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Math Help - Venn Diagrams

  1. #1
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    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?
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  2. #2
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    Hello, wahhdoe!

    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 . . .


    . . \begin{array}{c||c|c||c|}<br />
& \text{Cheese} & \sim\text{Cheese} & \text{Total} \\ \hline\hline<br />
\text{Salted} & & & 48 \\ \hline \sim\text{Salted} & & & 27 \\ \hline\hline<br />
\text{Total} & 35 & 40 & 75\\ \hline \end{array}



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

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



    Reading across, we can fill in the empty cells:

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



    Reading down, we have two equations:

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



    The chart becomes:

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


    Now you can answer the question.

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