If 1 is added to both numerator and denominator of a fraction, the fraction becomes 1/2. If 8 is added to both the fraction becomes 2/3. What is the fraction
2(n + 1) = d + 1 ... 2n + 2 = d + 1
3(n + 8) = 2(d + 8) ... 3n + 24 = 2d + 16
solve the system of equations for n and d
To solve this problem, let's start by setting up an equation for the given information.
Let's say the original fraction is represented by 'x/y', where 'x' is the numerator and 'y' is the denominator.
According to the first condition, if 1 is added to both the numerator and denominator of the original fraction, the new fraction becomes 1/2. Mathematically, this can be written as:
(x + 1) / (y + 1) = 1/2
Simplifying this equation, we get:
2(x + 1) = y + 1
2x + 2 = y + 1
2x - y = -1 --(Equation 1)
Now, let's move on to the second condition. We are given that if 8 is added to both the numerator and denominator of the original fraction, the new fraction becomes 2/3. In equation form, this can be written as:
(x + 8) / (y + 8) = 2/3
Simplifying this equation, we get:
3(x + 8) = 2(y + 8)
3x + 24 = 2y + 16
3x - 2y = -8 --(Equation 2)
Now, we have a system of equations (Equation 1 and Equation 2) that we can solve to find the values of 'x' and 'y'.
Multiplying Equation 1 by 3 and Equation 2 by 2, we get:
6x - 3y = -3
6x - 4y = -16
By subtracting the first equation from the second, we can eliminate 'x':
(6x - 4y) - (6x - 3y) = -16 - (-3)
-y = -13
y = 13
Substituting the value of 'y' back into Equation 1:
2x - 13 = -1
2x = 12
x = 6
Therefore, the fraction is 6/13.