Question

.Ten years ago, the age of a person’s mother was three times the age of her son. Ten years hence, the mother’s age will be two times the age of her son. The ratio of their present ages will be:​

Answer 1

Answer: Let the age of the son ten years ago be equal to x years. Therefore the age of the mother ten years ago will be equal to 3x. Following this logic, we see that the present age of the son will be equal to x + 10 years and that of the mother will be equal to 3x + 10 years.

The second condition says that ten years from the present, the mother’s age will be twice that of her son. After ten year’s the mother’s age will be 3x + 10 + 10 years and that of the son will be x + 10 + 10 years. As per the question we have:

(3x + 10) +10 = 2 [(x + 10) + 10] or (3x + 20) = 2[x + 20]. In other words, we can write:

x = 20 years. Thus the present age of the mother = 3(20) + 10 = 70 years. Also the present age of the son = 20 + 10 = 30 years. Thus the ratio is 7:3 and the correct option is 7:3.

Step-by-step explanation:

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Answer 2

Step-by-step explanation:

$\bf{\underline{\underline\red{Given:-}}}$

• Ten years ago, age of mother was three times the age of her son.

• Ten years hence, mother's age will be twice the age of her son.

$\bf{\underline{\underline\blue{To\:Find:-}}}$

• The ratio of the present ages of mother and son.

$\bf{\underline{\underline\green{Solution:-}}}$

Let the present age of son be x

And present age of Mother be y

According to the 1st condition:-

Ten years ago:-

Age of son = x - 10

Age of mother = y - 10

Given,Ten years ago, age of mother was three times the age of her son.

So:-

$\implies \sf y - 10 = 3(x - 10)$

$\implies \sf y - 10 = 3x - 30$

$\implies \sf y - 3x= - 30 + 10$

$\implies \sf - 3x + y = - 20$

$\implies \sf - ( -3x + y = - 20)$

$\implies \sf 3x - y = 20......(i)$

According to the 2nd condition:-

Ten years hence

Age of son = x + 10

Age of mother = y + 10

Given,Ten years hence, mother's age will be twice the age of her son.

So:-

$\implies \sf y + 10 = 2(x + 10)$

$\implies \sf y + 10 =2x + 20$

$\implies \sf y - 2x = 20 - 10$

$\implies \sf - 2x + y= 10.....(ii)$

Adding equations (i) and (ii)

$\implies \sf 3x - y + (- 2x+ y)= 20 + 10$

$\implies \sf 3x - y- 2x + y= 30$

$\implies \sf x= 30$

Substituting x = 30 in equation (i)

$\implies \sf 3x - y= 20$

$\implies \sf 3(30) - y= 20$

$\implies \sf 90 - y= 20$

$\implies \sf - y= 20 - 90$

$\implies \sf - y= - 70$

$\implies \sf y= 70$

The present age of son = x = 30

The present age of mother = y = 70

The ratio of ages of mother to son

$\sf = \dfrac{70}{30}$

$\sf = \dfrac{7}{3}$

$\sf \implies Mother : Son= 7 :3$

$\underline{ \boxed{ \purple { \rm\therefore The \: ratio \: of \: their \: present \: ages= 7 :3}}}$