Question

7. The sum of a numerator and denominator of a fraction is 18. If the denominator is
increased by 2, the fraction reduces to 1/3. Find the fraction.
​

Answer 1

$\sf \bf \huge {\boxed {\mathbb {QUESTION}}}$

$\tt 7. \:The\: sum \:of \:a\: numerator\: and\: denominator\: of\: a \: fraction\\\tt is\: 18. \: If\: the\: denominator\: is \:increased\: by \: 2, \\\tt \: the\: fraction\: reduces\: to\:\dfrac{1}{3}. Find\: the \:fraction.$

$\sf \bf \huge {\boxed {\mathbb {ANSWER}}}$

${\color {springgreen} \underline {\sf Let \:numerator \:be\: x}}$

${\color {springgreen} \underline {\sf Let\: denominator\: be \:y}}$

$\sf \bf {\boxed {\mathbb {GIVEN}}}$

• $\implies {\boxed {\color {teal} {\bf x+y=18 \longrightarrow (i)}}}$

$\sf \bf \implies \dfrac{x}{y+2}=\dfrac{1}{3}$

$\sf \bf \implies 3x=y+2$

• $\implies {\boxed {\color {teal} {\bf x=\dfrac{y+2}{3} \longrightarrow (ii)}}}$

$\sf \bf {\boxed {\mathbb {TO\:FIND}}}$

• $\sf \bf The \:fraction \bigg(\dfrac{x}{y}\bigg)$

$\sf \bf {\boxed {\mathbb {SOLUTION}}}$

$\implies {\color {indigo} {\bf x+y=18 \longrightarrow (i)}}$

$\implies {\color {indigo} {\bf x=\dfrac{y+2}{3} \longrightarrow (ii)}}$

${\underbrace {\overbrace {\color {orange} {\sf Substitute \:the \:value \:of \:equation \:(ii) \:in\:equation \:(i)}}}}$

$\sf \bf \implies \dfrac{y+2}{3}+y=18$

$\sf \bf \implies \dfrac{y+2+3y}{3}=18$

$\sf \bf \implies y+2+3y=18\times 3$

$\sf \bf \implies 4y+2=54$

$\sf \bf \implies 4y=54-2$

$\sf \bf \implies 4y=52$

$\sf \bf \implies y=\dfrac{52}{4}$

$\sf \bf \implies y=\dfrac{\cancel{52}}{\cancel{4}}$

$\implies{\boxed {\boxed {\color {blue} {\sf \bf y=13}}}}$

${\underbrace {\overbrace {\color {orange} {\sf Substitute \:the \:value \:of \:y \:in\:equation \:(i)}}}}$

$\sf \bf \implies x+y=18$

$\sf \bf \implies x+13=18$

$\sf \bf \implies x=18-13$

$\implies{\boxed {\boxed {\color {blue} {\sf \bf x=5}}}}$

${\color {black}{\underbrace \color{red}{\underline \color{red}{\sf \therefore The\:fraction \:\dfrac{x}{y} \:is\:\dfrac{5}{13}}}}}$

$\sf \bf \huge {\boxed {\mathbb {HOPE \:IT \:HELPS \:YOU}}}$

___________________________________________

$\sf \bf \huge {\boxed {\mathbb {EXTRA\:INFORMATION}}}$

${\color {aqua} {\sf Identities}}$

$\sf \bf {(a+b)}^{2}={a}^{2}+2ab+{b}^{2}$

$\sf \bf {(a-b)}^{2}={a}^{2}-2ab+{b}^{2}$

$\sf \bf (a+b) (a-b) ={a}^{2}-{b}^{2}$

Answer 2

Answer:

$⟹4y+2=54</p><p></p><p>\sf \bf \implies 4y=54-2⟹4y=54−2</p><p></p><p>\sf \bf \implies 4y=52⟹4y=52</p><p></p><p>\sf \bf \implies y=\dfrac{52}{4}⟹y=452</p><p></p><p>\sf \bf \implies y=\dfrac{\cancel{52}}{\cancel{4}}⟹y=452</p><p></p><p>\implies{\boxed {\boxed {\color {blue} {\sf \bf y=13}}}}⟹y=13</p><p></p><p>{\underbrace {\overbrace {\color {orange} {\sf Substitute \:the \:value \:of \:y \:in\:equation \:(i)}}}}Substitutethevalueofyinequation(i)</p><p></p><p>\sf \bf \implies x+y=18⟹x+y=18</p><p></p><p>\sf \bf \implies x+13=18⟹x+13=18</p><p></p><p>\sf \bf \implies x=18-13⟹x=18−13</p><p></p><p>\implies{\boxed {\boxed {\color {blue} {\sf \bf x=5}}}}⟹x=5</p><p></p><p>$