MathMate attempted to help but did not show how the problem is answered. It only confused the problem. Could someone else please help? Thanks.
The denominator of a fraction is 4 more than the numerator. If both the denominator and the numerator are increased by 1, the resulting fraction equals 1/2. What is the original fraction?
**Need to show all work**
The fraction is x/(x+4), where
(x+1)/(x+5) = 1/2
That can be rewritten
x+5 = 2(x+1)
which reduces to
x = 3
Therefore the fraction is 3/7
I used trial and error to come up with 3/7.
I started with 1/5 and 2/6, which of course doesn't work.
To solve this problem, let's break it down into steps:
Step 1: Define the variables
Let's assume the numerator is represented by the variable 'n' and the denominator is represented by the variable 'd'.
Step 2: Translate the given information into equations
From the problem statement, we are given two pieces of information:
1) The denominator is 4 more than the numerator, so we can write the equation as:
d = n + 4
2) If both the numerator and denominator are increased by 1, the resulting fraction is 1/2. We can write this as:
(n + 1) / (d + 1) = 1/2
Step 3: Solve the equations
We now have two equations in two variables. We can substitute the value of 'd' from equation 1) into equation 2) to solve for 'n'.
Substituting d = n + 4 in equation 2):
(n + 1) / ((n + 4) + 1) = 1/2
(n + 1) / (n + 5) = 1/2
Cross multiplying we get:
2(n + 1) = n + 5
Expanding the equation:
2n + 2 = n + 5
Moving all variables to one side:
2n - n = 5 - 2
n = 3
Step 4: Find the value of 'd'
Using equation 1):
d = n + 4
d = 3 + 4
d = 7
Step 5: Check the solution
We can substitute the values of n and d in the second equation to check if it indeed gives us the correct answer.
(n + 1) / (d + 1) = 1/2
(3 + 1) / (7 + 1) = 4/8
4/8 = 1/2 (which is true)
Step 6: Determine the original fraction
The original fraction is n/d = 3/7.