Question

the numerator of a fraction is greater than its denominator by 4 and 5 is added to both the numerator and denominator the fraction will be equivalent to 5 by 4 find the fraction​

Answer 1

$\Large {\underline { \sf \purple{Clarification :}}}$

Here,we are given that the numerator of a fraction is greater than its denominator by 4. Also, if 5 is added to both the numerator and denominator the fraction will be equivalent to 5/4. We have to find the fraction.

We'll first assume the numerator and the denominator as two variables, say x and y. After that, as per the given question we'll form linear equations. Then, by using cross multiplication method and transposition method, we'll find the value of x and y.

$\bf \red { \dag }$ Cross multiplication :

• This is the method used to solve a linear equation having variables and constants.
• In this method, we cross multiply the values and equate the linear equation.

$\bf \red { \dag }$ Transposition method :

• This is the method used to solve a linear equation having variables and constants.
• In this method, we transpose the values from R.H.S to L.H.S and vice-versa and changes its sign while transposing to find the value of the unknown value.

$\Large {\underline { \sf \purple{Explication \: of \: Steps :}}}$

$\underline{\pmb{ \sf {\maltese \; \; \; Given \: Information \: : \; \; \; }}}$

• The numerator of a fraction is greater than its denominator by 4.

• 5 is added to both the numerator and denominator the fraction will be equivalent to 5 by 4

$\underline{\pmb{ \sf {\maltese \; \; \; To \: calculate \: : \; \; \; }}}$

• The fraction.

$\underline{\pmb{ \sf {\maltese \; \; \; Calculations \: : \; \; \; }}}$

Let,

• Numerator = x
• Denominator = y

So,

$\bigstar \: \boxed{\sf {Fraction = \dfrac{x}{y} }}$

$\underline{ \sf {\maltese \; \; \; According \: to \: the \: question : \; \; \; }}$

$\longmapsto$ Numerator = Denominator + 4

$\longmapsto \sf {x = y + 4 }$

Let it be the equation (1).

Also,

$\longrightarrow \bf { \dfrac{x+5}{y+5} = \dfrac{5}{4} }$

$\underline{ \sf \purple {\maltese \; \; \; Finding \: its \: denominator : \; \; \; }}$

Substitute the value of x from the equation (1) .

$\longrightarrow \sf { \dfrac{y+4+5}{y+5} = \dfrac{5}{4} }$

$\longrightarrow \sf { \dfrac{y+9}{y+5} = \dfrac{5}{4} }$

• By cross multiplication,

$\longrightarrow \sf { 4(y+9) = 5(y+5) }$

• By distributive property,

$\longrightarrow \sf { 4(y)+4(9) = 5(y)+5(5) }$

$\longrightarrow \sf { 4y + 36 =5y + 25 }$

$\longrightarrow \sf { 36 -25=5y - 4y }$

$\longrightarrow \boxed {\sf { 11 = y }}$

Therefore, denominator (y) is 11.

$\underline{ \sf \purple {\maltese \; \; \; Finding \: its \: numerator : \; \; \; }}$

From the equation (1),

$\longrightarrow \sf {x = y + 4 }$

$\longrightarrow \sf {x = 11 + 4 }$

$\longrightarrow \boxed{\sf {x = 15 }}$

Therefore, numerator (x) is 15.

$\underline{ \sf \purple {\maltese \; \; \; Finding \: the \: fraction : \; \; \; }}$

$\bigstar \: \boxed{\sf {Fraction = \dfrac{x}{y} }}$

$\longrightarrow \sf \purple {Fraction = \dfrac{15}{11} }$

Therefore, the fraction is 15/11.