Question

There are 26 students in class. All of the students speak either English or Japanese, and some speak both languages. 18 can speak English and 13 can speak Japanese. 1)How many can speak both English and Japanese 2) how many speak English but not Japanese?

Answer 1

We know that, out of 26 students, 18 can speak english and 13 can speak japanese.

We have to find:

1) How many can speak both English and Japanese

2) How many speak English but not Japanese?

We will use set theory to answer this questions.

The whole set has a size of 26 students.

It can be subdivided in the following sets:

1) We will find how many students speak both languages (eng ∪ jap).

The total number of students can be then expressed as the sum of the students that speak english and the students that speak japanes minus the number of students that speak both:

$S=S_{eng}+S_{jap}-S_{eng\cup jap}$

NOTE: we substract the number of students that speak both languages beacuse we would be adding them twice, as they are both included in the set "speak english"and in the set "speak japanese".

Then, we can replace and calculate as:

$\begin{gathered} S=S_{eng}+S_{jap}-S_{eng\cup jap} \\ 26=18+13-S_{eng\cup jap} \\ S_{eng\cup jap}=18+13-26 \\ S_{eng\cup jap}=5 \end{gathered}$

2) We have to find how many speak English but not Japanese.

Then, we concentrate in the set of students that speak english: it has a total of 18 students.

But this set also includes the students that speak both, that are 5 students.

Then, we can calculate the number of students that speak only english as:

$S_{only\text{ }eng}=S_{eng}-S_{eng\cup jap}=18-5=13$

Answer:

1) 5 students speak both languages

2) 13 students speak only english