Question

The denominator of a fraction is 3 less than the numerator.
Increasing the numerator by 4and decreasing the denominator by 3, we get the number 3
Find the fraction.​

Answer 1

Given:–

• The denominator of a fraction is 3 less than the numerator.
• Increasing the numerator by 4 and decreasing the denominator by 3, we get the number 3.

To Find:–

• What is the fraction ?

Solution:–

Let the fraction be x/y

• $\sf{\dfrac{Numerator}{Denominator} = \dfrac{x}{y}}$

Case...❶

✰ The denominator of a fraction is 3 less than the numerator.

$\bf{\longmapsto y = x - 3....1)}$

Case...❷

✰ Increasing the numerator by 4 and decreasing the denominator by 3, we get the number 3.

$\sf{\longmapsto \dfrac{x + 4}{y -3} = 3 }$

$\sf{\longmapsto x + 4 = 3(y -3)}$

$\sf{\longmapsto x + 4 = 3y - 9 }$

$\sf{\longmapsto x -3y = -9-4}$

$\bf{\longmapsto x -3y = -13....2) }$

Put the value of 'y' from equation 1) in equation 2)

$\sf{\longmapsto x -3(x -3) = -13}$

$\sf{\longmapsto x -3x +9 = -13}$

$\sf{\longmapsto -2x = -13 - 9 }$

$\sf{\longmapsto -2x = -22 }$

$\sf{\longmapsto x = \cancel\dfrac{-22}{-2} }$

${\large{\underbrace{\bf{\star \: x = 11 \: \star}}}}$

Put the value of 'x' in equation 1)

$\sf{\longmapsto y = x -3 }$

$\sf{\longmapsto y = 11-3 }$

${\large{\underbrace{\bf{\star \: y = 8 \: \star}}}}$

$\sf{\mapsto Fraction = \dfrac{x}{y}}$

$\bf{\star Fraction = \dfrac{11}{8}}$

Hence, the required fraction is 11/8

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Answer 2

Let the Numerator be x

So, denominator = x - 3

We know $\red {\rm Fraction = \dfrac{Numerator}{Denominator}}$

So, $\bf \dfrac{x}{x - 3}$

Now, according to Question

➥ $\rm \dfrac{x + 4}{x - 3 - 3} = 3$

➥ $\rm \dfrac{x + 4}{ x - 6} = \dfrac{3}{1}$

Doing Cross Multiplication

➥ $\rm 1( x + 4) = 3( x - 6)$

➥ $\rm x + 4 = 3x - 18$

➥ $\rm x - 3x = -18 - 4$

➥ $\rm \cancel{ -}2x = \cancel{-} 22$

➥ $\rm 2x = 22$

➥ $\rm x = \cancel{ \dfrac{ 22}{2} }$

➥ $\rm x = 11$

Thus, the value of x is 11

Now, we will subsitute it into our fraction

➦ $\sf \dfrac{x}{x - 3}$

➦ $\sf \dfrac{11}{11 - 3}$

➦ $\sf \dfrac{11}{8}$

So, the fraction is 11/8

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