Question

5. The numerator of a rational number is less than its denominator by 3. If the numerator becomes three times and the denominator is increases by 20, the new number becomes 1 Find the original number. 8​

Answer 1

Appropriate Question :-

The numerator of a rational number is less than its denominator by 3. If the numerator becomes three times and the denominator is increases by 20, the new number becomes 1/8. Find the original number.

$\large\underline{\sf{Solution-}}$

Given that,

The numerator of a rational number is less than its denominator by 3.

So, Let assume that

Numerator of a fraction be x

So,

Denominator of a fraction is x + 3

Thus,

$\rm \: Fraction \: = \: \dfrac{x}{x + 3}$

Now, further given that, If the numerator becomes three times and the denominator is increases by 20, the new number becomes 1/8.

So,

Numerator of a fraction = 3x

Denominator of a fraction = x + 3 + 20 = x + 23

So,

$\rm \: Fraction \: = \: \dfrac{3x}{x + 23}$

According to statement,

$\rm \: \dfrac{3x}{x + 23} = \dfrac{1}{8}$

$\rm \: 24x = x + 23$

$\rm \: 24x - x =23$

$\rm \: 23x =23$

$\rm\implies \:\rm \: x =\dfrac{23}{23} = 1$

So,

$\rm \: Fraction \: = \: \dfrac{x}{x + 3}$

On substituting the value of x, we get

$\rm \: Fraction \: = \: \dfrac{1}{1 + 3}$

$\rm\implies \:\rm \: Fraction \: = \: \dfrac{1}{4}$

Answer 2

Given : The numerator of a ration number is less than its denominator by 3 .If the numerator becomes three times and denominator is increased by 20 the number becomes 1/8 .

To Find : Find the Original number

$\\ \qquad{\rule{200pt}{3pt}}$

SolutioN : For Solving these types of questions we need to form the Equation first. And, by cross - Multiplying we can get the No. .Let's Solve :

$\red{❒}$ According to the Question :

$\longmapsto$ The numerator of a ration number is less than its denominator by 3 .So,

$\qquad \: \: {\underline{\overline{\boxed{\sf{ \bigg\{ \dfrac{Numerator}{Denominator} \bigg\} = \bigg\{ \dfrac{y}{y + 3} \bigg\} }}}}} \bigstar$

$\longmapsto$ the numerator becomes three times and denominator is increased by 20 the number becomes 1/8 .So,

$\qquad \: \: {\underline{\overline{\boxed{\sf{ \bigg\{ \dfrac{y \times 3}{(y + 3) + 20 } \bigg\} = \bigg\{ \dfrac{1}{8} \bigg\} }}}}} \bigstar$

$\red{❒}$ Let's Cross Multiply :

${\dashrightarrow{\qquad{\sf{ 8 \bigg( 3y \bigg) = 1 \bigg\{ \bigg( y + 3 \bigg) + 20 \bigg\} }}}} \\ \\ \\ \ {\dashrightarrow{\qquad{\sf{ 8 \bigg( 3y \bigg) = 1 \bigg( y + 23 \bigg) }}}} \\ \\ \\ \ {\dashrightarrow{\qquad{\sf{ 24y = y + 23 }}}} \\ \\ \\ \ {\dashrightarrow{\qquad{\sf{ 24y - y = 23 }}}} \\ \\ \\ \ {\dashrightarrow{\qquad{\sf{ 23y = 23 }}}} \\ \\ \\ \ {\dashrightarrow{\qquad{\sf{ y = \dfrac{23}{23} }}}} \\ \\ \\ \ {\dashrightarrow{\qquad{\sf{ y = \cancel\dfrac{23}{23} }}}} \\ \\ \\ \ {\qquad \: \: {\dashrightarrow{\underline{\boxed{\pmb{\frak{ y = 1 }}}}}}} \: {\purple{\bigstar}}$

$\red{❒}$ Calculating the Rational Number :

• Numerator = y = 1
• Denominator = y + 3 = 1 + 3 = 4

$\therefore \: \:$ Original Rational Number is 1/4 .

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