Question

If 1 is subtracted from the numerator and 2 is added to the denominator of

a fraction, the value obtained is 1/2. If 3 is added to its numerator and 2 is
subtracted from its denominator, the resulting value is 1 1/4. Find the fraction.

Answer 1

Let the fraction be represented by "a/b".

According to the problem statement, we can create two equations:

Equation 1: (a-1)/(b+2) = 1/2

Equation 2: (a+3)/(b-2) = 1 1/4 = 5/4

We can simplify Equation 1 by cross-multiplying to get:

2(a-1) = b+2

2a - 4 = b + 2

2a = b + 6 --- (Equation A)

Similarly, we can simplify Equation 2 by converting 1 1/4 to an improper fraction:

5/4 = 1 + 1/4 = 4/4 + 1/4 = (4+1)/4 = 5/4

So, Equation 2 becomes:

5(a+3) = 4(b-2)

5a + 15 = 4b - 8

5a = 4b - 23 --- (Equation B)

Now, we have two equations (Equation A and B) with two unknowns (a and b). We can solve for one variable in terms of the other and substitute in either equation to find the value of the other variable. Here, we will solve for 'b' in Equation A:

2a = b + 6

b = 2a - 6

Substitute this value of 'b' in Equation B:

5a = 4b - 23

5a = 4(2a - 6) - 23

5a = 8a - 47

3a = 47

a = 47/3

Substitute this value of 'a' in Equation A to find 'b':

2a = b + 6

2(47/3) = b + 6

94/3 - 6 = b

b = 76/3

Therefore, the fraction is (a/b) = 47/76.