SOLUTION: A basket contains k red and 7 green apples, where k is a constant. Two apples are picked randomly from the basket, one after the other without replacement. If the probability of pi

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Question 1210446: A basket contains k red and 7 green apples, where k is a constant. Two apples are picked randomly from the basket, one after the other without replacement. If the probability of picking two green apples is 7/12, find the:
(a) number of red apples in the basket
(b) total number of apples in the basket
Answer by ikleyn(53354) About Me  (Show Source):
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A basket contains k red and 7 green apples, where k is a constant. Two apples are picked randomly from the basket,
one after the other without replacement. If the probability of picking two green apples is 7/12, find the:
(a) number of red apples in the basket
(b) total number of apples in the basket
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Formula for the probability is  P = %287%2F%28k%2B7%29%29%2A%286%2F%28k%2B6%29%29,

so we have this equation

    
    %287%2F%28k%2B7%29%29%2A%286%2F%28k%2B6%29%29 = 7%2F12.


Cancel common factor 7 in both sides.  You will get

    %281%2F%28k%2B7%29%29%2A%286%2F%28k%2B6%29%29 = 1%2F12.


Cross-multiply to get

    (k+7)*(k+6) = 72.


Now you have the product of two consecutive integer numbers in the left side,
and this product equals 72.


For anyone who knows the multiplication table, it is clear that the product is 8*9 = 72,
so the lesser factor is 8

    k+6 = 8,

    k = 8-6 =2.


ANSWER.  (a) the number of green apples in the basket originally is 2;.

         (b) the total number of apples in the basket originally is 7+2 = 9.

Solved.