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Exact answers: fractions

Good practice states that only estimations and final answers should be rounded and it should be stated to what accuracy the solution has been rounded.
© National STEM Learning Centre
In some cases a rounded answer is not sufficient and the context of the problem requires an exact answer. When performing a calculation which requires a large number of steps it is important that the exact answer to one calculation is used in the next calculation. If rounded answers are used then ‘rounding errors’ can mount up leading to a final answer which can be significantly different to the exact answer.
If the calculations being performed were to calculate the dosage of a medicine, then the difference between the exact answer and an answer with rounding errors could be the difference between life and death.
Good practice states that only estimations and final answers should be rounded and it should be stated to what accuracy the solution has been rounded.
We’ll consider three different ways of expressing solutions as exact answers, starting with fractions, then in the next steps surds and pi.

Exact answers with fractions

For this activity you will need a calculator. We are going to explore what happens when we round answers and then use these rounded in subsequent calculations. This will show how rounding errors can accumulate leading to unacceptable degrees of inaccuracy. We will consider strategies to avoid accumulating such errors.

Task

Type into a calculator \(100 \div 13 \times 13\) and your answer gives a value of 100.
Now type in \(100 \div 13\) and press equals.
Some calculators will give an answer of \(\frac{100}{13}\). If it does, work out how to display this as a decimal. Most calculators will give 7.692307692.
Rounding to 2 decimal places as is common in many situations gives 7.69.
Clear your calculator and type in 7.69 x 13. This should give an answer of 100 but your calculator gives 99.97. What might be the issues with this?
Type in 7.69 and divide by 13 then round your answer to 2 decimal places. Do you get 0.59?
So we have taken 100 and divided by 13 twice, so multiplying 0.59 by 13 then 13 again should return us to 100. Try it. What do you get? 99.71?
We can continue this process: 0.59 divided by 13 gives 0.05 to 2 decimal places.
\(0.05 \times 13 \times 13 \times 13\) should give 100, but using the method with our calculator above, it gives 109.85 quite a way from 100.

Fractions as exact answers

10 divided by 2 gives an answer of 5: this is an exact answer.
10 divided by 4 gives an answer of 2.5: this is an exact answer.
10 divided by 7 gives an answer of 1.428571429: this is not an exact answer as this number is a recurring decimal, which goes on for ever.
Your calculator has rounded the answer to make it fit on the screen. The way to express this answer as an exact answer is as the fraction \(\frac{10}{7}\). We look more at fractions and recurring decimals in our accompanying course Maths subject knowledge: Fractions, decimals and percentages.

Teaching resource

This activity on errors uses examples in the context of shape and space to help students understand how big errors may be, and how errors accumulate when measurements are used in calculations.
© National STEM Learning Centre
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Maths Subject Knowledge: Understanding Numbers

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