Question

The ratio of two numbers is 3:4. When 5 is added to both, the new ratio
becomes 4:5. What are number?​

Answer 1

❍ Let's say, that the required two numbers be 3x and 4x respectively.

$\underline{\bigstar\:\boldsymbol{According~to~the~~ Question~:}}$

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• As per given condition, when five is added to both the numbers, then the new ratio becomes 4: 5.

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Therefore,

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$:\implies\sf \bigg\{\dfrac{3x + 5}{4x + 5}\bigg\} = \bigg\{\dfrac{4}{5}\bigg\} \\\\\\:\implies\sf 5\Big\{3x + 5\Big\} = 4\Big\{4x + 5\Big\} \\\\\\:\implies\sf 15x + 25 = 16x + 20 \\\\\\:\implies\sf 15x - 16x = 20 - 25 \\\\\\:\implies\sf \cancel{-}~x = \cancel{-}~5 \\\\\\:\implies\underline{\pink{\boxed{\pmb{\frak{x = 5}}}}}\:\bigstar$

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Hence,

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• First number, 3x = 3(5) = 15

• Second number, 4x = 4(5) = 20

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$\therefore{\underline{\textsf{Hence,~two~required~numbers~are~\textbf{15}~\sf{and}~\textbf{20}.}}}$

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$\rule{250px}{.3ex}$

V E R I F I C A T I O N :

• As it is given that, when five is added to both the numbers, then the new ratio becomes 4: 5.

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Therefore,

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$\dashrightarrow\sf \dfrac{3x + 5}{4x + 5} = \dfrac{4}{5} \\\\\\\dashrightarrow\sf \dfrac{3(5) + 5}{4(5)+5}=\dfrac{4}{5} \\\\\\\dashrightarrow\sf \dfrac{15 + 5}{20 + 5} = \dfrac{4}{5}\\\\\\\dashrightarrow\sf \cancel\dfrac{20}{25} = \dfrac{4}{5}\\\\\\\dashrightarrow\underline{\boxed{\pmb{\frak{\dfrac{4}{5} = \dfrac{4}{5}}}}}$

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$\qquad\quad\therefore{\underline{\purple{\textsf{\textbf{Hence, Verified!}}}}}$⠀⠀⠀

Answer 2

★ Let the numbers be 3k and 4k respectively.

$\:$

• It is given that, when 5 is added to both numbers the new ratio becomes 4 : 5.

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$\: \:$

$\bold{ \underline{★ \:According \: to \: the \: question \: : }}$

$\: \:$

$\sf : \implies \: \dfrac{3k + 5}{4k + 5} = \dfrac{4}{5} \\ \\ \\ \sf : \implies \: 5(3k + 5) = 4(4k + 5) \\ \\ \\ \sf : \implies \: 15k + 25 = 16k + 20 \\ \\ \\ \sf : \implies \: 16k - 15k = 25 - 20 \\ \\ \\ \sf : \implies \: \underline{\boxed{\sf\purple{k = 5}}} \: \bigstar$

$\: \:$

$\: \:$

• first number = 3k = 3(5) = 15
• second number = 4(k) = 4(5) = 20

$\: \:$

$\sf\therefore{\underline{Hence, \: the \: two \: numbers \: are \: \bold{15} \: and \: \bold{20} \: respectively.}}$