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Pair of Linear Equations in Two Variables
NCERT Exemplar Ch 03
CBSE_NCERT_EXEMPLAR_CH03
Grade 10

Question:

A fraction becomes $\dfrac{9}{11}$ if $2$ is added to both the numerator and the denominator. If $3$ is added to both the numerator and the denominator, it becomes $\dfrac{5}{6}$. Find the fraction.

Step-by-Step Solution

Key Concept: Let fraction be $\dfrac{x}{y}$. Form linear equations $\dfrac{x+2}{y+2} = \dfrac{9}{11}$ and $\dfrac{x+3}{y+3} = \dfrac{5}{6}$.
Stepwise Solution:

Condition 1: $11(x + 2) = 9(y + 2) \Rightarrow 11x + 22 = 9y + 18 \Rightarrow 11x - 9y = -4$ -- (eq 1). [1.0 Mark]

Condition 2: $6(x + 3) = 5(y + 3) \Rightarrow 6x + 18 = 5y + 15 \Rightarrow 6x - 5y = -3$ -- (eq 2). [1.0 Mark]

From eq 2: $x = \dfrac{5y - 3}{6}$. Substitute into eq 1: $11\left(\dfrac{5y - 3}{6}\right) - 9y = -4 \Rightarrow 55y - 33 - 54y = -24 \Rightarrow y = 9$.
Then $x = \dfrac{45 - 3}{6} = 7$. Required fraction is $\dfrac{7}{9}$. [1.0 Mark]

Marking Scheme:

• Forming first linear equation $11x - 9y = -4$: 1.0 Mark
• Forming second linear equation $6x - 5y = -3$: 1.0 Mark
• Solving for $x = 7, y = 9$ and stating fraction $7/9$: 1.0 Mark

Correct Answer:
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