# Region method on Venn Diagram

There are $34$ students in a group that there are the people who speak german and french and $14$ people who speak minimum a language (French or German). $23$ people who speak maximum a language.

• How many students who speak minimum a language are there?

So, how do we solve this question with region method?

Let's give numbers for the regions while applying it on the Venn Diagram.

I mean like

$$\sum_{i=i} r(i) = 34$$

$$r(1) + r(2) + r(3) + r(4) = 34$$

$$r(1) + r(2) = 14$$

Hope you got what i mean.

Regards

## 1 Answer

I am assuming references of regions as: $r(1)$ to only French speaking students, $r(2)$ to only German speaking students, $r(3)$ to both language speakers and $r(4)$ to neither language speakers.

$$\sum_{i=1} r(i) = 34$$

$$r(1) + r(2) + r(3) + r(4) = 34$$

$$r(1) + r(2) +r(3)= 14$$ (since minimum one language means at least one language)

$$r(4) + r(1) +r(2)= 23$$ (since maximum one language means at most one language)

From the above equations, we get-

$r(3) = 11$ (students that speak both languages)

$r(4)=20$ (students that speak neither language)

$r(1)+r(2)=3$ (students that speak only one language).

I am not sure what the question is asking but this is all that can be ascertained by what is given in the question (given that I understood it correctly).

Calculations-

$$r(1) + r(2) +r(3)= 14$$

Adding $r(4)$ to both sides,

$$r(1) + r(2) +r(3)+r(4)= 14+r(4)$$

$$34=14+r(4)$$

$$r(4)=20$$

$$r(4) + r(1) +r(2)= 23$$

Adding $r(3)$ to both sides,

$$r(1) + r(2) +r(3)+r(4)= 23+r(3)$$

$$34=23+r(3)$$

$$r(3)=11$$

Replacing $r(3)$ and $r(4)$ in first equation by their values,

$$r(1) + r(2) + 11 + 20 = 34$$

$$r(1) + r(2) = 3$$

• Also, don't call it region method. It's just a way of getting it done without lots of notations and to avoid confusion. :P – Maadhav Gupta Nov 3 '17 at 22:16
• What is this method called by? Also, why didn't you do it like your last answer? – Cargobob Nov 4 '17 at 8:11
• @Korrigi It's not a recognised method. It's just a way to get the question solved. You probably won't find it anywhere. This just simplifies calculations into parts. Formally, in set theory, the union-intersection notation is used. You can use this 'method' to check your answer or maybe to answer mcqs without much hassle, but formally it is not accepted, maybe you can use it to justify your answer but it's generally neither recognised nor accepted. I just use it because it is easy to understand. – Maadhav Gupta Nov 4 '17 at 8:12
• @Korrigi I just didn't make the Venn Diagram here, as the regions are few here. – Maadhav Gupta Nov 4 '17 at 8:13
• However, you didn't do it like your last answer. I didn't mean that. – Cargobob Nov 4 '17 at 8:13