The numerator of a fraction is 7 less than the denominator if three is added to the numerator and 9 is subtracted from the denominator the new fraction is equal to 3/2. Find the original fraction.
if the numerator is x, the original fraction is
x/(x+7)
after modification,
(x+3)/(x-2) = 3/2
2(x+3) = 3(x-2)
2x+6 = 3x-6
x = 12
original fraction: 12/19
modified fraction: 15/10 = 3/2
To solve this problem, we can start by setting up equations based on the given information.
Let's assume the original fraction is x/y, where x is the numerator and y is the denominator.
According to the problem, the numerator of the fraction is 7 less than the denominator. This can be expressed as:
x = y - 7
If three is added to the numerator and nine is subtracted from the denominator, the new fraction is equal to 3/2. This can be expressed as:
(x + 3) / (y - 9) = 3/2
Now, we have a system of two equations:
x = y - 7
(x + 3) / (y - 9) = 3/2
To solve this system, we can use substitution. Substitute the value of x from the first equation into the second equation:
[(y - 7) + 3] / (y - 9) = 3/2
Simplify the equation:
(y - 4) / (y - 9) = 3/2
To get rid of the denominators, we can cross-multiply:
2(y - 4) = 3(y - 9)
Expand and simplify:
2y - 8 = 3y - 27
Bring like terms to one side:
2y - 3y = -27 + 8
-y = -19
Multiply both sides by -1 to isolate y:
y = 19
Now, substitute the value of y back into the first equation to find x:
x = y - 7
x = 19 - 7
x = 12
Therefore, the original fraction is 12/19.