Question

The ratio of a man and his wife is $4:3$. After $4$ years this ratio will be $9:7$. If at the time of marriage this ratio was $5:3$, then how many years ago they married?
A. $12years$
B. $8years$
C. $10years$
D. $15years$

Answer 1

Hint: In the given question fist assume the present age of husband and wife is $x$, then the ratio of age after $4$ years is given, we will make equations according to this condition and put it equal to the ratio. Hence we will get the value of $x$ now we will put this value in assumed value and get the present age of husband and wife. The second condition is the ratio at the time of marriage is given; now we will make an equation according to this and put it equal to the ratio. Hence we will get how many years ago they married.

Complete step by step Answer:

Let the present age will be $4x{\text{ }}and{\text{ }}3x$
According to the question:
After $4$ years the husband age $ = 4x + 4..........\left( 1 \right)$
And wife age$ = 3x + 4.........\left( 2 \right)$
Given that after $4$ years ratio will be $9:7$
Divide equation $1{\text{ }}by{\text{ }}2$ and put is equal to $9:7$
We get:
$
   \Rightarrow \dfrac{{4x + 4}}{{3x + 4}} = \dfrac{9}{7} \\
   \Rightarrow 7\left( {4x + 4} \right) = 9\left( {3x + 4} \right) \\
   \Rightarrow 28x + 28 = 27x + 36 \\
   \Rightarrow 28x - 27x = 36 - 28 \\
   \Rightarrow x = 8 \\
 $
Thus we get the value of $x$
Now put the value of $x$in present age of husband and wife:
We get:
Husband age $ = 4 \times 8 = 32$ years
Wife age=$3 \times 8 = 24$ years
Hence we get the present age of husband and wife.
Now again according to the question:
At the time of marriage, this ratio was$5:3$
Let the marriage took place $y$ years back
Now by given conditions:
We get:
$
   \Rightarrow \dfrac{{32 - y}}{{24 - y}} = \dfrac{5}{3} \\
   \Rightarrow 3\left( {32 - y} \right) = 5\left( {24 - y} \right) \\
   \Rightarrow 96 - 3y = 120 - 5y \\
   \Rightarrow 96 - 120 = - 5y + 3y \\
   \Rightarrow 24 = 2y \\
   \Rightarrow y = 12 \\
 $
Thus the marriage took place $12$ years back.
Hence the correct answer is option A.

Note: In the given problem first we have to assume the value of the present age of husband and wife, without doing this we cannot find the present age of them. Then we have to make two equations one for age after four years because the present age is given and second for at the time of marriage according to the given conditions. Put the ratio of equations of age after four years of the present age of husband and wife equal to the given ratio of age after four years. Then we have to make equations for the age at the time of marriage and put the ratio of both ages at the time of marriage equality to the given ration. Thus we get the correct answer.