep going until the two rounded values are different.1 s.f.: lower bound 4, upper bound 4 — the same
2 s.f.: lower bound 3.8, upper bound 3.9 — different - Choose the most accurate value on which they agree. The bounds agree to 1 s.f. but not to 2 s.f. We cannot be sure of any more digits.4 (to 1 s.f.)
Check. Any value between 3.846 and 3.912 rounds to 4 to 1 s.f. ✓
Watch out. If the bounds do not agree to 1 s.f., the answer cannot be given to any useful accuracy with this data.
Method. Round the lower bound and the upper bound to different numbers of significant figures. The answer can only be given to an accuracy at which both bounds give the same number.
- Write the two bounds. The true value is between these.lower bound = 0.4551, upper bound = 0.4612
- Round both to 1, 2, 3… significant figures and compare. Keep going until the two rounded values are different.1 s.f.: lower bound 0.5, upper bound 0.5 — the same
2 s.f.: lower bound 0.46, upper bound 0.46 — the same
3 s.f.: lower bound 0.455, upper bound 0.461 — different - Choose the most accurate value on which they agree. The bounds agree to 2 s.f. but not to 3 s.f. We cannot be sure of any more digits.0.46 kg (to 2 s.f.)
Check. Any value between 0.4551 and 0.4612 rounds to 0.46 to 2 s.f. ✓
Watch out. If the bounds do not agree to 1 s.f., the answer cannot be given to any useful accuracy with this data.
Method. Round the lower bound and the upper bound to different numbers of significant figures. The answer can only be given to an accuracy at which both bounds give the same number.
- Write the two bounds. The true value is between these.lower bound = 1204, upper bound = 1231
- Round both to 1, 2, 3… significant figures and compare. Keep going until the two rounded values are different.1 s.f.: lower bound 1000, upper bound 1000 — the same
2 s.f.: lower bound 1200, upper bound 1200 — the same
3 s.f.: lower bound 1200, upper bound 1230 — different - Choose the most accurate value on which they agree. The bounds agree to 2 s.f. but not to 3 s.f. We cannot be sure of any more digits.1200 (to 2 s.f.)
Check. Any value between 1204 and 1231 rounds to 1200 to 2 s.f. ✓
Watch out. If the bounds do not agree to 1 s.f., the answer cannot be given to any useful accuracy with this data.
Key words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.
Method. Find the bounds of every measurement, then choose the combination that makes the area as big (upper bound) or as small (lower bound) as possible.
- Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.side = 12.5 cm (to 1 decimal place): unit = 0.1, half a unit = 0.05
lower bound = 12.5 − 0.05 = 12.45, upper bound = 12.5 + 0.05 = 12.55 - Decide which bounds to use. To get the biggest possible area, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest area, do the opposite.side: a bigger side makes the area bigger, so for the upper bound of the area use the upper bound of side, and for the lower bound of the area use the lower bound.
- Calculate the upper bound of the area. Put the bounds you chose into the calculation. Use a calculator if the numbers are not easy.12.55 × 12.55 = 157.5025 cm²
Check. Check the bounds make sense: the area found from the original values is 156.25 cm², which lies between the lower bound and the upper bound. ✓
Watch out. Use the upper bound of every measurement for the upper bound of the answer, and the lower bound of every measurement for the lower bound. Do not mix them in a sum or product.
Method. Find the upper bound of the total mass and compare it with 7.2 kg.
- Find the upper bound of each mass. Half a unit: 0.005 for 2 d.p., and 0.05 for 1 d.p.2.35 + 0.005 = 2.355 and 4.8 + 0.05 = 4.85
- Add the upper bounds. The biggest total uses the biggest masses.2.355 + 4.85 = 7.205 kg
- Compare with the limit. 7.205 > 7.2
- Answer. Values just below the upper bound are possible.Yes, the total could be more than 7.2 kg (anything from just over 7.2 up to just under 7.205).
Check. 2.354 + 4.849 = 7.203 > 7.2, and both masses round correctly (2.354 to 2.35, 4.849 to 4.8). ✓
Watch out. Do not just add the given values. The bounds can push the total over the limit.
Method. Work out Ben’s value, then try the combination that should give the biggest answer, and compare.
- Find the bounds. Nearest whole number means half a unit of 0.5.a: 11.5 to 12.5. b: 3.5 to 4.5
- Ben’s method. UB of a divided by UB of b.12.5 ÷ 4.5 = 2.78 (3 s.f.)
- The better choice. To make a division as big as possible, divide the biggest a by the smallest b.12.5 ÷ 3.5 = 3.57 (3 s.f.)
- Compare. The second answer is bigger.3.57 > 2.78, so Ben’s value is not the upper bound. Dividing by the biggest b makes the answer too small.
Check. The upper bound is UB of a divided by LB of b. ✓
Watch out. When you divide, a small divisor gives a big answer.
Method. Think about what makes a subtraction answer as small as possible, and test with numbers.
- Think about the answer. The difference a − b is smallest when a is as small as it can be and we take away as much as possible.smallest a, biggest b
- Test with numbers. Let a = 8 and b = 3, both to the nearest whole number. Then a is from 7.5 and b up to 3.5.7.5 − 3.5 = 4 (using UB of b)
7.5 − 2.5 = 5 (using LB of b) - Compare. 4 is smaller than 5.Taking away the upper bound of b gives the smaller answer.
Check. Subtracting a bigger number always gives a smaller answer. ✓
Watch out. Draw the two numbers on a number line if it helps.
Key words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.
Method. Find the bounds of every measurement, then choose the combination that makes the length as big (upper bound) or as small (lower bound) as possible.
- Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.area = 24 cm² (to the nearest whole number): unit = 1, half a unit = 0.5
lower bound = 24 − 0.5 = 23.5, upper bound = 24 + 0.5 = 24.5
width = 4 cm (to the nearest whole number): unit = 1, half a unit = 0.5
lower bound = 4 − 0.5 = 3.5, upper bound = 4 + 0.5 = 4.5 - Decide which bounds to use. To get the biggest possible length, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest length, do the opposite.area: a bigger area makes the length bigger, so for the upper bound of the length use the upper bound of area, and for the lower bound of the length use the lower bound.
width: a bigger width makes the length smaller, so for the upper bound of the length use the lower bound of width, and for the lower bound of the length use the upper bound. - Calculate the upper bound of the length. Put the bounds you chose into the calculation. Use a calculator if the numbers are not easy.24.5 ÷ 3.5 = 7 cm
Check. Check the bounds make sense: the length found from the original values is 6 cm, which lies between the lower bound and the upper bound. ✓
Watch out. For a subtraction or a division, the bound you need is not the same one each time: to make a − b biggest you need the upper bound of a but the lower bound of b. The same is true for dividing by b.
Key words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.
Method. Find the bounds of every measurement, then choose the combination that makes the product as big (upper bound) or as small (lower bound) as possible.
- Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.a = 40 (to the nearest ten): unit = 10, half a unit = 5
lower bound = 40 − 5 = 35, upper bound = 40 + 5 = 45
b = 60 (to the nearest ten): unit = 10, half a unit = 5
lower bound = 60 − 5 = 55, upper bound = 60 + 5 = 65 - Decide which bounds to use. To get the biggest possible product, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest product, do the opposite.a: a bigger a makes the product bigger, so for the upper bound of the product use the upper bound of a, and for the lower bound of the product use the lower bound.
b: a bigger b makes the product bigger, so for the upper bound of the product use the upper bound of b, and for the lower bound of the product use the lower bound. - Calculate the lower bound of the product. Put the other bounds into the calculation.35 × 55 = 1925
Check. Check the bounds make sense: the product found from the original values is 2400, which lies between the lower bound and the upper bound. ✓
Watch out. Use the upper bound of every measurement for the upper bound of the answer, and the lower bound of every measurement for the lower bound. Do not mix them in a sum or product.
Key words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.
Method. Find the bounds of every measurement, then choose the combination that makes the total length as big (upper bound) or as small (lower bound) as possible.
- Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.plank length = 2.4 m (truncated to 1 decimal place): unit = 0.1
lower bound = 2.4 (truncating never rounds up, so the value shown is the smallest possible), upper bound = 2.4 + 0.1 = 2.5
number of planks = 5 is an exact value, so it has no error: lower bound = upper bound = 5 - Decide which bounds to use. To get the biggest possible total length, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest total length, do the opposite.plank length: a bigger plank length makes the total length bigger, so for the upper bound of the total length use the upper bound of plank length, and for the lower bound of the total length use the lower bound.
- Calculate the upper bound of the total length. Put the bounds you chose into the calculation. Use a calculator if the numbers are not easy.2.5 × 5 = 12.5 m
- Calculate the lower bound of the total length. Put the other bounds into the calculation.2.4 × 5 = 12 m
Check. Check the bounds make sense: the total length found from the original values is 12 m, which lies between the lower bound and the upper bound. ✓
Watch out. Use the upper bound of every measurement for the upper bound of the answer, and the lower bound of every measurement for the lower bound. Do not mix them in a sum or product.
Method. Find the lower bound of x ÷ y and compare it with 3.
- Find the bounds. Half a unit of 0.1 is 0.05.x: 6.25 to 6.35
y: 2.05 to 2.15 - Find the lower bound of the quotient. The smallest answer comes from the smallest x divided by the biggest y.6.25 ÷ 2.15 = 2.907 (4 s.f.)
- Compare with 3. 2.907 is less than 3, and values just above the lower bound are possible.2.907 < 3
- Write the conclusion. The lower bound is 2.91 (3 s.f.), which is less than 3, so x ÷ y could be less than 3.
Check. 6.3 ÷ 2.1 = 3 exactly, but with x = 6.26 and y = 2.14 we get 2.925 < 3, and both round correctly. ✓
Watch out. Use the lower bound of x and the upper bound of y to make a division as small as possible.