# QTS Numeracy Model Solutions – Test 2, Q13

In this series of blog posts, we look at some of the detailed methods that you could use to tackle the questions on the QTS Numeracy practice papers from the Department for Education.

In this post, we look at question 13 from practice test 2. The full practice paper can be downloaded here​.

## The Question

To inform her choice of reading materials, a primary teacher looked at the spread of reading ages in her class. The scatter graph shows the actual age and reading age of 21 pupils in the class.

What proportion of the class have the same reading age as their actual age? Give your answer as a decimal to one decimal place.

## Worked Solution

The first step in tackling any question on your QTS Skills Tests is making sure you are clear what the question is actually asking. In this case, it is asking what proportion of the class have the same reading age as their actual age (rather than just the number of pupils who have it the same).

Proportion compares one part to a whole, and it is typically expressed in one of three ways – as a fraction, as a decimal, or as a percentage. In this case, the question makes clear that you need to give your answer as a decimal.

To work out the proportion as a decimal, you will need to divide the number of pupils that have the same reading age and actual age by the total number of pupils in the class.

The question tells you that the total number of pupils in the class is 21.

This just leaves you to work out the number of pupils that have the same reading age and actual age, and to do this you need to use the scatter graph.

Each of the points on the scatter graph represents a different pupil in the class, and you can use the graph to find out both the reading age and the actual age of the pupil. The horizontal axis at the bottom of the graph represents the actual age of the pupil, so if you draw a straight line down from the point to this axis, the value you read off will be the actual age (the thicker lines labelled 5, 6, 7 and 8 represent years and the thinner lines between them represent months). The vertical axis at the side of the graph represents the reading age of the pupil, so if you draw a straight line across from the point to this axis, you will read off the reading age (again, the thicker lines are years and the thinner ones are months).

For example, look at the red dot furthest to the left on the scatter graph. This point represents a pupil who when you look down to the horizontal axis shows an actual age of 5 years and 9 months, and when you look across to the vertical axis shows a reading age of 6 years and 5 months.

For this question you need the pupils that have exactly the same reading age and actual age. Whilst you could look at each point and compare the two ages, this is not the quickest way to do it. The scatter graph that you have been given has a blue diagonal line drawn from the bottom left of the graph to the top right (not all scatter graphs you come across will have such a line, but it is very useful when they do). This line is drawn to show where the two ages are the same, so all the points above the line are closer to the vertical axis and have a higher reading age than actual age and all the points below the line are closer to the horizontal axis and have a higher actual age than reading age.

You are looking for the pupils with exactly the same reading age and actual age, and these pupils are represented by points that are exactly on the blue diagonal line.

There are 6 such points on the line (6y 1m, 6y 4m, 6y 5m, 6y 7m, 6y 9m and 6y 10m).

This means that 6 pupils out of a class of 21 have the same reading age and actual age, so you are now in a position to work out what proportion of the class this is.

6 ÷ 21 = 0.28571429…

When you have worked out your answer, it is good to check back with the question what format they want your answer to be given in. It says, ​”Give your answer as a decimal to one decimal place.”

This means that your answer should only keep one digit after the decimal place. The long decimal you worked out is in between 0.2 and 0.3, so you need to decide whether to round it down to 0.2 or round it up to 0.3. To do this, look at the next digit in the decimal (in this case the ‘8’) and if this digit is a 5 or above then round up, if it is a 4 or below then round down. In this case we need to round up, so the answer will be 0.3.