The sum of the numerator and denominator of a fraction is 3 less than twice the denomiator.If the numerator and denomiator are decreased by 1,the numerator becomes half the denominator. Determine the fraction
https://www.jiskha.com/display.cgi?id=1372233030
To determine the fraction, let's break down the problem step by step.
Let's assume the numerator of the fraction is represented by the variable "n" and the denominator is represented by the variable "d."
Step 1: Understand the given information.
- The sum of the numerator and denominator is 3 less than twice the denominator.
This can be mathematically represented as:
n + d = 2d - 3
- If both the numerator and denominator are decreased by 1, the numerator becomes half the denominator.
This can be represented as:
(n - 1) = (1/2)(d - 1)
Step 2: Set up a system of equations.
Based on the given information, we have two equations:
n + d = 2d - 3 (Equation 1)
(n - 1) = (1/2)(d - 1) (Equation 2)
Step 3: Solve the system of equations.
Let's solve the system of equations using substitution or elimination.
From Equation 1, we can rearrange it to isolate n:
n = 2d - 3 - d
n = d - 3 (Equation 3)
Substitute Equation 3 into Equation 2:
(d - 3 - 1) = (1/2)(d - 1)
(d - 4) = (1/2)(d - 1)
Multiply both sides of the equation by 2 to eliminate the fraction:
2(d - 4) = d - 1
2d - 8 = d - 1
Simplify the equation:
d = 7
Substitute the value of d into Equation 3 to solve for n:
n = 7 - 3
n = 4
Therefore, the numerator is 4 and the denominator is 7.
The fraction is 4/7.