Question

which of the following is correct (I) 5(y2-1)=5y2_1 (ii) 2(y_3)=2y_3 (iii) x+5)/5=x+1 (iv) (5x+1)2=25×2+10x+1​

Answer 1

Given:

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The ratio of numerator and denominator of the rational number is 2:7.

If 4 is added to the numerator and 2 is subtracted from the denominator the rational number becomes 10/19.

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To find:

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Original number?

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Solution:

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☯ Let Numerator and Denominator of rational no. be 2x and 7x respectively.

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$\underline{\bigstar\:\boldsymbol{According\:to\:the\:question\::}}$

If 4 is added to the numerator and 2 is subtracted from the denominator the rational number becomes 10/19.

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$:\implies\sf \dfrac{2x + 4}{7x - 2} = \dfrac{10}{19}$

$:\implies\sf 19(2x + 4) = 10(7x - 2)$

$:\implies\sf 38x + 76 = 70x - 20$

$:\implies\sf 38x - 70x = - 20 - 76$

$:\implies\sf - 32x = - 96$

$:\implies\sf x = \cancel{ \dfrac{96}{32}}$

$:\implies{\boxed{\sf{\purple{x = 3}}}}\;\bigstar$

Therefore,

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Numerator of required rational no., 2x = 2 × 3 = 6

Denominator of required rational no., 5x = 7 × 3 = 21

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$\therefore$ Hence, The original number is 6/21.

Answer 2

$\large \orange{\underline{ \orange{\underline{\bold{\purple{\overbrace{\pink{ \underbrace{ |\:\:\:\: \rm{ \mathfrak{ \red{answer}} \:\:\:\:|} }}}}}}}}}$

$\large \orange{\underline{ \orange{\underline{\bold{\purple{\overbrace{\pink{ \underbrace{ |\:\:\:\: \rm{ \mathfrak{ \red{ \huge{answer}}} \:\:\:\:|} }}}}}}}}}$

$\rm{ \bold{ \green{\underline{ \pink{ \underline{ \red{ \underbrace{ \blue{ \:\:\:\:\:\: \dfrac{6}{21} \:\:\:\:\:\: }}}}}}}}}$

$\rm{ \green{ \underline{ \red{ \overbrace{ \pink{ \mathfrak{ \:\:\:explanation\:in\:details \:\: \downarrow}}}}}}}$

$\large\underline\bold{ \mathcal{ \pink{GIVEN:}}}$

$\dashrightarrow ratio \: of\: numerator\:and\: denominator\: is\: 2:7 \\ \dashrightarrow If \:4 \:is\: added\: to\: the \: \\ numerator\: and \:2 \:is \:subtracted\: \\ from \:the\: denominator\: the \:rational\: number \:becomes \: 10/19.$

$\large\underline\bold{\red{ \mathcal{TO\: Find:}}}$

$\rm{\dashrightarrow \underline{ \overline{ \red{original \:number . }}} }$

$\large\underline\bold{ \underline{ \mathfrak{ \purple{Solving:}}}}$

$\bf{ \purple{ \underbrace{\red{let\:numberaontor\:and\: denominator\:be \: 2x\:and\:7 x .}}}}$

• If 4 is added to the numerator and 2 is subtracted from the denominator the rational number becomes 10/19.

$:\implies\sf \dfrac{2x + 4}{7x - 2} = \dfrac{10}{19}\\ \\ \\ \\ :\implies\sf 19(2x + 4) = 10(7x - 2)\\ \\:\implies\sf 38x + 76 = 70x - 20\\ \\:\implies\sf - 32x = - 96\\ \\:\implies\sf x = \cancel{ \dfrac{96}{32}}\\ \\ :\implies \overbrace{ \underbrace{ \overline{ \underline{ \purple{x= 3}}}} } \\ \\ \underline{ \dashrightarrow numerator= 2x= 2\times 3= 6 } \\ \\ \underline{denominator\:= 7x= 7\times 3 = 21 }\\ \\: \implies \dfrac{numerator}{denominator} = \dfrac{6}{21}$

$\rm{ \bold{\red{ \underline{ \pink{ \underline{ \overline{ \purple{ \overline{ \green{ \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:}}}}}}}}}}$

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