What is the smallest number of beans?

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A bag contains some red, some yellow and some blue beans.
The probability of picking a red bean at random is 0.2.
It is known that the ratio of yellow to blue beans is $5:7$.
What is the smallest number of beans?

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Solution

From the question we know $P(\mathrm{colour}=\mathrm{red})=0.2$,
but we also know that $P(\mathrm{colour}=\mathrm{red})=\frac{r}{n}$,
where $r$ is the number of red beans and $n$ is the total number of beans.
So $r=\frac{n}{5}$ (1).

We also know that $r+y+b=n$ (2),
where $y$ is the number of yellow beads and $b$ is the number of blues.

From the ratio given $\frac{y}{b}=\frac{5}{7}$,

so $y=\frac{5b}{7}$ (3).

Substituting equations (1) and (3) into (2) we get $\frac{n}{5}+\frac{5b}{7}+b=n$.

$b=\frac{7n}{15}$ (4),

and $y=\frac{n}{3}$ (5).

Using (1), (4) and (5) to create a ratio,

$\frac{n}{5}:\frac{n}{3}:\frac{7n}{15}$

$3:5:7$.

So, the smallest number of beans is $3+5+7=15$.

-oOo-

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