Question

if 2 is added to both the numerator and denominator then the new fraction becomes 9/11 if 3 is added to both numerator and denominator , it becomes 5/6 find the fraction​

Answer 1

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✦ Required Answer:

♦️ GiveN:

• When 2 is added to both numerator and denominator, fraction becomes 9/11
• When 3 is added to both numerator and denominator, fraction becomes 5/6

♦️ To FinD:

• The original fraction....?

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✦ How to solve ?

The question is mentioning a fraction, so it would have numerator and denominator which are unknown, So we can take two variables ( Let x and y) for the numerator and denominator respectively. Then, we will frame two equations and solving simultaneously.

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✦ Solution:

We have consider numerator as x and denominator as y

Then the fraction would be x/y

Case-1: When 2 is added,

• New numerator = x + 2
• New Denominator = y + 2

$\large{\underline{\red{\rm{According \:to \:question}}}}$

$\large{ \rm{ \longrightarrow \: \frac{x + 2}{y + 2} = \frac{9}{11}}} \\ \\ \dag{\rm{ \green{ \: cross \: multiplying}}} \\ \\ \large{ \rm{ \longrightarrow \: 11(x + 2) = 9(y + 2)}} \\ \\ \large{ \rm{ \longrightarrow \: 11x + 22 = 9y + 18}} \\ \\ \large{ \rm{ \longrightarrow \: 11x - 9y = 18 - 22}} \\ \\ \large{ \rm{ \longrightarrow \: 11x - 9y = - 4..............(1)}}$

Case-2: When 3 is added,

• New numerator = x + 3
• New denominator = y + 3

$\large{\underline{\red{\rm{According \:to \:question}}}}$

$\large{ \rm{ \longrightarrow \: \frac{x + 3}{y + 3} = \frac{5}{6}}} \\ \\ \large{ \rm{ \longrightarrow \: 6(x + 3) = 5(y + 3)}} \\ \\ \large{ \rm{ \longrightarrow \: 6x + 18 = 5y + 15}} \\ \\ \large{ \rm{ \longrightarrow \: 6x - 5y = 15 - 18}} \\ \\ \large{ \rm{ \longrightarrow \: 6x - 5y = - 3.............(2)}}$

From (1), we can write it as

$\large{ \rm{ \longrightarrow \: 11x = - 4 + 9y}} \\ \\ \large{ \rm{ \longrightarrow \: x = \frac{9y - 4}{11} }}$

Substitute x in eq.(2)

$\large{ \rm{ \longrightarrow \: 6( \frac{9y - 4}{11}) - 5y = - 3}} \\ \\ \large{ \rm{ \longrightarrow \: \frac{54y - 24}{11} - 5y = - 3}} \\ \\ \large{ \rm{ \longrightarrow \: \frac{54y - 24 - 55y}{11} = - 3}} \\ \\ \large{ \rm{ \longrightarrow \: \frac{ - y - 24}{11} = - 3}} \\ \\ \rm{ \dag{ \green{cross \: multiplying}}} \\ \\ \large{ \rm{ \longrightarrow \: - y - 24 = - 33}} \\ \\ \large{ \rm{ \longrightarrow \: - y = - 33 + 24}} \\ \\ \large{ \longrightarrow{ \boxed{ \rm{ \red{y = 9}}}}}$

Putting the value of y in eq.(1),

$\large{ \rm{ \longrightarrow \: x = \frac{9y - 4}{11}}} \\ \\ \large{ \rm{ \longrightarrow \: x = \frac{9(9) - 4}{11} }} \\ \\ \large{ \rm{ \longrightarrow \: x = \frac{77}{11}}} \\ \\ \large{ \longrightarrow{ \boxed{ \rm{ \red{x = 7}}}}}$

Hence, The fraction is $\Large{\rm{\frac{7}{9}}}$

$\large{ \rm{ \therefore{ \underline{ \green{Hence \: solved \: \dag}}}}}$

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Answer 2

hope it helps you mate ((: