Question 350818
problem 1. the sum of the reciprocals of two numbers is 7. The larger reciprocal exceeds the  smaller by 7/3. Find the numbers.
let x be one number
y be the other number
1/x +  1/y =7
x+y/xy=7
x+y=7xy.......................1
..
1/x-1/y = 7/3
y-x / xy = 7/3
3(y-x)=7xy
3y-3x=7xy .....................2
multiply 1 by 3 and add to 2
3x+3y = 21xy
Add
6y=28xy
x= 3/14
...
14/3 +1/y =7
1/y = 7 - 14/3
1/y =7/3
y = 3/7

Reciprocal of x & y when added = 7

problem 2. if the numerator and the denominator of a certain fraction are both increased by 3, the  resulting fractions equals 2/3. if the numerator and the denominator are both  decreased by 2, the resulting fraction equals 1/2. determine the fraction.
let the fraction be x/y
x+3/y+3 = 2/3
3(x+3)=2(y+3)
3x+9=2y+6
3x-2y=-3.....................1
x-2  / y-2  = 1/2
2(x-2)=y-2
2x-4 = y-2
2x-y = 2......................2
multiply equation 1 by -2 & equation 2 by 3 
-6x+4y=6
6x-3y=6
y=12
plug value of y in equation 1
3x-24 =-3
3x=24-3
3x=21
x=7 
problem 3. a certain fraction has a value 3/4. If its numerator is decreased by 7 and its denominator increased by 4, the resulting fraction has the value of 1/2. find the original fraction. 

...
x/y = 3/4 
x=3y/4.................1

x-7  / y+4  = 1/2
2(x-7)=y+4
2x-14=y+4
2x-y=18................2
plug value of x in equation 2
2*(3y/4)-y =18
3y/2-y =18
(3y-2y )/ 2 =18
y = 2*18
y=36
plug value of y in equation 1
2x-36=18
2x=36+18
2x=54
x=27

....
C H E C K 

27/36 = 3/4
AAAh1