A positive integer is 5 less than another.

Question

If 4 times the reciprocal of the smaller integer is subtracted from 2 times the reciprocal of the larger integer, then the result is −3/5. Find all pairs of integers that satisfy this condition.

Answer 1

 0 < X = Y-5        so X is smaller than Y, with both positive

 

2(1/y) - 4(1/x) = -3/5

 

2(1/y) - 4(1/(y-5)) = -3/5

 

 

 

The common denominator is y(y-5)

 

2(y-5) - 4y = (-3/5)(y)(y-5)  <--- multiplies both sides by the common denominator

 

2y - 10 - 4y = (-3/5)(y)(y-5) <--- distributive

 

-10 - 2y =  (-3/5)(y)(y-5)  <--- combines like terms

 

2y+10 = (3/5)(y)(y-5) <--- multiplies both sides by -1

 

2y + 10 = (3/5)(y^2 - 5y) <--- distributes the y

 

10y + 50 = 3(y^2 - 5y) <--- multiplies both sides by 5 to clear away the fraction

 

10y + 50 = 3y^2 - 15y <--- distributes the 3

 

0 = 3y^2 - 15y - 10y - 50 <--- everything to right side

 

0 = 3y^2 - 25y - 50

 

0 = ( 3y    +   5 )(y    -   10   )

 

3y+5 = 0 results in a negative fraction

 

y-10 = 0 ---> y = 10

 

The integers are 5 and 10

 

check: 2(1/10) - 4(1/5) = 1/5 - 4/5 = -3/5  <--- 4 times the reciprocal of the smaller 

                                                                                   subtracted from

                                                                       2 times the reciprocal of the larger

                                                                                          is -3/5