Question

the sum of the numerater and the denominator if a fraction is equal to 7. four times the numerator is 8 less than 5 times the denominator. find the fraction​

Answer 1

Given :

• The sum of the numerater and the denominator if a fraction is equal to 7.
• Four times the numerator is 8 less than 5 times the denominator.

To Find :

• The required Fraction.

Solution :

Let the numerator of the fraction be x.

Let the denominator of the fraction be y.

Fraction = $\bold{\dfrac{x}{y}}$

Case 1 :

The sum of the quantities, numerator and denominator of the fraction is 7.

Equation :

$\sf{\longrightarrow{x+y=7}}$

$\sf{x=7-y\:\:\:\:\:(1)}$

Case 2 :

Four times the numerator of the fraction is 8 less than 5 times the denominator.

Equation :

$\sf{\longrightarrow{4x=5y-8}}$

$\sf{\longrightarrow{4(7-y)=5y-8}}$

$\bold{\big[From\:equation\:(1)\:x\:=\:7-y\big]}$

$\sf{\longrightarrow{28-4y=5y-8}}$

$\sf{\longrightarrow{28+8=5y+4y}}$

$\sf{\longrightarrow{36=9y}}$

$\sf{\longrightarrow{\dfrac{36}{9}=y}}$

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

Substitute, y = 4 in equation (1),

$\sf{\longrightarrow{x=7-y}}$

$\sf{\longrightarrow{x=7-4}}$

$\sf{\longrightarrow{x=3}}$

$\large{\boxed{\sf{Numerator\:=\:x\:=\:3}}}$

$\large{\boxed{\sf{Denominator\:=\:y\:=\:4}}}$

$\large{\boxed{\sf{Fraction\:=\:\dfrac{x}{y}=\dfrac{3}{4}}}}$

Answer 2

$\huge\underline\mathbb\red{GIVEN:-}$

• The sum of the numerater and the denominator is 7
• Four times the numerator is 8 less than 5 times the denominator

$\huge\underline\mathbb\red{TO\:FIND:-}$

Find the fraction = ?

$\huge\underline\mathbb\red{SOLUTION:-}$

Let the fraction be $\large\bf{\frac{x}{y}}$

According to question :-

The sum of the numerater and the denominator is equal to 7

$\implies$$\bf\blue{ x + y = 7....1)}$

$\implies$$\bf{x = 7-y}$$\bf{(from..1)}$

four times the numerator is 8 less than 5 times the denominator

$\implies$ $\bf\blue{4x = 5y - 8...2)}$

putting the value of x,

$\implies$$\bf{ 4(7-y)= 5y - 8}$

$\implies$ $\bf{28 - 4y = 5y - 8}$

$\implies$$\bf{ 28 + 8 = 4y + 5y}$

$\implies$ $\bf{36 = 9y}$

$\implies$$\bf{y = \frac{36}{9}}$

$\large{\boxed{\bf{\red{y = 4}}}}$

put the value of 'y' in equation 1)

$\implies$$\bf{x + 4 = 7}$

$\implies$$\bf{ x = 7-4}$

$\large{\boxed{\bf{\red{x= 3}}}}$

Thus, the fraction becomes $\bf{\frac{3}{4}}$

$\large\underline\mathbb\red{FINAL \: ANSWER:-}$

$\huge{\boxed{\bf{\red{Fraction = \frac{x}{y}=\frac{3}{4}}}}}$

$\large\underline\mathbb\red{VERIFICATION:-}$

Put the values of x and y in equation 1)

$\implies$$\bf\blue{ x + y = 7}$

$\implies$$\bf{ 3+ 4 = 7)}$

$\implies$$\large\bf{ 7 = 7}$

Put the values of x and y in equation 2)

$\implies$ $\bf\blue{4x = 5y - 8}$

$\implies$ $\bf{4(3) = 5(4) - 8}$

$\implies$ $\bf{12 = 20 - 8}$

$\implies$ $\large\bf{12 = 12}$

$\large\underline\mathrm\red{VERIFIED:-}$