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.


From the question we know P(colour=red)=0.2,
but we also know that P(colour=red)=rn,
where r is the number of red beans and n is the total number of beans.
So r=n5 (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 yb=57,

so y=5b7 (3).

Substituting equations (1) and (3) into (2) we get n5+5b7+b=n.

b=7n15 (4),

and y=n3 (5).

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



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


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