Question

The ratio of the age of a man and his wife is 4: 3. After 4 years, this ratio will be 9 : 7. If at
the time of the marriage, the ratio was 5:3, then how many years ago they were married?
(a) 12 years
(d) 15 years
(b) 8 years
(c) 10 years​

Answer 1

❍ 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

⠀⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

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

⠀⠀⠀⠀⠀

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

⠀⠀⠀⠀⠀

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

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.

⠀⠀⠀⠀

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

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

Answer 2

Answer:

Given :-

• The ratio of the age of a man and his wife is in the ratio of 4 : 3. After 4 years, their ratio is 9 : 7. At the time of the marriage, the ratio was 5 : 3.

To Find :-

• How many years ago they were marriage.

Solution :-

Let, the present age of man or husband be 4x years

And, the present age of his wife will be 3x years

After 4 years ago :

Age of man or husband is 4x + 4 years

And, the age of his wife is 3x + 4 years

According to the question,

↦ $\sf \dfrac{4x + 4}{3x + 4} =\: \dfrac{9}{7}$

By doing cross multiplication we get,

↦ $\sf 7(4x + 4) =\: 9(3x + 4)$

↦ $\sf 28x + 28 =\: 27x + 36$

↦ $\sf 28x - 27x =\: 36 -28$

➠ $\sf\bold{\pink{x =\: 8\: years}}$

Hence, the required ages of man and his wife is :

✪ Present age of man or husband :

↦ $\sf 4x\: years$

↦ $\sf 4(8)\: years$

↦ $\sf 4 \times 8\: years$

➦ $\sf\bold{\green{32\: years}}$

And,

✪ Present age of his wife :

↦ $\sf 3x\: years$

↦ $\sf 3(8)\: years$

↦ $\sf 3 \times 8\: years$

➦ $\sf\bold{\green{24\: years}}$

Hence, the present age of husband is 32 years and the present age of his wife is 24 years.

Now,

Let, the their marriage took place Y years back.

The time of the marriage, the ratio of man and his wife is 5 : 3.

Again, according to the question,

⇒ $\sf \dfrac{32 - Y}{24 - Y} =\: \dfrac{5}{3}$

By doing cross multiplication we get,

⇒ $\sf 5(24 - Y) =\: 3(32 - Y)$

⇒ $\sf 120 - 5Y =\: 96 - 3Y$

⇒ $\sf 120 - 96 =\: - 3Y + 5Y$

⇒ $\sf 24 =\: 2Y$

⇒ $\sf \dfrac{\cancel{24}}{\cancel{2}} =\: Y$

➠ $\sf\bold{\red{Y =\: 12\: years}}$

$\therefore$ They were marriage 12 years back.

Hence, the correct options is option no (a) 12 years.