Given:
The ratio of the present ages of Manthan and his mother is 3 ∶ 7 respectively
The ratio of the present ages of Manthan and his father is 5 ∶ and 14 respectively.
Calculation:
Let's denote Manthan's age as Ma, his mother's age as Mo, and his father's age as F.
According to the question,
Manthan's age to his mother's age forms a ratio of 3 : 7.
⇒ \(\dfrac{Ma}{Mo} = \dfrac{3}{7} \)
Manthan's age to his father's age forms a ratio of 5 : 14.
⇒ \(\dfrac{Ma}{F} = \dfrac{5}{14} \)
From these two equations, we can express the Mother's age and the Father's age in terms of Manthan's age:
⇒ Mo = \( \dfrac{7}{3} \) × Ma
⇒ F = \( \dfrac{14}{5} \) × Ma
Now we need to find the ratio of the Mother's age to the Father's age:
⇒ \(\dfrac{Mo}{F} \)= (\( \dfrac{7}{3} \)× Ma)/(\(\dfrac{14}{5} \) × Ma)
⇒ \(\dfrac{Mo}{F} \)= (\( \dfrac{7}{3} \))/(\(\dfrac{14}{5} \)) = (7 × 5)/(14 × 3) = \(\dfrac{35}{42} \) = \(\dfrac{5}{6} \)
∴ The ratio between the present ages of Manthan's mother and his father is 5:6.