When the numerator and denominator of a given fraction are increased by 2 and 1,...
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When the numerator and denominator of a given fraction are increased by 2 and 1, respectively, then the resultant fraction becomes 2/3. If the denominator is reduced by 1 and the numerator is raised by 1, then the resulting fraction becomes 3/4. What was the initial fraction?
Show answer & explanation
Let the initial fraction be x/y. From the first condition: (x+2)/(y+1) = 2/3, which gives 3x - 2y = -2. From the second condition: (x+1)/(y-1) = 3/4, which gives 4x - 3y = -1. Solving this system of linear equations yields x = 2 and y = 5, so the initial fraction is 2/5.
Step-by-step Derivation:
Let the initial fraction be x/y.
Condition 1: Numerator increased by 2, denominator increased by 1 → fraction becomes 2/3
(x + 2)/(y + 1) = 2/3
3(x + 2) = 2(y + 1)
3x + 6 = 2y + 2
3x - 2y = -4 ... (Equation 1)
Condition 2: Numerator increased by 1, denominator reduced by 1 → fraction becomes 3/4
(x + 1)/(y - 1) = 3/4
4(x + 1) = 3(y - 1)
4x + 4 = 3y - 3
4x - 3y = -7 ... (Equation 2)
Solving the system:
From Equation 1: 3x - 2y = -4 → Multiply by 4: 12x - 8y = -16
From Equation 2: 4x - 3y = -7 → Multiply by 3: 12x - 9y = -21
Subtract: (12x - 8y) - (12x - 9y) = -16 - (-21)
y = 5
Substitute y = 5 into Equation 1:
3x - 2(5) = -4
3x - 10 = -4
3x = 6
x = 2
Initial fraction = 2/5
Verification:
• Condition 1: (2+2)/(5+1) = 4/6 = 2/3 ✓
• Condition 2: (2+1)/(5-1) = 3/4 ✓