Solving Age Ratio Problems: Finding Present Age

This problem involves finding the present age of a person based on age ratios at different times and a relationship between two individuals' ages. We are given the initial ratio of ages of A and B, the ratio after a certain number of years, and the age difference between C and B.

Setting Up the Equations for Age Ratios

Let the present age of A be $8x$ years and the present age of B be $9x$ years, according to the given ratio of 8:9.

After 9 years:

• Age of A will be $8x + 9$ years.
• Age of B will be $9x + 9$ years.

The problem states that after 9 years, the ratio of their ages will be 19:21. We can write this as an equation:

\begin{equation} \frac{8x + 9}{9x + 9} = \frac{19}{21} \end{equation}

Solving for the Unknown Variable

Now, we need to solve this equation for $x$. We can cross-multiply:

\begin{align*} 21(8x + 9) &= 19(9x + 9) \\ 168x + 189 &= 171x + 171 \end{align*}

Next, we rearrange the terms to isolate $x$:

\begin{align*} 189 - 171 &= 171x - 168x \\ 18 &= 3x \end{align*}

Finally, solve for $x$:

\begin{equation} x = \frac{18}{3} = 6 \end{equation}

Calculating Present Ages of A and B

Now that we have the value of $x$, we can find the present ages of A and B:

• Present age of A = $8x = 8 \times 6 = 48$ years.
• Present age of B = $9x = 9 \times 6 = 54$ years.

Finding the Present Age of C

The problem states that C is 3 years younger than B. To find the present age of C, we subtract 3 from B's present age:

\begin{equation} \text{Present age of C} = \text{Present age of B} - 3 \end{equation}

\begin{equation} \text{Present age of C} = 54 - 3 = 51 \text{ years} \end{equation}

Therefore, the present age of C is 51 years.

Let's verify the ratio of A and B's ages after 9 years: $\frac{57}{63}$. Dividing both numerator and denominator by 3, we get $\frac{19}{21}$, which matches the ratio given in the problem. This confirms our value of $x$ is correct.

Conclusion

Based on the age ratios and the relationship between B and C's ages, the present age of C is 51 years.

Revision Table: Key Steps in Age Problems

Additional Information: Understanding Age and Ratio Concepts

Age problems often involve setting up linear equations based on given conditions related to age differences, ratios, or sums of ages at different points in time. The key is to correctly translate the word problem into algebraic expressions and equations.

• Ratio: A ratio compares two quantities. If the ratio of A's age to B's age is $m:n$, it means for some value $x$, A's age is $mx$ and B's age is $nx$.
• Age Progression: If someone's current age is $A$, their age after $y$ years will be $A+y$, and their age $y$ years ago was $A-y$.
• Age Difference: The difference between two people's ages remains constant over time. If A is $k$ years older than B today, A will still be $k$ years older than B in any number of years.

These fundamental concepts help in solving various types of age-related word problems encountered in quantitative aptitude sections of exams.