Question

The denominator of a rational number is greater than its numerator by 4. If the numerator is increased by 7 and denominator is decreased by 2 ,the new number brcomes 6/5. Find the original number

Answer 1

$\large \dag$ Question :-

The denominator of a rational number is greater than its numerator by 4. If the numerator is increased by 7 and denominator is decreased by 2 , the new number becomes 6/5. Find the original number.

$\large \dag$ Answer :-

$\red\dashrightarrow\underline{\underline{\sf  \green{The   \: Original \:  Number  \: is \:  \frac{23}{27} }} }$

$\large \dag$ Step by step Explanation :-

Let Numerator and Denominator of Original Fraction be :

• Numerator = x

As Per the question denominator of the rational number is greater than its numerator by 4 so,

• Denominator should be = x + 4

So ,

$\text{Original Fraction = } \frac{\text x}{\text x + 4}$

❒ When numerator is increased by 7 and denominator is decreased by 2 :

$\\ \rm \text{Fraction Becomes } :  \frac{x +7}{x + 2}$

⏩ According To Question :

$\\ \large \blue \bigstar  \:   \red{ \bf  \frac{x +7}{x + 2} =  \frac{6}{5}  }$

$:\longmapsto \rm 6(x  +  2) = 5(x + 7)$

$:\longmapsto \rm 6x  + 12 = 5x + 35$

$:\longmapsto \rm 6x  -  5x = 35-12$

$\purple{ \large :\longmapsto  \underline {\boxed{{\bf x = 23} }}}$

$\blue\dashrightarrow\underline{\underline{\sf  \orange{Numerator  \: of \:  Original  \: Fraction = 23}} }$

☆ As denominator of the rational number is greater than its numerator by 4

$\rm\therefore \:  Denominator  = 23   + 4$

$\blue\dashrightarrow\underline{\underline{\sf  \orange{Denominator  \: of \:  Original  \: Fraction = 27}} }$

Therefore,

$\large\underline{\pink{\underline{\frak{\pmb{ Original  \: Fraction  =  \dfrac{23}{27} }}}}}$

Answer 2

Given : The denominator of a rational number is greater than its numerator by 4. If the numerator is increased by 7 and denominator is decreased by 2 ,the new number becomes 6/5. Find the original number

Need to Find : The original fraction

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★ Cᴏɴᴄᴇᴘᴛ :

According to the question, first we need to assume any variable then the denominator of the fraction is 4 greater then the numerator after which the numerator increases by 7 and the denominator decreases by 2. So, after doing the changes we get the new fraction which is 6/5. Thereby getting the value of the variable and then putting the values instead of the variable we can get the answer

★ Sᴏʟᴜᴛɪᴏɴ :

Let us assume the Numerator be a

Hence,

$\star \quad \small \boxed{ \frak{ \pink{Original \: Fraction} = \green{ \frac{Numerator}{Denominator}}}} \quad \star$

$\rightarrow \frak{Original \: Fraction = \frac{a}{a + 4}}$

Now, 7 is added to the numerator and 2 is subtracted in the denominator

$\rightarrow \frak{Original \: Fraction = \blue{\frac{a + 7}{a + 4 - 2}}}$

$\rightarrow \frak{Original \: Fraction = \blue{\frac{a + 7}{a + 2}}}$

Now after which the new fraction becomes 6/5

$\dashrightarrow \frak{New \: Fraction = \frac{6}{5}}$

So, Original Fraction = New Fraction

$\small\hookrightarrow \frak{ \frac{a + 7}{a + 2} = \frac{6}{5}} \\ \\ \small\hookrightarrow \frak{5(a + 7) = 6(a + 2)} \\ \\ \small\hookrightarrow \frak{5a + 35 = 6a + 12} \\ \\ \small\hookrightarrow \frak{6a - 5a = 35 - 12} \\ \\ \small\star \quad \underline{ \boxed{ \frak{ \red{a = 23}} }}$

★ Fɪɴᴅɪɴɢ Oʀɪɢɪɴᴀʟ Fʀᴀᴄᴛɪᴏɴ :

$\hookrightarrow \frak{ Original \: Fraction = \frac{a}{a + 4}}$

$\hookrightarrow \frak{ Original \: Fraction = \frac{23}{23 + 4}}$

$\small{ \mathbb{ \red{HENCE \: THE \: ORIGINAL \: FRACTION \: \: IS } }} \: \: \underline{\boxed{ \pink{\frak{ \frac{23}{27} }}}}$