The value of a certain fraction becomes 1/5 if one is added to its numerator. If one is taken from its denominator, its value becomes 1/7. Find the fraction by setting up a pair of simultaneous equations and solving them
To find the fraction, let's denote the numerator as "x" and the denominator as "y". We can set up the following pair of equations based on the given information:
Equation 1: (x + 1) / y = 1/5
Equation 2: x / (y - 1) = 1/7
Now, let's solve these equations simultaneously to find the values of x and y.
Step 1: Simplify Equation 1
Multiply both sides of Equation 1 by 5y to eliminate the fractions:
5y(x + 1) / y = 5y(1/5)
5x + 5y = y
Step 2: Simplify Equation 2
Multiply both sides of Equation 2 by 7(y - 1) to eliminate the fractions:
7(y - 1)(x / (y - 1)) = 7(y - 1)(1/7)
7x = y - 1
Step 3: Solve the simultaneous equations
We now have a system of two linear equations:
5x + 5y = y
7x = y - 1
To solve this system, we can use substitution or elimination method. Let's use the elimination method by multiplying both sides of Equation 1 by 7:
35x + 35y = 7y
7x = y - 1
Now, we can subtract the second equation from the first equation to eliminate "y":
35x + 35y - 7x = 7y - (y - 1)
28x + 35y - 7y = 6y + 1
28x + 28y = 6y + 1
Step 4: Continue solving for x and y
Rearrange the equation:
28x = 6y - 28y + 1
28x = -22y + 1
x = (-22y + 1) / 28
Now substitute the value of x into one of the original equations, let's use Equation 1:
(x + 1) / y = 1/5
[(-22y + 1) / 28 + 1] / y = 1/5
Simplify:
[(-22y + 1 + 28) / 28] / y = 1/5
[(-22y + 29) / 28] / y = 1/5
(-22y + 29) / 28y = 1/5
Cross-multiply:
5(-22y + 29) = 28y
Expand the equation:
-110y + 145 = 28y
Combine like terms:
110y + 28y = 145
Simplify:
138y = 145
Solve for y:
y = 145 / 138
y = 1.05
Now substitute the value of y back into x = (-22y + 1) / 28 to find x:
x = (-22(1.05) + 1) / 28
x = (-23.1 + 1) / 28
x = -22.1 / 28
x = -0.79
Therefore, the fraction would be -0.79 / 1.05, which can be simplified as -79/105.
(n+1)/d = 1/5
n/(d-1) = 1/7
5(n+1) = d
7n = d-1
5n - d = -5
7n - d = -1
2n = 4
n=2
d=15
Check:
n/d = 2/15
3/15 = 1/5
2/14 = 1/7