Question

The denominator of a fractions is 4 more than twice the numerator. When both numerator anddenominator are decreased by 6, then the denominator becomes 12 times the numerator. Findthe fraction.

Answer 1

Step-by-step explanation:

$\tt \underline{ \pink{Given:}}$

The denominator of a fractions is 4 more than twice the numerator. When both numerator and denominator are decreased by 6, then the denominator becomes 12 times he numerator.

$\tt \underline{ \pink{To \: Find}}$

The fraction

$\tt{ \underline{ \pink{Solution:}}}$

$\pink{ \tt{ Let}}$

• Numerator be x
• Denominator=2x+4

$\sf \: \N ew \:numerator = x - 6 \\ \sf\N ew \: denominator = 2x + 4 - 6 = 2x - 2$

Now denominator has become 12 times the numerator. So, if we multiply 12 to the numerator after decreasing 6 from it, it will be equal to the denominator after decreasing 6 from it.

$\tt \: \red{ ACQ}$

$\therefore \tt \: 12(x - 6) = 2x - 2 \\ \tt \leadsto12x - 70 = 2x - 2 \\ \tt \leadsto \: 12x - 2x = - 2 + 72 \\ \tt \leadsto10x = 70 \\ \tt \leadsto \: x = \frac{ \cancel{70}}{ \cancel{10} } \\ \green{\underline{ \tt \: x = 7}}$

Now, numerator=x=7

Denominator=2x+4=2×7+4=14+4=18

$\blue{ \tt \: Required \: fraction = \frac{7}{18} }$

Answer 2

$\sf\small\underline{Let:-}$

$\tt{\leadsto The\: numerator\:_{(fraction)}=x}$

$\tt{\leadsto The\:denominator\:_{(fraction)}=y}$

$\tt{\leadsto The\: Fraction=\dfrac{x}{y}}$

$\sf\small\underline{To\: Find:-}$

$\tt{\leadsto The\: Fraction\:_{(number)}=?}$

$\sf\small\underline{Solution:-}$

To solve this question at 1st we have to set equation then calculate the Numerator and denominator. After that substituting the value of Numerator and denominator to calculate the fractional number.

$\sf\small\underline{Given\:in\:case-(i):-}$

$\rm{\implies Denominator=2(Numerator)+4}$

$\rm{\longrightarrow y=2x+4}$

$\rm{\longrightarrow 2x-y=-4-----(i)}$

$\sf\small\underline{Given\:in\:case-(ii):-}$

$\rm{\implies Denominator-6=12(Numerator-6)}$

$\rm{\longrightarrow y-6=12(x-6)}$

$\rm{\longrightarrow 12x-y=66----(ii)}$

• In eq (i) × by 12 and eq (ii) × by 2:-

$\rm{\longrightarrow 24x-12y=-48}$

$\rm{\longrightarrow 24x-2y=132}$

• By Solving we get here :-

$\rm{\longrightarrow -10y=-180}$

$\rm{\longrightarrow 10y=180}$

$\rm{\longrightarrow y=18}$

• Putting the value y= 18 in eq (i):-

$\rm{\longrightarrow 2x-y=-4}$

$\rm{\longrightarrow 2x-18=-4}$

$\rm{\longrightarrow 2x=-4+18}$

$\rm{\longrightarrow x=7}$

$\sf\small\underline{Hence,\:the\: fraction=\frac{7}{18}}$