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Mother is 25 years older than her son. Find the age of the son if, after 8 years, the ratio of son’s age to mother’s age will be \[\dfrac{4}{9}\].

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Hint: Assign variables to mother’s age and son’s age. Write the equations relating to their age in the present and after 8 years and solve them to find the son’s age.

Complete step by step answer:
Let the present age of the mother be x years and the present age of the son be y years.
It is given that a mother is 25 years older than her son. Then, we have:

\[x = y + 25............(1)\]

After eight years, the ratio of son’s age to the mother’s age is 4:9. We know that the age of the mother after 8 years is (x + 8) and the age of the son after 8 years is (y + 8). Hence, we have:

\[\dfrac{{y + 8}}{{x + 8}} = \dfrac{4}{9}............(2)\]

We now have two equations with two unknowns, hence, we substitute equation (1) in equation (2), to obtain the age of the son.

\[\dfrac{{y + 8}}{{(y + 25) + 8}} = \dfrac{4}{9}\]

Simplifying the term in the denominator, we get:

\[\dfrac{{y + 8}}{{y + 33}} = \dfrac{4}{9}\]

Cross-multiplying, we get:

\[9(y + 8) = 4(y + 33)\]

Multiplying inside the brackets, we obtain:

\[9y + 72 = 4y + 132\]

Taking all terms containing y to the left-hand side and all constant terms to the right hand side, we have:

\[9y - 4y = 132 - 72\]

Simplifying, we get:

\[5y = 60\]

Solving for y, we get:

\[y = \dfrac{{60}}{5}\]

\[y = 12\]

Hence, the present age of the son is 12 years.

Note: After you obtain the age of the son, find the age of the mother and substitute in the equations to check if the answer is correct. The key in such questions is to translate the statements of the problem into mathematical statements.
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