Question

the numerator and the denominator of a fraction are in the ratio3:2. if 3 is the added to the numerator and 2 is subtracted from the denominator, a new fraction is formed whose value is 9/4 . find the orginal fraction​

Answer 1

❍ Let's say, that the numerator and denominator of the fraction be 3x and 2x respectively.

Hence,

• Fraction is = 3x/2x

⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━⠀⠀⠀⠀

⠀⠀

$\underline{\bigstar\:{\pmb{\textsf{According to the given Question :}}}}$

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• If 3 is added to the numerator and 2 is subtracted from the denominator, a new fraction is formed whose value is ⁹⁄₄. After converting it into fraction, we get —

⠀⠀

$\dashrightarrow\sf\bigg\{\dfrac{3x + 3}{2x - 2}\bigg\} = \bigg\{\dfrac{9}{4}\bigg\} \\\\\\\dashrightarrow\sf 4\Big\{3x + 3\Big\} = 9\Big\{2x - 2\Big\} \\\\\\\dashrightarrow\sf 12x + 12 = 18x - 18 \\\\\\\dashrightarrow\sf 18x - 12x = 12 + 18 \\\\\\\dashrightarrow\sf 6x = 30 \\\\\\\dashrightarrow\sf x = \cancel\dfrac{30}{6} \\\\\\\dashrightarrow\underline{\boxed{\pmb{\frak{\pink{x = 5}}}}}\;\bigstar$

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Therefore,

⠀⠀⠀▪︎ Numerator of the fraction, 3x = 3(5) = 15

⠀⠀⠀▪︎ Denominator of the fraction, 2x = 2(5) = 10

⠀⠀

$\therefore{\underline{\sf{Hence,\; the\; required\; original\; fraction\; is\; {\pmb{\frak{\dfrac{15}{10}}}.}}}}$

Answer 2

Answer:

Given :-

• The numerator and the denominator of a fraction are in the ratio of 3 : 2.
• 3 is added to the numerator and 2 is subtracted from the denominator.
• A new fraction is formed whose value is 9/4.

To Find :-

• What is the original fraction.

Solution :-

Let,

$\mapsto$ Numerator = 3x

$\mapsto$ Denominator = 2x

Hence,

$\leadsto$ The original fraction will be :

$\implies \sf \dfrac{Numerator}{Denominator}$

$\implies \sf\bold{\green{\dfrac{3x}{2x}}}$

According to the question,

$\longrightarrow \sf \dfrac{Numerator\: +\: 3}{Denominator\: -\: 2} =\: \dfrac{9}{4}$

$\longrightarrow \sf \dfrac{3x + 3}{2x - 2} =\: \dfrac{9}{4}$

By doing cross multiplication we get,

$\longrightarrow \sf 9(2x - 2) =\ 4(3x + 3)$

$\longrightarrow \sf 18x - 18 =\: 12x + 12$

$\longrightarrow \sf 18x - 12x =\: 12 + 18$

$\longrightarrow \sf 6x =\: 30$

$\longrightarrow \sf x =\: \dfrac{\cancel{30}}{\cancel{6}}$

$\longrightarrow \sf x =\: \dfrac{5}{1}$

$\longrightarrow \sf\bold{\purple{x =\: 5}}$

Hence, the original fraction is :

$\dashrightarrow \sf \dfrac{3x}{2x}$

$\dashrightarrow \sf \dfrac{3(5)}{2(5)}$

$\dashrightarrow \sf \dfrac{3 \times 5}{2 \times 5}$

$\dashrightarrow \sf\bold{\red{\dfrac{15}{10}}}$

${\normalsize{\bold{\underline{\therefore\: The\: original\: fraction\: is\: \dfrac{15}{10}\: .}}}}$

VERIFICATION :-

$\implies \sf \dfrac{3x + 3}{2x - 2} =\: \dfrac{9}{4}$

By putting x = 5 we get,

$\implies \sf \dfrac{3(5) + 3}{2(5) - 2} =\: \dfrac{9}{4}$

$\implies \sf \dfrac{3 \times 5 + 3}{2 \times 5 - 2} =\: \dfrac{9}{4}$

$\implies \sf \dfrac{15 + 3}{10 - 2} =\: \dfrac{9}{4}$

$\implies \sf \dfrac{\cancel{18}}{\cancel{8}} =\: \dfrac{9}{4}$

$\implies \sf\bold{\pink{\dfrac{9}{4} =\: \dfrac{9}{4}}}$

Hence, Verified.