Question

if 2 is added to the numerator and denominator it becomes 9/10 and if 3 is substracted from the numerator and denominator it becomes 4/5 find the fractions

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Answer 1

Answer:-

Let the fraction be x/y.

Given:

If 2 is added to the numerator and denominator, the fraction becomes 9/10.

$\implies \sf\frac{x + 2}{y + 2} = \frac{9}{10} \\ \\ \implies \sf \: 10(x + 2) = 9(y + 2) \\ \\ \implies \sf \: 10x + 20 = 9y + 18 \\ \\ \implies \sf \: 10x = 9y + 18 - 20 \\ \\ \implies \sf \: 10x = 9y - 2 \\ \\ \implies \sf \: x = \frac{9y - 2}{10} \: \: - - \: \: equation \: (1)$

Also,

The fraction becomes 4/5 when 3 is subtracted from numerator and denominator.

$\: \implies \sf \: \frac{x - 3}{y - 3} = \frac{4}{5} \\ \\ \implies \sf \: 5(x - 3) = 4(y - 3) \\ \\ \implies \sf \: 5x - 15 = 4y - 12$

Substitute the value of x from equation (1).

$\: \implies \sf \: 5 \bigg( \dfrac{9y - 2}{10} \bigg) - 15 = 4y - 12 \\ \\ \implies \sf \: \frac{9y - 2 - 15(2)}{2} = 4y - 12 \\ \\ \implies \sf \: 9y - 2 - 30 = 2(4y - 12) \\ \\ \implies \sf \: 9y - 32 = 8y - 24 \\ \\ \implies \sf \: 9y - 8y = - 24 + 32 \\ \\ \implies \sf \red {y = 8}$

Substitute the value of y in equation (1).

$\: \implies \sf \: x = \frac{9(8) - 2}{10} \\ \\ \implies \sf \: x = \frac{72 - 2}{10} \\ \\ \implies \sf \: x = \frac{70}{10} \\ \\ \implies \sf \red{x = 7}$

∴ The required fraction x/y is 7/8.

Answer 2

Answer:

$Let the fraction be x/y.</p><p></p><p>Given:</p><p></p><p>If 2 is added to the numerator and denominator, the fraction becomes 9/10.</p><p></p><p>\begin{gathered}\implies \sf\frac{x + 2}{y + 2} = \frac{9}{10} \\ \\ \implies \sf \: 10(x + 2) = 9(y + 2) \\ \\ \implies \sf \: 10x + 20 = 9y + 18 \\ \\ \implies \sf \: 10x = 9y + 18 - 20 \\ \\ \implies \sf \: 10x = 9y - 2 \\ \\ \implies \sf \: x = \frac{9y - 2}{10} \: \: - - \: \: equation \: (1)\end{gathered}⟹y+2x+2=109⟹10(x+2)=9(y+2)⟹10x+20=9y+18⟹10x=9y+18−20⟹10x=9y−2⟹x=109y−2−−equation(1)</p><p></p><p>Also,</p><p></p><p>The fraction becomes 4/5 when 3 is subtracted from numerator and denominator.</p><p></p><p>\begin{gathered}\: \implies \sf \: \frac{x - 3}{y - 3} = \frac{4}{5} \\ \\ \implies \sf \: 5(x - 3) = 4(y - 3) \\ \\ \implies \sf \: 5x - 15 = 4y - 12 \\\end{gathered}⟹y−3x−3=54⟹5(x−3)=4(y−3)⟹5x−15=4y−12</p><p></p><p>Substitute the value of x from equation (1).</p><p></p><p>\begin{gathered}\: \implies \sf \: 5 \bigg( \dfrac{9y - 2}{10} \bigg) - 15 = 4y - 12 \\ \\ \implies \sf \: \frac{9y - 2 - 15(2)}{2} = 4y - 12 \\ \\ \implies \sf \: 9y - 2 - 30 = 2(4y - 12) \\ \\ \implies \sf \: 9y - 32 = 8y - 24 \\ \\ \implies \sf \: 9y - 8y = - 24 + 32 \\ \\ \implies \sf \red {y = 8}\end{gathered}⟹5(109y−2)−15=4y−12⟹29y−2−15(2)=4y−12⟹9y−2−30=2(4y−12)⟹9y−32=8y−24⟹9y−8y=−24+32⟹y=8</p><p></p><p>Substitute the value of y in equation (1).</p><p></p><p>\begin{gathered}\: \implies \sf \: x = \frac{9(8) - 2}{10} \\ \\ \implies \sf \: x = \frac{72 - 2}{10} \\ \\ \implies \sf \: x = \frac{70}{10} \\ \\ \implies \sf \red{x = 7}\end{gathered}⟹x=109(8)−2⟹x=1072−2⟹x=1070⟹x=7</p><p></p><p>∴ The required fraction x/y is 7/8.</p><p></p><p>$