Problema Solution
you are given three integers whose sum is 16. The product of thes three numbers is 90. Find the sum of the reciprocals of these numbers.please provide the your sum as a fraction in simplified form
Answer provided by our tutors
Let 'x', 'y' and 'z' represent the given integers.
x + y + z = 16
xyz = 90
We need to find 1/x + 1/y + 1/x.
From x + y + z = 16 we have x<16, y<16 and z<16.
On the other hand the factors of 90 are:
90 = 3*3*2*5
90 written as a product of 3 factors is:
3*3*10 = 90 and also 3 + 3 + 10 = 16
3*6*5 = 90 and also 3 + 6 + 5 = 14
3*2*15 = 90 and also 3 + 2 + 15 = 20
9*5*2 = 90 and also 9 + 5 + 2 = 16
10*9*1 = 90 and also 10 + 9 + 1 = 20
For 3, 3 and 10 the sum of the reciprocals is:
1/3 + 1/3 + 1/10 = 2/3 + 1/10 = (2*10 + 1*3)/30 = 23/30
For 9, 5 and 2 the sum of the reciprocals is:
1/9 + 1/5 + 1/2 = (10+16+45)/90 = 71/90