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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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