Question

In a rational number, twice the numerator is 2 more than the denominator. If 3 is added to each, the numerator and the denominator, the new fraction is 2/3. Find the original number.​

Answer 1

Answer :

• Original number is 7/12.

Given :

• In a rational number, twice the numerator is 2 more than the denominator
• If 3 is added to each the numerator and the denominator, the new fraction is 2/3

To find :

• Original number

Solution :

Given, twice the numerator is 2 more than the denominator so,

• Let the numerator be x
• Denominator be 2x - 2

We know that,

• Fraction = Numerator/Denominator

➞ x/2x - 2

And also Given , if 3 added to each the numerator and denominator so ,

• x + 3 / 2x - 2 + 3 = 2/3

➞ x + 3 / 2x - 2 + 3 = 2/3

➞ (x + 3) / 2x + 1 = 2/3

Now , cross multiplying we get,

➞ 3(x + 3) = 2(2x + 1)

➞ 3x + 9 = 4x + 2

➞ 3x - 4x = 2 - 9

➞ -x = - 7

➞ x = 7

Then,

• Numerator is 7
• Denominator = 2x - 2 = 2(7) - 2 = 12

Finding the original number :

➞ Fraction = Numerator/Denominator

➞ 7/12

Hence , Original number is 7/12.

Verification :

➞ x/2x - 2

➞ 7/2(7) - 2

➞7/14 - 2

➞7/12

Hence , Verified.

Answer 2

☯ Let the numerator be x and the denominator be (2x -2).

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• ${\underbrace{\underline{\sf{\bf{\red{Fraction\:=\:{\dfrac{Numerator}{Denominator}}}}}}}}$

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$\:\:\:\:\:\:\:\leadsto\:\sf{\dfrac{x}{(2x\:-\:2)}}$

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• So, the numerator and denominator are increased by 3, then fraction is ⅔.

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$\:\:\:\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{(x\:+\:3)}{(2x\:-\:2\:+\:3)}}\:=\:{\dfrac{2}{3}}$

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$\:\:\:\:\:\:\:\:\dashrightarrow\:\sf{\dfrac{(x\:+\:3)}{(2x\:+\:1)}}\:=\:{\dfrac{2}{3}}$

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• By Cross - Multiplying We Get ;

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$\:\:\:\:\:\::\:\Longrightarrow\:\sf{3(x\:+\:3)\:=\:2(2x\:+\:1)}$

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$\:\:\:\:\:\::\:\Longrightarrow\:\sf{3x\:+\:9\:=\:4x\:+\:2}$

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$\:\:\:\:\:\::\:\Longrightarrow\:\sf{3x\:-\:4x\:=\:2\:-\:9}$

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$\:\:\:\:\:\::\:\Longrightarrow\:\sf{-x\:=\:-7}$

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$\:\:\:\:\:\::\:\Longrightarrow\:\sf{x\:=\:7}$

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• The numerator is x = 7, denominator is (2x - 2) = [2(7) - 2] = 14 - 2 = 12.

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• And the fraction is $\:\sf{\dfrac{numerator}{denominator}}\:$=$\:\sf{\dfrac{7}{12}}$

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$\:\:\:\:\:\:\:\leadsto\:\sf{\dfrac{x}{(2x\:-\:2)}}$

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$\:\:\:\:\:\:\:\leadsto\:\sf{\dfrac{7}{2}}(7)\:-\:2$

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$\:\:\:\:\:\:\:\leadsto\:\sf{\dfrac{7}{14}}\:-\:2$

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$\:\:\:\:\:\:\:\leadsto\:\sf{\dfrac{7}{12}}$

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