11 Plus Exams Forum

I simply cannot solve this percentages question. I have searched online and this forum, but cannot find an example or an explanation.
It strikes me as ridiculously hard for an 11 + question and is in one of the Bond books for the CEM test.

In a school of 500 pupils, 20% learn Latin, 40% learn Spanish and 60% learn French. If half of the pupils learn two languages and no one learns more than two, how many pupils do not learn a language?

I have worked out the number of pupils studying each language and the number who study two languages, but we cannot get further than this!
300 learn French,
100 learn Latin,
200 learn Spanish
250 learn two languages.

I have drawn Venn diagrams, worked out fractions and clearly missing something obvious! Help much appreciated.

Work out the total of langauages studied which is 600. (300+200+100).

250 do two, so they count for 500 of the total.

Does that help get you there?

Thanks Tinkers.That helps so much. I had both of those figures - just couldn’t ‘see’ what to do next.

Am I right in thinking that this is not a typical 11 + percentages question for CEM?

That bit I can’t help with as my DD did GL so I have no experience with CEM. Hopefully someone else with far more CEM experience than me can tell you.

Hi,

Both DS & DD did CEM (2017, 2014) and neither of them remember any questions like that, DS said that he was expecting similar questions but there was nothing like that at all!
DD can't remember very much at all as it was 3 years ago but doesn't recall anything along those lines

Thank goodness for that. I had checked through GCSE and IGCSE maths materials and papers and not found anything similar.

Venn diagrams are in the Primary curriculum but not limked to %.

While you need a basic understanding of percentages to get to the numbers of pupils, to solve the problem you need to be able to figure out the logic of it. I don't think this is a typical "percentages" question, but might be a typical logic problem requiring clear analytical thinking and being able to translate the question into a logical approach.

The "trick" to this one is to see that there are three groups of pupils:
Those doing 2 languages (none do more, we are told) =250
Those doing 1 = ?
Those doing none = the question being asked.
The question asks for the number doing none, but to get there you have to work out how many do only 1 first, which is not directly asked for nor given.

An approach to getting your head round problems like this is to realise that what is being counted is not really pupils, but the intersection of a pupil and a language. A good mental proxy for this is the text book the pupil has, so we can say:
There are 300 French text books, 100 Latin, 200 Spanish = total of 600 text books shared between the pupils studying one or more languages.
250 pupils learn two languages so they have two books each. This is 500 of the total number of books, leaving 100 for the pupils studying only 1
250 + 100 = 350, leaving 150 studying none.

Hope this helps.

Thanks Adam. That is very clear, helpful explanation - particularly the use of textbooks.