SOLUTION: The difference between two integers is 5. If the reciprocal of the smaller is added to twice the reciprocal of the larger the result is 23/66. Find the two integers.

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Question 1043768: The difference between two integers is 5. If the reciprocal of the smaller is added to twice the reciprocal of the larger the result is 23/66. Find the two integers.
Found 3 solutions by josgarithmetic, Edwin McCravy, MathTherapy:
Answer by josgarithmetic(39631)   (Show Source): You can put this solution on YOUR website!

Answer by Edwin McCravy(20066)   (Show Source): You can put this solution on YOUR website!


Let y = the larger integer
Let x = the smaller integer





Substitute (5+x) for y in







LCD = 66x(5+x)

Multiply each numerator by every factor of
the denominator which is not under it:









Subtract 115x from both sides



Subtract 23x2 from both sides

to get 0 on the right side



Arrange left side in descending order:



Multiply through by -1:



There may be a way to factor that but
when the numbers are that big, it's
easier to use the quadratic formula,
especially if you have a calculator:











Using the +







Using the -







The problem calls for an integer.  That's not
an integer, so we can discard that value of x.

So x = 6

Substitute into







Answers: Larger integer = 11, Smaller integer = 6

Edwin
Answer by MathTherapy(10559)   (Show Source): You can put this solution on YOUR website!




The difference between two integers is 5. If the reciprocal of the smaller is added to twice the reciprocal of the larger the result is 23/66. Find the two integers.
Let the larger be L

Then smaller = L – 5

We then get: _______

66L + 2(66)(L – 5) = 23L(L – 5) -------- Multiplying by LCD, 66L(L – 5)











23L(L – 11) – 60(L – 11) = 0

(L – 11)(23L – 60) = 0

L, or   	   OR        L =  (ignore)

       

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