Sec 1 Maths · Approximation and estimation

Rounding errors questions with answers

Practice questions on rounding errors, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

The height of an HDB block is $130$ m, correct to the nearest 10 m. Complete the statement: the actual height of the block is less than $\ldots\ldots$ m.

Step 1 of 2

The actual value can be up to half a rounding unit either side

Half of

10

= 5

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Questions

Question 1Least possible value; the value it is less than
The mass of a parcel is $350$ g, correct to the nearest 10 g. Write down the least possible mass of the parcel.
Show the worked solution
1. Step 1: The actual value can be up to half a rounding unit either sideHalf of $10$ g $= 5$ g
2. Step 2: Least possible value: subtract half a unit $350 - 5 = 345$ g
Question 2Range of possible values
The length of a rod is $24$ cm, correct to the nearest cm. Given that the actual length of the rod is $l$ cm, write down the range of possible values of $l$ .
Show the worked solution
1. Step 1: The actual value can be up to half a rounding unit either sideHalf of $1$ cm $= 0.5$ cm
2. Step 2: Write the range: the least possible value, and the value it must be less than $23.5 \leq l < 24.5$
Question 3Greatest possible difference from the actual value
The distance between two MRT stations is $3.7$ km, correct to 1 decimal place. Write down the greatest possible difference between the actual distance and $3.7$ km.
Show the worked solution
1. Step 1: The greatest possible difference is half of the unit the value is rounded toHalf of $0.1$ km $= 0.05$ km
Question 4Percentage error of an estimate
By writing each number correct to 1 significant figure, $52.3 \times 0.0189$ is estimated to be $1$ . Express the difference between this estimate and the exact value as a percentage of the exact value, giving your answer correct to 3 significant figures.
Show the worked solution
1. Step 1: Find the exact value (calculator) $52.3 \times 0.0189 = 0.98847$
2. Step 2: Error = | estimate − exact value |Error $= 1 - 0.98847$ $= 0.01153$
3. Step 3: Percentage error = error ÷ exact value × 100%Percentage error $= \dfrac{0.01153}{0.98847} \times 100\%$ $= 1.16644\ldots\%$ $= 1.17\%$ (3 s.f.)
Question 5A count to n significant figures: smallest and largest
The number of books in a library is $5730$ , correct to 3 significant figures. Find the smallest and the largest possible number of books.
Show the worked solution
1. Step 1: Which place is the last significant figure in? (count 3 figures from the left) $5730$ to 3 significant figures is rounded to the nearest $10$ .
2. Step 2: Anything from half a unit below up to (not including) half a unit above rounds to itHalf of $10$ is $5$ .
3. Step 3: Smallest: take half a unit away $\text{Smallest} = 5730 - 5$ $\phantom{\text{Smallest}} = 5725$
4. Step 4: Largest WHOLE number: add half a unit, then take 1 away (that number itself would round up) $\text{Largest} = 5730 + 5 - 1$ $\phantom{\text{Largest}} = 5734$ ( $5735$ itself rounds up to $5740$ ).

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Questions

Question 1Least possible value; the value it is less than

The mass of a parcel is $350$ g, correct to the nearest 10 g. Write down the least possible mass of the parcel.

Show the worked solution

Step 1: The actual value can be up to half a rounding unit either sideHalf of $10$ g $= 5$ g
Step 2: Least possible value: subtract half a unit $350 - 5 = 345$ g

Question 2Range of possible values

The length of a rod is $24$ cm, correct to the nearest cm. Given that the actual length of the rod is $l$ cm, write down the range of possible values of $l$ .

Show the worked solution

Step 1: The actual value can be up to half a rounding unit either sideHalf of $1$ cm $= 0.5$ cm
Step 2: Write the range: the least possible value, and the value it must be less than $23.5 \leq l < 24.5$

Question 3Greatest possible difference from the actual value

The distance between two MRT stations is $3.7$ km, correct to 1 decimal place. Write down the greatest possible difference between the actual distance and $3.7$ km.

Show the worked solution

Step 1: The greatest possible difference is half of the unit the value is rounded toHalf of $0.1$ km $= 0.05$ km

Question 4Percentage error of an estimate

By writing each number correct to 1 significant figure, $52.3 \times 0.0189$ is estimated to be $1$ . Express the difference between this estimate and the exact value as a percentage of the exact value, giving your answer correct to 3 significant figures.

Show the worked solution

Step 1: Find the exact value (calculator) $52.3 \times 0.0189 = 0.98847$
Step 2: Error = | estimate − exact value |Error $= 1 - 0.98847$ $= 0.01153$
Step 3: Percentage error = error ÷ exact value × 100%Percentage error $= \dfrac{0.01153}{0.98847} \times 100\%$ $= 1.16644\ldots\%$ $= 1.17\%$ (3 s.f.)

Question 5A count to n significant figures: smallest and largest

The number of books in a library is $5730$ , correct to 3 significant figures. Find the smallest and the largest possible number of books.

Show the worked solution

Step 1: Which place is the last significant figure in? (count 3 figures from the left) $5730$ to 3 significant figures is rounded to the nearest $10$ .
Step 2: Anything from half a unit below up to (not including) half a unit above rounds to itHalf of $10$ is $5$ .
Step 3: Smallest: take half a unit away $\text{Smallest} = 5730 - 5$ $\phantom{\text{Smallest}} = 5725$
Step 4: Largest WHOLE number: add half a unit, then take 1 away (that number itself would round up) $\text{Largest} = 5730 + 5 - 1$ $\phantom{\text{Largest}} = 5734$ ( $5735$ itself rounds up to $5740$ ).