Question

The numerator and denominator of a rational number are in the ratio 3:4. If the denominator is increased by 3, the ratio becomes 3:5. Find the rational number.
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Answer 1

$\textbf {\underline {\underline {Step-By-Step\: Explanation:}}}$

$\sf\blue {Given:}$

• Numerator and denominator are in the ratio 3 : 4
• Denominator is increased by 3, ratio becomes 3:5

$\sf\blue {To\:Find:}$

• The rational number

$\huge\underline\green {\tt Solution:}$

Let the common factor be x

So,

Numerator = 3x

Denominator = 4x

➡ It is given that, Denominator is increased by 3. So, Denominator = 4x + 3

Now,

$\sf\pink {\underline {According\:to\:Question}}$

$\dfrac {3x}{4x + 3 }$ = $\dfrac {3}{5}$

➡ After cross multiplying :

We get,

➡ 15x = 3 ( 4x + 3 )

➡ 15x = 12x + 9

➡ 15x - 12x = 9

➡ 3x = 9

➡ x = $\dfrac {9}{3}$

•°• x = 3

Now,

Numerator = 3x = 3 × 3 = 9

Denominator = 4x = 4 × 3 = 12

Hence,

The required rational number is $\sf\red{\dfrac {9}{12}}$

Answer 2

Given:

Numerator and denominator of a fraction is in the ratio 3:4

If the Denominator is increased by three the ratio becomes 3:5

To find:

The fraction

Explanation:

$\sf let \: numerator\: and \: Denominator \: be \: x \: and \: y$

According to the given info,

$\longrightarrow \sf \: \dfrac{x}{y} = \dfrac{3}{4} \\ \\ \longrightarrow \sf 4x = 3y \\ \\ \longrightarrow \sf4x - 3y = 0 - - - - -(1)</p><p>$

If the Denominator is increased by 3

Denominator is increased by 3Ratio becomes 3:5

$\longrightarrow \sf \dfrac{x}{y + 3} = \dfrac{3}{5} \\ \\ \longrightarrow \sf5x = 3(y + 3) \\ \\ \longrightarrow \sf5x = 3y + 9 \\ \\ \longrightarrow \sf5x - 3y = 9 - - - - - - (2)$

$\underline{ (2) - (1)}$

$\longrightarrow \sf5x - 3y - (4x - 3y) = 9 - 0 \\ \\ \longrightarrow \sf5x - 3y - 4x + 3y = 9 \\ \\ \longrightarrow \sf5x \cancel{ - 3y} - 4x \cancel{ + 3y} = 9 \\ \\ \longrightarrow \sf \boxed{ \sf \: x = 9}$

Substituting value of x in (1)

$\longrightarrow \sf4(9) - 3y = 0 \\ \\ \longrightarrow \sf36 - 3y = 0 \\ \\ \longrightarrow \sf3y = 36 \\ \\ \longrightarrow \sf \: y = \dfrac{ \cancel{ 36}}{ \cancel 3} \\ \\ \longrightarrow \boxed {\sf \: y = 12}$

$\longrightarrow \boxed{ \sf \: fraction = \dfrac{x}{y} = \dfrac{9}{12} = \dfrac{3}{4} }$