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.

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

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)