Question

In a group of$950$ persons, $750$ can speak Hindi and$460$ can speak English. Find:

(i) How many can speak both Hindi and English?

(ii) How many can speak Hindi only?

(iii) How many can speak English only?

Answer 1

Step 1: Find the number of people that can speak both Hindi and English.

Given: $\mathrm{n}\left(\mathrm{H}\right)=750,\mathrm{n}\left(\mathrm{E}\right)=460$ and the total number of people $\mathrm{n}(\mathrm{H}\cup \mathrm{E})=950$

Let, $\mathrm{n}\left(\mathrm{H}\right)$ denote the number of people who can speak Hindi , $\mathrm{n}\left(\mathrm{E}\right)$ denote the number of people who can speak English and $\mathrm{n}(\mathrm{H}\cap \mathrm{E})$ denote the number of people who can speak both Hindi and English.

We know that,

$$\begin{gathered} n(\mathrm{H}\cap \mathrm{E})=\mathrm{n}\left(\mathrm{H}\right)+\mathrm{n}\left(\mathrm{E}\right)-\mathrm{n}(\mathrm{H}\cup \mathrm{E}) \\ =750+460-950 \\ =1210-950 \\ =260 \\ \therefore \mathrm{n}(\mathrm{H}\cap \mathrm{E})=260 \end{gathered}$$

Therefore, $260$ people can speak both Hindi and English.

Step 2: Find the number of people that can speak Hindi only.

This can be found by subtracting the number of people who speak Hindi and the number of people who speak both Hindi and English language.
So, the number of people who speak Hindi only are $=\mathrm{n}\left(\mathrm{H}\right)-\mathrm{n}(\mathrm{H}\cap \mathrm{E})$

$$\begin{gathered} =750-260 \\ =490 \end{gathered}$$

Therefore, $490$ people can speak Hindi only.

Step 3: Find the number of people that can speak English only.

This can be found by subtracting the number of people who speak English only and the number of people who speak both Hindi and English language.
So, the number of people who speak English only are $=\mathrm{n}\left(\mathrm{E}\right)-\mathrm{n}(\mathrm{H}\cap \mathrm{E})$

$$\begin{gathered} =460-260 \\ =200 \end{gathered}$$

Therefore, $200$ people can speak English only.

Hence, $260$ people can speak both Hindi and English, $490$ people can speak Hindi only and $200$ people can speak English only.