Number
Grade 6-7Upper & Lower Bounds
A rounded measurement hides a range of possible true values, and upper and lower bounds questions ask you to work with that range instead of the rounded value itself. This lesson covers finding the bounds and error interval of a rounded measurement, and choosing the correct combination of bounds when adding, subtracting, multiplying or dividing two measurements.
What you need to know
- A value rounded to the nearest 10 could be up to 5 either side of the stated value.
- The lower bound is included; the upper bound is the value you never quite reach.
- For the largest possible result, use the largest values, but take care with subtraction and division.
- Maximum of a difference uses upper minus lower; maximum of a quotient uses upper divided by lower.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Finding the upper and lower bounds of a rounded value
A rounded value could have been rounded from anywhere within half of the rounding unit either side of it. Subtract half the rounding unit for the lower bound, and add half the rounding unit for the upper bound.
A length of 8 cm, correct to the nearest cm, has a rounding unit of 1 cm, so half of that is 0.5 cm. The lower bound is 8 - 0.5 = 7.5 cm, and the upper bound is 8 + 0.5 = 8.5 cm.
Writing an error interval
An error interval writes the range of possible true values as an inequality, using the lower bound with a "less than or equal to" sign and the upper bound with a strict "less than" sign, since the value would round up to the next value at the upper bound itself.
A number, n, is 27 correct to the nearest whole number: the error interval is 26.5 ≤ n < 27.5.
Maximising or minimising a sum or difference
To maximise a sum, add the upper bounds of both values. To minimise a difference, subtract the upper bound of the smaller value from the lower bound of the larger value, since that combination gives the smallest possible gap between them.
Two masses of 15 kg and 9 kg, both to the nearest kg: the minimum possible difference is the lower bound of 15 minus the upper bound of 9, which is 14.5 - 9.5 = 5 kg.
Maximising or minimising a product or quotient
To maximise a product, multiply the upper bounds of both values. To maximise a quotient, divide the upper bound of the numerator by the lower bound of the denominator, since dividing by the smallest possible denominator gives the largest possible result.
A car travels 150 miles, correct to the nearest 10 miles, in 3 hours, correct to the nearest whole hour: the maximum possible average speed is the upper bound of distance divided by the lower bound of time, which is 155 ÷ 2.5 = 62 mph.
Worked examples
Three exam-style questions, fully solved
A length is measured as 8 cm, correct to the nearest cm. Write down the upper and lower bounds of the length.
Easy- 1.Find half of the rounding unit: half of 1 cm is 0.5 cm
- 2.Subtract for the lower bound: 8 - 0.5
- 3.Add for the upper bound: 8 + 0.5
Answer: Lower bound 7.5 cm, upper bound 8.5 cm
A car travels 150 miles, correct to the nearest 10 miles, in 3 hours, correct to the nearest whole hour. Work out the maximum possible average speed of the car.
Medium- 1.Find the upper bound of the distance: 155 miles
- 2.Find the lower bound of the time: 2.5 hours
- 3.Divide the upper bound of distance by the lower bound of time: 155 ÷ 2.5
Answer: 62 mph
A rectangular garden has a length of 12 m and a width of 7 m, both correct to the nearest metre. Is the area of the garden guaranteed to be at least 74 m²? Use the lower bound of the area to justify your answer.
Hard- 1.Find the lower bound of the length: 11.5 m
- 2.Find the lower bound of the width: 6.5 m
- 3.Multiply the lower bounds to find the smallest possible area: 11.5 × 6.5 = 74.75 m²
Answer: Yes, since even the smallest possible area, 74.75 m², is at least 74 m²
Practice
6 questions, marked instantly
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Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
A length is 8 cm to the nearest centimetre. What is the lower bound?
A mass is 40 kg to the nearest 10 kg. What is the upper bound?
A time is 12.4 seconds to 1 decimal place. What is the lower bound?
A rectangle has sides of 6 cm and 4 cm, each measured to the nearest centimetre. What is the upper bound for its area?
x = 5.0 correct to 1 decimal place. What is the upper bound of x?
A bag of sugar weighs 1 kg to the nearest 100 g. What is the least it could weigh, in grams?
Avoid these
The mistakes students make most often
Using the whole rounding unit instead of half of it when finding the upper and lower bounds.
Writing an error interval with a "less than" sign on the lower bound instead of "less than or equal to", or the wrong way round.
When minimising a difference, using the upper bounds of both values instead of the lower bound of the larger value and the upper bound of the smaller value.
When maximising a quotient, dividing by the upper bound of the denominator instead of the lower bound, which gives a smaller result instead of the maximum.
When asked to justify whether a claim is guaranteed to be true, using the wrong bound to test it, instead of the bound that represents the worst-case scenario for the claim.
FAQ
Questions students and parents ask
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