Question

the present ages of ajay and vijay are in ratio 4:5. Four years from now their ages will be in ratio 5:6. Find their present age. No spam ​

Answer 1

❍ The Given ratio of present ages of Ajay and Vijay is 4: 5. So, Let's say the present ages of Ajay and Vijay are 4x and 5x respectively.

☆ Four years from now their ages —

• Ajay's age after four years = (4x + 4)
• Vijay's age after four years = (5x + 4)

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

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• Four years from now their ages (Ajay's age and Vijay's age) will be in the ratio of 5: 6.

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Therefore,

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$:\implies\sf \bigg(\dfrac{4x + 4}{5x + 4} \bigg) = \bigg(\dfrac{5}{6}\bigg) \\\\\\:\implies\sf 6(4x + 4) = 5(5x + 4) \\\\\\:\implies\sf 24x + 24 = 25x + 20\\\\\\:\implies\sf 24x - 25x = 20 - 24 \\\\\\:\implies\sf -x = -4\\\\\\:\implies\underline{\boxed{\frak{\pmb{\purple{x = 4}}}}}\;\bigstar$

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Hence,

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• Ajay's present age = 4x = 4(4) = 16 years

• Vijay's present age = 5x = 5(4) = 20 years

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$\therefore{\underline{\sf{Hence, their\: present\;ages\:are\;\bf{16\; years\;\&\;20\:years }.}}}$

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

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

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• It is given as, Four years from now their ages (Ajay's age and Vijay's age) will be in the ratio of 5: 6. So, let's verify their ages :

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$\dashrightarrow\sf \dfrac{4x + 4}{5x + 4} = \dfrac{5}{6} \\\\\\\dashrightarrow\sf \dfrac{4(4) + 4}{5(4) + 4} = \dfrac{5}{6} \\\\\\\dashrightarrow\sf \dfrac{16 + 4}{20 + 4} = \dfrac{5}{6}\\\\\\\dashrightarrow\sf \cancel\dfrac{20}{24} = \dfrac{5}{6} \\\\\\\dashrightarrow\underline{\boxed{\frak{\dfrac{5}{6} = \dfrac{5}{6}}}}$

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

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Answer 2

Given : The present ages of ajay and vijay are in ratio 4:5 & four years from now their ages will be in ratio 5:6.

Exigency to find : The Present age of Ajay and Vijay .

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❒ Let's consider the Present ages of Ajay and Vijay be 4x and 5x yrs , respectively .

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:According \: to \: \: the \: Question \: \::}}$

⠀⠀⠀⠀⠀━━━ Four years from now the ages of Ajay and Vijay will be in ratio 5:6.

$\qquad \longmapsto \sf \dfrac{ Present \:Age\:of\:Ajay\: + 4 }{ Present \:Age\:of\:Vijay\: + 4 } = \dfrac{5}{6}$

$\qquad \longmapsto \sf \dfrac{ 4x\: + 4 }{ 5x\: + 4 } = \dfrac{5}{6}$

⠀⠀⠀⠀⠀By Cross Multiplication :

$\qquad \longmapsto \sf \dfrac{ 4x\: + 4 }{ 5x\: + 4 } = \dfrac{5}{6}$

$\qquad \longmapsto \sf 6(4x\: + 4) = 5( 5x\: + 4)$

$\qquad \longmapsto \sf 24x\: + 24 = 5(5x\: + 4)$

$\qquad \longmapsto \sf 24x\: + 24 = 25x\: + 20$

$\qquad \longmapsto \sf 24x\: = 25x\: + 20 - 24$

$\qquad \longmapsto \sf 24x\: = 25x\: - 4$

$\qquad \longmapsto \sf 24x\: - 25x = \: - 4$

$\qquad \longmapsto \sf -x\: = \: - 4$

$\qquad \longmapsto \sf \cancel{-} x\: = \: \cancel{-} 4$ [ "-ve" sign will be eliminated from both side]

$\qquad \longmapsto \frak{\underline{\purple{\:x = 4\:yrs }} }\:\:\:\bigstar$

Therefore,

• Present age of Ajay is 4x = 4(4) = 16 yrs .
• Present age of Vijay is 5x = 5(4) = 20 yrs .

Therefore,

⠀⠀⠀⠀⠀$\therefore {\underline{ \sf {\:Present \:age\:of\:Ajay \:and\:Vijay \:is}\:\bf{16\:yrs \:and\:20\:yrs}\:\:\sf{,respectively}}}$

$\rule200{1.5}$

$\large {\boxed{\sf{\mid{\overline {\underline {\star \: Verification \::}}}\mid}}}$

Given that :

⠀⠀⠀⠀⠀━━━ Four years from now the ages of Ajay and Vijay will be in ratio 5:6.

• The Present ages of Ajay and Vijay are 16 yrs & 20 yrs ,respectively.

$\qquad \longmapsto \sf \dfrac{ Present \:Age\:of\:Ajay\: + 4 }{ Present \:Age\:of\:Vijay\: + 4 } = \dfrac{5}{6}$

$\qquad \longmapsto \sf \dfrac{ 16\: + 4 }{ 20\: + 4 } = \dfrac{5}{6}$

$\qquad \longmapsto \sf \dfrac{ 20 }{ 24 } = \dfrac{5}{6}$

$\qquad \longmapsto \sf \cancel {\dfrac{ 20 }{ 24 }} = \dfrac{5}{6}$

$\qquad \longmapsto \sf \dfrac{ 5 }{ 6 } = \dfrac{5}{6}$

$\qquad \longmapsto \frak{\underline{\purple{\: \dfrac{5}{6} = \dfrac{5}{6}}} }\:\:\:\bigstar$

⠀⠀⠀⠀⠀$\therefore {\underline {\bf{ Hence, \:Verified \:}}}$

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