the denominator of a fraction exceeds the numerator by 7, if the numerator is increased by 3, and the denominator is increased by 5, the resulting fraction equals 1/2. Find the original fraction.
Let's break down the problem step by step:
Let the numerator of the fraction be x.
Therefore, the denominator of the fraction is x + 7.
According to the second part of the problem:
If we increase the numerator by 3, it becomes x + 3.
If we increase the denominator by 5, it becomes x + 7 + 5, which simplifies to x + 12.
The resulting fraction is 1/2, so we can set up the following equation:
(x + 3) / (x + 12) = 1/2
To solve this equation, we can cross multiply:
2(x + 3) = x + 12
Expanding the equation:
2x + 6 = x + 12
Subtracting x from both sides:
2x - x + 6 = 12
x + 6 = 12
Subtracting 6 from both sides:
x = 6
So, the numerator of the original fraction is 6.
The denominator is x + 7, which is 6 + 7 = 13.
Therefore, the original fraction is 6/13.
To find the original fraction, let's first set up a system of equations based on the given information.
Let's assume the numerator of the original fraction is "x" and the denominator is "y".
From the first statement, we know that the denominator exceeds the numerator by 7:
y = x + 7 (Equation 1)
From the second statement, we know that if the numerator is increased by 3 and the denominator is increased by 5, the resulting fraction is 1/2:
(x + 3)/(y + 5) = 1/2 (Equation 2)
To solve the system of equations, we can substitute Equation 1 into Equation 2:
(x + 3)/(x + 7 + 5) = 1/2
Simplifying this equation:
(x + 3)/(x + 12) = 1/2
Cross multiplying:
2(x + 3) = x + 12
2x + 6 = x + 12
Subtracting x from both sides:
2x - x + 6 = 12
x + 6 = 12
Subtracting 6 from both sides:
x = 12 - 6
x = 6
Now that we have found the value of x, we can substitute it back into Equation 1 to find the value of y:
y = x + 7
y = 6 + 7
y = 13
Therefore, the original fraction is 6/13.
so?
original fraction is x/(x+7)
when changed,
(x+3)/(x+7+5) = 1/2
2(x+3) = x+12
2x+6 = x+12
x = 6
original fraction is 6/13
new fraction is 9/18 = 1/2