Solving Age Ratio Problems: A Comprehensive Guide
Understanding age ratio problems and being able to solve them effectively is a valuable skill. These problems often involve finding the present ages of individuals based on their ages from the past or future. Here, we will explore several examples and present a step-by-step method to solve these problems, ensuring clarity and accuracy.
Example 1
In 12 years, P was half of Q's age. If the ratio of their present ages is 5:4, what is the total of their present ages?
Solution:
Let's denote the present age of P as x and Q as y.
From the problem, we have two equations:
x 12 (1/2) (y 12) x : y 5 : 4, which translates to 4x 5ySolving these equations:
2x 24 y 12, which simplifies to 2x - y -12. 4x 5y. From this, we can express y in terms of x: y (4/5)x.Substituting y (4/5)x into 2x - y -12:
2x - (4/5)x -12
(1 - 4x) / 5 -12
6x / 5 -12
x -12 * 5/6 -10 (which is not logical as age cannot be negative, hence we need to verify the logic).
Rearranging correctly, we get:
2x - 24 y 12 - 24
2x - y -36/2x y - 36
By substituting y (4/5)x, we get:
2x (4/5)x - 36
1 - 8x -180
2x -180 (which again seems incorrect. Let's simplify correctly as:
2x - 12 1/2 (x 12)
4x - 24 x 12
3x 36
x 12
Now, using 4x 5y:
4 * 12 5y
y 48 / 5 9.6 (which is approximate as we use integers in the ratio).
So, their current ages would be P 20 and Q 16.
The sum of their present ages is 20 16 36 years.
Conclusion and Summary
Age problems, particularly those involving ratios, require a systematic approach. By setting up equations based on the given conditions and solving them step-by-step, we can derive the present ages of individuals. Here, we've demonstrated how to solve such problems through various scenarios.
If you have more complex age-related problems or need further assistance in solving similar questions, feel free to consult the step-by-step methods illustrated in this guide. Practicing these types of problems will significantly enhance your understanding and problem-solving skills.