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Kelly is getting some snacks for a party. a) (1)/(4) of her guests hate peanut butter. Of these, (2)/(3) also hate cola. What fraction of her guests hate both peanut butter and cola? b) (5)/(12) of the guests who don't hate peanut butter are vegetarians. What fraction of Kelly's guests are vegetarians who don't hate peanut butter

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Kelly is getting some snacks for a party.
a) (1)/(4) of her guests hate peanut butter. Of these, (2)/(3) also hate cola.
What fraction of her guests hate both peanut butter and cola?
b) (5)/(12) of the guests who don't hate peanut butter are vegetarians.
What fraction of Kelly's guests are vegetarians who don't hate peanut butter

Kelly is getting some snacks for a party. a) (1)/(4) of her guests hate peanut butter. Of these, (2)/(3) also hate cola. What fraction of her guests hate both peanut butter and cola? b) (5)/(12) of the guests who don't hate peanut butter are vegetarians. What fraction of Kelly's guests are vegetarians who don't hate peanut butter

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RalphElite · Tutor for 8 years

Answer

(a) The fraction of guests that hate both, peanut butter and cola, is \(\frac{1}{6}\). <br /><br />(b) The fraction of Kelly's guests that are vegetarians and do not hate peanut butter is \(\frac{5}{16}\).

Explain

## Step 1:<br />We obtain the fraction of Kelly's guests who hate both peanut butter and cola if \(\frac{1}{4}\) of her guests hate peanut butter, and of those, \(\frac{2}{3}\) also hate cola. <br /><br />We achieve this by multiplication because, mathematically, the "and" corresponds to multiplication when it comes to probabilities and fractions.<br /><br />### \(\frac{1}{4} \times \frac{2}{3}=\frac{1 \times 2}{4 \times 3} = \frac{2}{12} = \frac{1}{6}\) <br /><br />## Step 2:<br />Let's find the fraction of guests that are vegetarians and do not hate peanut butter. <br /><br />Here, \(\frac{5}{12}\) of her guests who don't hate peanut (that is one minus \(\frac{1}{4}\) the hateful guests or \(\frac{3}{4}\) are don't hate peanut butter) are vegetarians.<br /><br />So we multiply <br /><br />### \(\frac{3}{4} \times \frac{5}{12}=\frac{3 \times 5}{4 \times 12}= \frac{15}{48} = \frac{5}{16}\)
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