# QTS Numeracy Model Solutions – Test 2, Q21

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 21 from practice test 2. The full practice paper can be downloaded here​.

## The Question

A teacher produced the following table to show the marks achieved in an end of year geography test by pupils in three Year 7 classes. Tick all the true statements:

– Some pupils in Class A achieved less than 12%.
– At least one pupil in Class C achieved less than 20%.
– All pupils in Class B achieved at least 40%.

## Worked Solution

This question on averages and range probably causes issues for more students than any other question on the practice numeracy tests – so if you are struggling with it, don’t worry – you are not alone!

The question gives you three statements and asks you to tick all of the statements that are true. The best way to approach a question like this is to look at the statements one by one and decide whether each is true or false.

​Some pupils in Class A achieved less than 12%.

You need to work out whether this is possible based on the data given in the table for class A.

You are told that the range is 60, the median is 50 and the mode is 72. What this means is that the difference between the lowest and highest scoring pupils in the class is 60. The middle pupils scored 50 and the most common score was 72.

Think about what would happen if the lowest scoring pupil scored less than 12% (for example 11%). This would mean the highest scoring pupil scored 60 more than this, so the highest score would be 71% (or even lower if the lowest score was less than 11%) – but this cannot be the case as the most common score was 72% so the statement has to be FALSE.

​​​At least one pupil in Class C achieved less than 20%.​​​

For this class, the table shows that the range is 85, the median is 60 and the mode is 72.

This time, think about what would happen if the statement were false. This would mean that nobody scored lower than 20%, so the lowest score would be 20% (or even higher).

You are told that the range is 85, meaning that the difference between the highest score and the lowest score is 85.

If the lowest score is 20 then the highest score would have to be 20 + 85 = 105% (and it would be even higher if the lowest score was bigger than 20%) – but again this is impossible so the statement must be FALSE.

All pupils in Class B achieved at least 40%

The table shows that in Class B, the range is 28, the median is 50 and the mode is 68. You can approach this in a similar way to how you approached the first statement.

Think about what would happen if the statement was false and the lowest scoring pupil scored less than 40% (for example 39%). The difference between this and the highest scoring pupil would need to be 28 (the range), so the score of the highest scoring pupil would be 39 + 28 = 67 (or lower, if the lowest scoring pupil scored less than 39%).

However, the table shows you that the most common score (mode) is 68%, which cannot be the case if the highest score is 67%. This means that nobody can have scored less than 40% and the statement must be TRUE.

## Final Answer: False, False, True

Note – I have made a point of being thorough in this method to ensure you understand each point as we go. When you sit your QTS Skills Tests, your process will probably be much quicker as these techniques start to become second nature to you.

## Further Help

If you require any further help with questions like this, we have created a selection of resources to provide all the help you need.

Revision Book – The Guide to the QTS Skills Tests book devotes a whole chapter to each QTS numeracy topic, with detailed methods, worked examples and plenty of practice questions for you to have a go at. The book also includes three practice tests and fully worked solutions to every question.

Practice Tests – Want more practice tests to see how ready you are? Take a look at our selection of practice papers for the QTS numeracy test (including some totally free papers as our gift to you).

Revision Cheat Sheets – Our cheat sheets boil down everything you need to know to just the key points on the topic. They are a perfect resource for your last minute revision!