The sum of the numerator and denominator of a fraction is 3 less than twice the denominator. if the numerator and denominator are decreased by 1, the numerator becomes half the denominator. determine the fraction

General
1

the sum of the numerator and denominator of a fraction is 3 less than twice the denominator. if the numerator and denominator are decreased by 1, the numerator becomes half the denominator. determine the fraction.

2

The sum of the numerator and denominator of a fraction is 3 less than twice the denominator. If the numerator and denominator are decreased by 1, the numerator becomes half the denominator. Determine the fraction.

Answer:
To determine the fraction, let’s denote the numerator by ( N ) and the denominator by ( D ). We need to set up two equations based on the information provided.

  1. Formulate the First Equation:

    • The sum of the numerator and the denominator is 3 less than twice the denominator.
      This gives us the equation:
      N + D = 2D - 3
    • Simplify this equation:
      N + D = 2D - 3 \implies N = D - 3 \tag{1}

  2. Formulate the Second Equation:

    • If the numerator and denominator are decreased by 1, the numerator becomes half the denominator:
      N - 1 = \frac{1}{2} (D - 1)
    • Multiply both sides by 2 to clear the fraction:
      2(N - 1) = D - 1
    • Simplify this equation:
      2N - 2 = D - 1 \implies 2N - D = 1 \tag{2}

  3. Solve the System of Equations:

    • Substitute equation (1) into equation (2):
      2(D - 3) - D = 1
      • Simplify and solve for ( D ):
        2D - 6 - D = 1 \implies D - 6 = 1 \implies D = 7
    • Now, use equation (1) to find ( N ):
      N = D - 3 \implies N = 7 - 3 \implies N = 4

Final Answer:
The fraction is \frac{4}{7}