Question

245 If the numerator of a fraction is increased by 2 and the denominator by 1 it becomes 1. Again if the numerator is decreased by 4 and the denominator by 2 it
becomes 1/2. Find the fraction.​

Answer 1

Step-by-step explanation:

Let the fraction be yx

As per the given statements, y−4x+2=2

⇒x+2=2y−8

⇒x−2y=−10 --- (1)

Also, y+2x−1=31

⇒3x−3=y+2

⇒3x−y=5 --- (2)

Multiplying eqn (2) by 2 and subtracting eqn (1) from it, we get

2(3x−y)−(x−2y)=2(5)−(−10)

⇒5x=20

⇒x=4

Substituting this value of x in eqn (2), we get

3(4)−y=5

⇒y=12−5=7

∴x=4;y=7

So, x+y=4+7=11

Answer 2

Given

• If the numerator of a fraction is increased by 2 and the denominator by 1 it becomes 1.
• If the numerator is decreased by 4 and the denominator by 2 it becomes 1/2.

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To find

• Required fraction.

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Solution

• Let numerator be x and denominator be y.

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$\sf\longrightarrow{Required\: fraction = \dfrac{x}{y}}$

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★ If the numerator of a fraction is increased by 2 and the denominator by 1 it becomes 1.

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$\tt\longmapsto{\dfrac{x + 2}{y + 1} = 1}$

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$\tt\longmapsto{x + 2 = y + 1}$

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$\tt\longmapsto{x - y = -1}$⠀⠀⠀... [1]

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★ If the numerator is decreased by 4 and the denominator by 2 it becomes 1/2.

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$\tt\longmapsto{\dfrac{x - 4}{y - 2} = \dfrac{1}{2}}$

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$\tt\longmapsto{2(x - 4) = y - 2}$

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$\tt\longmapsto{2x - 8 = y - 2}$

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$\tt\longmapsto{2x - y = 6}$⠀⠀... [2]

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$\: \: \: \: \: \: \: \: \: \:\underline{\sf{\red{Subtracting\: [1]\: from\: [2]}}}$

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$\tt:\implies{2x - y - (x - y) = 6 - (-1)}$

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$\tt:\implies{2x - \cancel{y} - x + \cancel{y} = 6 + 1}$

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$\tt:\implies{x = 7}$

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$\: \: \: \: \: \: \: \: \: \:\underline{\sf{\red{Putting\: the\: value\: of\: x\: in\: [1]}}}$

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$\tt:\implies{7 - y = -1}$

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$\tt:\implies{- y = -1 - 7}$

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$\tt:\implies{y = 8}$

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Hence,

• The required fraction is $\sf{\dfrac{7}{8}}$