Question

The ratio of the age of a man and his wife is 4:3. at the time of marriage the ratio was 5:3 and after 4 years this ratio will become 9:7. how many years ago were they married?

Answer 1

❍ Lets consider the Present age of man and his wife be 4x and 3x respectively.

Then, After 4 years,

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Husband's age = (4x + 4) years

Wife's age = (3x + 4) years

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$\begin{gathered}\underline{\bigstar\:\boldsymbol{According\:to \:the \:given \:Condition,}}\\\end{gathered}$

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After 4 years, The ratio of the age of a man and his wife will be 9 : 7.

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$\begin{gathered}:\implies\sf \dfrac{4x + 4}{3x + 4} = \dfrac{9}{7}\\\\\\ :\implies\sf 7(4x + 4) = 9(3x + 4)\\\\\\ :\implies\sf 28x + 28 = 27x + 36\\\\\\ :\implies\sf 28x - 27x = 36 - 28\\\\\\ :\implies{\underline{\boxed{\frak{\purple{x = 8}}}}}\:\bigstar\\\\\end{gathered}$

Therefore,

The Present age of Man, 4x = 4 × 8 = 32 years

And, The Present age of his wife, 3x = 3 × 8 = 24 years

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¤ Now, Lets assume that their marriage took place in "T years" back.

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Then, By given Condition,

The ratio of the age of a man and his wife at the time of marriage is 5:3.

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$\begin{gathered}:\implies\sf \dfrac{32 - T}{24 - T} = \dfrac{5}{3}\\\\\\ :\implies\sf 3(32 - T) = 5(24 - T)\\\\\\ :\implies\sf 96 - 3T = 120 - 5T\\\\\\ :\implies\sf 5T - 3T = 120 - 96\\\\\\:\implies\sf 2T = 24\\\\\\:\implies\sf T = \cancel{\dfrac{24}{2}}\\\\\\ :\implies{\underline{\boxed{\frak{\pink{T = 12\:years}}}}}\:\bigstar\\\\\end{gathered}$

$\therefore\:{\underline{\sf{Hence,\:\sf{They\:were\:married}\:\pmb{12\:years}\:\sf{ago}.}}}$

Answer 2

❍ Lets consider the Present age of man and his wife be 4x and 3x respectively.

Then, After 4 years,

⠀⠀⠀⠀⠀

Husband's age = (4x + 4) years

Wife's age = (3x + 4) years

⠀⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

$\begin{gathered}\begin{gathered}\underline{\bigstar\:\boldsymbol{According\:to \:the \:given \:Condition,}}\\\end{gathered} \end{gathered}$

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After 4 years, The ratio of the age of a man and his wife will be 9 : 7.

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$\begin{gathered}\begin{gathered}:\implies\sf \dfrac{4x + 4}{3x + 4} = \dfrac{9}{7}\\\\\\ :\implies\sf 7(4x + 4) = 9(3x + 4)\\\\\\ :\implies\sf 28x + 28 = 27x + 36\\\\\\ :\implies\sf 28x - 27x = 36 - 28\\\\\\ :\implies{\underline{\boxed{\frak{\purple{x = 8}}}}}\:\bigstar\\\\\end{gathered} \end{gathered}$

Therefore,

The Present age of Man, 4x = 4 × 8 = 32 years

And, The Present age of his wife, 3x = 3 × 8 = 24 years

⠀⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

¤ Now, Lets assume that their marriage took place in "T years" back.

⠀⠀⠀⠀

Then, By given Condition,

The ratio of the age of a man and his wife at the time of marriage is 5:3.

⠀⠀⠀⠀

$\begin{gathered}\begin{gathered}:\implies\sf \dfrac{32 - T}{24 - T} = \dfrac{5}{3}\\\\\\ :\implies\sf 3(32 - T) = 5(24 - T)\\\\\\ :\implies\sf 96 - 3T = 120 - 5T\\\\\\ :\implies\sf 5T - 3T = 120 - 96\\\\\\:\implies\sf 2T = 24\\\\\\:\implies\sf T = \cancel{\dfrac{24}{2}}\\\\\\ :\implies{\underline{\boxed{\frak{\pink{T = 12\:years}}}}}\:\bigstar\\\\\end{gathered} \end{gathered}$

$\therefore\:{\underline{\sf{Hence,\:\sf{They\:were\:married}\:\pmb{12\:years}\:\sf{ago}.}}}$