SOLUTION: The smaller of two numbers is two-thirds of the larger, and the sum of their reciprocals is 1/6. What are the numbers?

Question 229118: The smaller of two numbers is two-thirds of the larger, and the sum of their reciprocals is 1/6. What are the numbers?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
S = smaller number
L = larger number

S = 2L/3 (equation 1)

1/S + 1/L = 1/6 (equation 2)

Multiply both sides of equation 2 by S*L.

1/S + 1/L = 1/6 becomes:

L + S = (S*L)/6

Multiply both sides of this equation by 6 to get:

6L + 6S = S*L

Replace S with (2L)/3 to get:

6L + 6*(2L)/3 = L*2L/3

Remove Parentheses to get:

6L + 12L/3 = 2L^2/3

Multiply both sides of equation by 3 to get:

18L + 12L = 2L^2

Combine like terms to get:

30L = 2L^2

Divide both sides of equation by 2 to get:

15L = L^2

Subttract 15L from both sides of equation to get:

L^2 - 15L = 0

Factor L to get:

L*(L-15) = 0

Solve for L to get:

L = 0
or:
L = 15

Substitute these in original equations to get:

S = 2L/3 becomes:

S = 2*0/3 = 0
or:
S = 2*15/3 = 10

When S = 0 and L = 0, then 1/S + 1/L = 1/6 becomes:

1/0 + 1/0 = 1/6 which is false so L = 0 is not a valid answer and can be rejected.

When S = 10 and L = 15, then 1/10 + 1/15 = 1/6.

Multiply both sides of of this equation by 30 to get:

3 + 2 = 5 which is true so L = 15 is good.

Your answer is L = 15 and S = 10.

The larger number is 15 and the smaller number is 10.

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