# Venn Diagrams

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• Apr 23rd 2010, 05:46 AM
wahhdoe
Venn Diagrams
I have a question with 3 parts, I have done the first 2 parts but i cannot make sense of the last part and was wondering if someone could please help me?

Q: 75 pupils, 48 pupils like ready salted, 35 like cheese & onion

i) if 7 pupils do not like either flavor, how many like both?
A: 15

ii) *what is the greatest possible number of pupils who could like both?
A: 35

**what is the smallest possible number of pupils who could like both?
A: 8

Now this is the part that i cannot figure out:

The number of pupils who like both is twice the number who do not like either;

iii) what fraction of the pupils who like ready salted do not like cheese & onion?
• Apr 23rd 2010, 07:35 AM
Soroban
Hello, wahhdoe!

Quote:

75 pupils: 48 pupils like ready salted, 35 like cheese & onion.

i) If 7 pupils do not like either flavor, how many like both?
. . . Answer: 15

iia) What is the greatest possible number of pupils who could like both?
. . . Answer: 35

iib) What is the smallest possible number of pupils who could like both?
. . . Answer: 8

All correct . . . Good work!

Now this is the part that i cannot figure out:

The number of pupils who like both is twice the number who do not like either;

iii) What fraction of the pupils who like ready salted do not like cheese & onion?

I often use a chart instead of a Venn diagram . . .

. . $\displaystyle \begin{array}{c||c|c||c|} & \text{Cheese} & \sim\text{Cheese} & \text{Total} \\ \hline\hline \text{Salted} & & & 48 \\ \hline \sim\text{Salted} & & & 27 \\ \hline\hline \text{Total} & 35 & 40 & 75\\ \hline \end{array}$

We are told: .$\displaystyle \text{(both)} \;=\;2 \times \text{(neither)}$

. . $\displaystyle \begin{array}{c||c|c||c|} & \text{Cheese} & \sim\text{Cheese} & \text{Total} \\ \hline\hline \text{Salted} & {\color{red}2x} & & 48 \\ \hline \sim\text{Salted} & & {\color{red}x} & 27 \\ \hline\hline \text{Total} & 35 & 40 & 75\\ \hline \end{array}$

Reading across, we can fill in the empty cells:

. . $\displaystyle \begin{array}{c||c|c||c|} & \text{Cheese} & \sim\text{Cheese} & \text{Total} \\ \hline\hline \text{Salted} & 2x & {\color{red}48-2x} & 48 \\ \hline \sim\text{Salted} & {\color{red}27-x} & x & 27 \\ \hline\hline \text{Total} & 35 & 40 & 75\\ \hline \end{array}$

Reading down, we have two equations:

. . $\displaystyle \begin{Bmatrix}2x + (27-x) &=& 35 \\ (48-2x) + x &=& 40 \end{Bmatrix} \quad\Rightarrow\quad x \:=\:8$

The chart becomes:

. . $\displaystyle \begin{array}{c||c|c||c|} & \text{Cheese} & \sim\text{Cheese} & \text{Total} \\ \hline\hline \text{Salted} & 16 & 32 & 48 \\ \hline \sim\text{Salted} & 19 & 8 & 27 \\ \hline\hline \text{Total} & 35 & 40 & 75\\ \hline \end{array}$

Now you can answer the question.