Two Fractions have Denominators

Question

Set up a linear system and solve it using elimination to determine the two numerators.

Two fractions have denominators 3 and 4 and their sum is 17/2. If the numerators are switched, the sum is 3/2.

Answer 1

The given information translates into

 

N/3 + M/4 = 17/2

N/4 + M/3 = 3/2

 

where N and M are integers.  Solving by Cramer's rule gives

 

N ≅ 50.6

M ≅ -33.4

 

The values are not integers, and hence the problem, as currently stated, has no solution.

 

If the problem is restated as simply solving the system below for x and y, with no condition demanding that x and y be integers, then the problem has a solution.

 

x/3 + y/4 = 17/2
x/4 + y/3 = 3/2

Answer 2

Two fractions have denominators 3 and 4 and their sum is 17/2. If the numerators are switched, the sum is 3/2.

 

Let the fractions be a/3 and b/4; then a/3 + b/4 = 17/2, with a & b assumed to be integers.

Also, switching numerators, we know  b/3 + a/4  =3/2.

To solve this system, multiply each by 12, throughout, and then we have this system:

 4a + 3b = 102

 4b + 3a = 18.

Now solve this in any of the usual ways.  

 

Warning: if either of a or b is not an integer, then the problem is defective!