Question

The sum of the numerator and the denominator of a fraction is 8 if 3 is added to both the numerator and the denominator the fraction becomes 3 upon 4 find the fraction

Answer 1

$\bf{\huge{\underline{\boxed{\sf{\blue{Answer:}}}}}}$

$\bf{\underline{\underline{\sf{\green{Given:}}}}}$

• The sum of the numerator and the denominator of a fraction is 8
• If 3 is added to both the numerator and the denominator the fraction becomes$\frac{3}{4}$

$\bf{\underline{\underline{\sf{\green{To\:find:}}}}}$

• $\bf\sf{The\:fraction}$

$\bf{\underline{\underline{\sf{\green{Solution:}}}}}$

Let the Numerator of the fraction be x

Let the Denominator of the fraction be y

Fraction = $\bf\sf\frac{x}{y}$

$\bf{\underline{\underline{\sf{\pink{As\:per\:first\:condition:}}}}}$

• The sum of the numerator and the denominator of a fraction is 8

Representing the condition mathematically to obtain our first equation.

=> Numerator + Denominator = 8

=> x + y = 8 ---> 1

$\bf{\underline{\underline{\sf{\pink{As\:per\:second\:condition:}}}}}$

• If 3 is added to both the numerator and the denominator the fraction becomes$\frac{3}{4}$

Numerator = x + 3

Denominator = y + 5

Fraction = $\bf\sf\frac{3}{4}$

Representing the condition mathematically.

=> $\sf\frac{x+3}{y+3}$ = $\sf\frac{3}{4}$

Cross multiplying,

=> $\sf{4(x+3)\:=\:3(y + 3) }$

=> $\sf{4x +12 \:=\:3y+9}$

=> $\sf{4x - 3y = 9 - 12}$

=> $\bf\sf{4x-3y=- 3}$ ----> 2

Multiply equation 1 by 4,

=>$\sf{ x + y = 8 }$

=> $\sf{4\:\times\:x\:+\:4\times\:y\:=\:4\times\:8}$

=> $\bf\sf{4x + 4y = 32}$ ---> 3

Solve equations 3 and 2 simultaneously by elimination method.

Subtract equation 3 from 2,

....$\sf{\:+\:4x\:+\:4y\:=\:32}$

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

----------------------------------

$\sf{7y\:\:=\:35}$

$\sf{y\:=}$$\sf\frac{35}{7}$

$\sf{y =5 }$

Substitute y = 5 in equation 1,

$\sf{x+y=8}$

$\sf{x + 5 = 8 }$

$\sf{x = 8 - 5 }$

$\bf\sf{x = 3}$

$\bf{\underline{\boxed{\sf{\purple{Numerator\:=\:x\:=\:3}}}}}$

$\bf{\underline{\boxed{\sf{\purple{Denominator\:=\:y\:=\:5}}}}}$

$\bf{\underline{\boxed{\sf{\purple{Fraction\:=\frac{3}{5}}}}}}$

$\bf{\huge{\underline{\boxed{\sf{\blue{Verification}}}}}}$

For first case :-

• The sum of the numerator and the denominator of a fraction is 8

Numerator = x = 3

Denominator = y = 5

=> x + y = 8

=> 3 + 5 = 8

=> 8 = 8

LHS = RHS.

For second case :-

• If 3 is added to both the numerator and the denominator the fraction becomes$\frac{3}{4}$

Numerator = x + 3 = 3 + 3 = 6

Denominator = y + 3 = 5 + 3 = 8

Fraction = $\bf\frac{3}{4}$

=> $\sf\frac{x+3}{y+3}$ = $\bf\frac{3}{4}$

=> $\sf\frac{6}{8}$ = $\bf\frac{3}{4}$

Dividing LHS by 2,

=> $\sf\frac{3}{4}$ = $\bf\frac{3}{4}$

LHS = RHS.

Hence Verified.

Answer 2

$\huge{\mathfrak{\underline{\underline{Answer :- }}}}$

$\huge{\bf{Fraction \: = \: \frac {3}{5}}}$

$\huge{\mathfrak{\blue{Explanation :-}}}$

$\large{\mathrm{\gray{Given :-}}}$

The sum of the numerator and the denominator of a fraction is 8.

If 3 is added to both the numerator and the denominator the fraction becomes 3/4.

$\large{\mathrm{\gray{Solution<strong> </strong><strong>:</strong><strong>-</strong>}}}$

Let the Numerator of the fraction be a

Let the Denominator of the fraction be b

Fraction = a/b

✯ Case 1

The sum of the numerator and the denominator of a fraction is 8

Representing the condition mathematically to obtain our first equation.

Numerator + Denominator = 8

a + b = 8 ........(1)

✯ Case 2

If 3 is added to both numerator and denominator then the fraction become 3/4.

Now,

A.T.Q

a + 3/ b + 3 = 3/4

✯ By cross multiplication

⟹ 4(a + 3) = 3(b + 3)

⟹ 4a + 12 = 3b + 9

⟹ 4a - 3b = 9 - 12

⟹ 4a - 3b = -3 ..........(2)

$\rule{200}{2}$

Now, we will substitute the value of a in equation1

a + b = 8

a = 8 - b

_________[Put values in eq 2]

⟹ 4(8 - b) - 3b = -3

⟹ 32 - 4b - 3b = -3

⟹ 32 - 7b = -3

⟹ -7b = -3 - 32

⟹ -7b = -35

⟹ 7b = 35

⟹ b = 35/7

⟹ b = 5

$\huge{\boxed{\boxed{\blue{Denominator (b) \: = \: 5}}}}$

______________[put values in eq 1]

⟹ a + b = 8

⟹a + 5 = 8

⟹a = 8 - 5

⟹a = 3

$\huge{\boxed{\boxed{\green{Numerator (a) \: = \: 3}}}}$

$\huge{\boxed{\boxed{\pink{Fraction \: = \: \frac{3}{5}}}}}$