a fraction is such that the denominator exceeds the numerator by 3.if both the numerator and numerator are reduced by 2,the fraction is decreased by 1/8.find the fraction
just put the words into math:
Original fraction: x/(x+3)
(x-2)/(x+3-2) = x/(x+3) - 1/8
Now just solve for x
X =5
Let's say the fraction is x/y, where y is greater than x and y = x + 3.
According to the given condition, if both the numerator and denominator are reduced by 2, the resulting fraction is decreased by 1/8.
Reducing the numerator by 2: (x - 2)
Reducing the denominator by 2: (y - 2) = (x + 3 - 2) = (x + 1)
The new fraction is: (x - 2) / (x + 1)
According to the given condition, this new fraction is decreased by 1/8:
(x/y) - (x - 2) / (x + 1) = 1/8
Now, let's solve the equation for x:
x/y - (x - 2) / (x + 1) = 1/8
Multiplying each term by 8(y + 1) to eliminate the fractions:
8(x + 1) - 8(x - 2) = y + 1
8x + 8 - 8x + 16 = y + 1
24 = y + 1
y = 24 - 1
y = 23
We know that y = x + 3, so:
23 = x + 3
x = 23 - 3
x = 20
Therefore, the fraction is 20/23.
Let's solve this step by step.
Step 1: Setting up the equation
Let's assume the numerator of the fraction is N and the denominator is D. Based on the given information, we can set up the following equation:
D = N + 3
Step 2: Solving the reduced fraction equation
When we reduce the numerator and denominator by 2, the new fraction is decreased by 1/8. We can write this as an equation:
(N - 2) / (D - 2) = (N / D) - 1/8
Step 3: Substitute the value of D from Step 1
Since we know that D = N + 3, we can substitute this value into the equation from Step 2:
(N - 2) / ((N + 3) - 2) = (N / (N + 3)) - 1/8
Step 4: Simplify and solve the equation
Let's simplify the equation by removing the parentheses and clearing the fractions:
(N - 2) / (N + 1) = (N / (N + 3)) - 1/8
(N - 2)(N + 3) = N(N + 1) - (N + 1)/8
(N^2 + N - 6) = (N^2 + N) - (N + 1)/8
Simplifying further:
N^2 + N - 6 = N^2 + N - (N + 1)/8
The N^2 and N terms cancel out:
-6 = - (N + 1)/8
To clear the fraction, we multiply both sides by 8:
-48 = - (N + 1)
Now, we can solve for N by multiplying both sides by -1 and simplifying:
48 = N + 1
47 = N
Step 5: Find the denominator
Using the value of N we found (N = 47), we can substitute it back into the equation from Step 1 to find the value of D:
D = N + 3
D = 47 + 3
D = 50
Therefore, the fraction is 47/50.