Upper and lower bounds, the biggest and smallest it could be
Learn how to find upper and lower bounds using the rounding bucket method. Worked examples, exam-style questions with answers.
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Download the PDF (623 KB)A plank is 2.4 m long to the nearest 10 cm. Could it actually be 2.46 m?
Upper and lower bounds feel hard because rounding hides real values. The rounding bucket holds all possible real lengths. You use the bucket edges to find the smallest and largest possible values.
The picture to keep in your head: The rounding bucket
Rounding drops every real length into a bucket. Everything from 2.35 m up to (but not including) 2.45 m lands in the '2.4 m' bucket. The edges of the bucket are the bounds: half the rounding step below and above. When you combine measurements, you pick from the edges of the buckets to make the answer as big, or as small, as it could be.
The method, every time
- Find half the unit size. Add and subtract it from the rounded number.
- Use the upper bound for multiplication and division. Use the lower bound for division.
- Use the upper bound for addition and subtraction. Use the lower bound for subtraction.
- Keep full values until the end. Only round at the end to find the final answer.
- Check your answer by using the bucket edges for the biggest and smallest possible results.
Words you'll see
- bounds
- Smallest and largest possible values from a rounded measurement.
- bucket
- A range of real values that round to a given number.
- unit
- The size of the rounding step, like 10 cm or 1 cm.
- truncation
- Cutting off the end of a number without rounding.
- interval
- The range of values between the lower and upper bounds.
Half a step each way
The rounding bucket holds all lengths that round to a given value. To find the error interval, you halve the rounding unit and add or subtract it. This gives the smallest and largest possible values. The error interval is written as lower ≤ x < upper.
Watch out
- Using the whole rounding unit instead of half. Always halve the rounding unit before adding or subtracting
- Writing ≤ at the upper end. Use < at the upper end of the error interval
Biggest × biggest makes the biggest
The rounding bucket holds all real lengths that round to a given value. To find the biggest area, use the biggest lengths from each bucket. For the largest area, pick the upper bounds of both measurements. This means using the biggest possible values for each measurement before multiplying.
Watch out
- Rounding the bounds before multiplying. Use the exact upper bounds before multiplying
- Using a lower and an upper bound together. Always use both upper or both lower bounds together
Big minus small, big ÷ small
The rounding bucket helps us find the biggest and smallest possible values. For big minus small or big ÷ small, use the upper bound of the first and lower bound of the second. This gives the biggest difference or speed. To find the smallest, swap the bounds.
Watch out
- Dividing upper by upper. Use upper for first and lower for second to get biggest result
- Rounding the final answer too early. Only round at the end to keep accuracy
Chopped, not rounded
Truncating chops digits off without rounding: 5.79 truncated to 1 decimal place is still 5.7. So everything from 5.7 up to (but not including) 5.8 truncates to 5.7. The bucket only goes upwards from the value, a whole step, not half a step each way.
Watch out
- Treating truncation like rounding. Truncation cuts off digits, not rounds them
- Giving more accuracy than the bounds support. Use the bounds as they are, don’t add extra digits
21 exam-style upper and lower bounds questions
Show your working: most marks are for the method. Marks are a guide to how much working an exam would expect. Questions written by Bloom Education in the style of GCSE (AQA, Edexcel, OCR) and Cambridge IGCSE papers.
A · Error intervals
- 1A number is rounded to the nearest 10. The result is 30. Write down the error interval for the original number.[2 marks]
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25 ≤ x < 35
The rounding step is 10. The lower bound is 30 - 5 = 25. The upper bound is 30 + 5 = 35. So 25 ≤ x < 35. - 2A number is rounded to the nearest whole number. The result is 7. Write down the error interval for the original number.[2 marks]
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6.5 ≤ x < 7.5
The rounding step is 1. The lower bound is 7 - 0.5 = 6.5. The upper bound is 7 + 0.5 = 7.5. So 6.5 ≤ x < 7.5. - 3A number is rounded to 1 decimal place. The result is 4.7. Write down the error interval for the original number.[2 marks]
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4.65 ≤ x < 4.75
The rounding step is 0.1. The lower bound is 4.7 - 0.05 = 4.65. The upper bound is 4.7 + 0.05 = 4.75. So 4.65 ≤ x < 4.75. - 4A number is rounded to 2 significant figures. The result is 3.6. Write down the error interval for the original number.[2 marks]
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3.55 ≤ x < 3.65
The rounding step is 0.1. The lower bound is 3.6 - 0.05 = 3.55. The upper bound is 3.6 + 0.05 = 3.65. So 3.55 ≤ x < 3.65. - 5A number is rounded to the nearest whole number. The result is 12. Write down the error interval for the original number.[2 marks]
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11.5 ≤ x < 12.5
The rounding step is 1. The lower bound is 12 - 0.5 = 11.5. The upper bound is 12 + 0.5 = 12.5. So 11.5 ≤ x < 12.5.
B · Sums and products
- 6A rectangle has length 8.2 cm and width 4.1 cm, both rounded to the nearest 0.1 cm. Calculate the upper and lower bounds of the perimeter.[3 marks]
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Upper bound of the perimeter: 24.8 cm. Lower bound of the perimeter: 24.4 cm.
Perimeter = 2 × (length + width). Upper bound: 2 × (8.25 + 4.15) = 2 × 12.4 = 24.8 cm. Lower bound: 2 × (8.15 + 4.05) = 2 × 12.2 = 24.4 cm. - 7A rectangle has length 8.2 cm and width 4.1 cm, both rounded to the nearest 0.1 cm. Calculate the upper and lower bounds of the area.[3 marks]
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Upper 34.2375 cm², lower 33.0075 cm²
Upper: 8.25 × 4.15 = 34.2375. Lower: 8.15 × 4.05 = 33.0075. - 820 identical bottles each have a mass of 0.5 kg, rounded to the nearest 0.1 kg. Find the upper and lower bounds of the total mass.[3 marks]
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Upper bound: 11.0 kg, Lower bound: 9.0 kg
Total mass = 20 × mass. Upper bound: 20 × 0.55 = 11.0 kg. Lower bound: 20 × 0.45 = 9.0 kg. - 9A square has side length 6.3 cm, rounded to the nearest 0.1 cm. Find the upper and lower bounds of the area.[3 marks]
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Upper bound: 40.3225 cm², Lower bound: 39.0625 cm²
Area = side × side. Upper bound: 6.35 × 6.35 = 40.3225 cm². Lower bound: 6.25 × 6.25 = 39.0625 cm².
C · Differences and division (Higher)
- 10A car travels 120 km (to the nearest km) in 2 hours (to the nearest 0.1 hour). Find the upper bound of its average speed, to 2 decimal places.[3 marks]
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61.79 km/h
Upper distance ÷ lower time = 120.5 ÷ 1.95 = 61.79… - 11A block has mass 2.5 kg (to 1 decimal place) and volume 0.35 m³ (to 2 decimal places). Find the lower bound of its density, to 2 decimal places.[3 marks]
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6.90 kg/m³
Lower mass ÷ upper volume = 2.45 ÷ 0.355 = 6.901… - 12Two rods are 1.4 m and 2.6 m long, both to the nearest 0.1 m. Find the upper bound of the difference in their lengths.[3 marks]
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1.3 m
Upper of the longer minus lower of the shorter: 2.65 - 1.35. - 13Two rods are 1.2 m and 2.3 m long, both to the nearest 0.1 m. Find the lower bound of the difference in their lengths.[3 marks]
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1.0 m
Lower of the longer minus upper of the shorter: 2.25 - 1.25. - 14A rectangle has area 50 cm² (to the nearest cm²) and width 4 cm (to the nearest cm). Find the upper bound of its length, to 2 decimal places.[3 marks]
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14.43 cm
Length = area ÷ width: upper area ÷ lower width = 50.5 ÷ 3.5 = 14.428…
D · Truncation and accuracy (Higher)
- 15A number is truncated to the nearest whole number. The result is 7. Write down the error interval for the original number.[3 marks]
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7 ≤ x < 8
Truncating chops off the decimals, so everything from 7 up to (not including) 8 becomes 7. - 16A number is truncated to 1 decimal place. The result is 4.7. Write down the error interval for the original number.[3 marks]
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4.7 ≤ x < 4.8
Truncating to 1 decimal place means cutting off after the first decimal. The number is between 4.7 and 4.799… The lower bound is 4.7, upper bound is 4.8. Error interval is 4.7 ≤ x < 4.8. - 17x = 3.4 and y = 1.2, both to 1 decimal place. Work out x ÷ y to a suitable degree of accuracy. Give a reason.[3 marks]
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3, to 1 significant figure
Upper 3.45 ÷ 1.15 = 3.0, lower 3.35 ÷ 1.25 = 2.68; both round to 3 at 1 s.f., but not at 2 s.f. - 18A number is truncated to the nearest 10. The result is 30. Write down the error interval for the original number.[3 marks]
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30 ≤ x < 40
Truncating to the nearest 10 chops off the units: everything from 30 up to (not including) 40 becomes 30.
E · Real problems (Higher)
- 19Will a 1.85 m sofa (nearest 5 cm) definitely fit in a 1.9 m gap (nearest 10 cm)? Explain using bounds.[4 marks]
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No, it may not fit
Longest sofa 1.875 m; narrowest gap 1.85 m. 1.875 > 1.85, so it might not fit. - 20How many 250 ml cups (nearest 10 ml) can definitely be filled from 5 litres (nearest 0.1 litre)?[4 marks]
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19 cups
Least water 4950 ml ÷ biggest cup 255 ml = 19.4…, so 19 cups for certain. - 21A 2.5 L container (nearest 0.1 L) is used to fill 1200 mL jugs (nearest 10 mL). How many jugs can definitely be filled?[4 marks]
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2 jugs
Least water 2450 ml ÷ biggest jug 1205 ml = 2.03…, so 2 jugs for certain.
Upper and lower bounds exam checklist
- Half the unit each way to find the bucket edges.
- Lower ≤ x < upper to show the range of possible values.
- Sum and product: upper with upper for the biggest result.
- Difference and division: upper with lower for the smallest result.
- Keep full values until the end to avoid errors.
- Truncation only goes up, not down, so watch out for that.
- Use the bucket edges to make the answer as big or as small as it could be.
Got stuck on a question?
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All of this page as a 10-page guide, with every answer at the back. No sign-up.
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