The ratio of present ages of A and B is 7 : 8. After 6 years from now, the ratio of their ages will be 8 : 9. If C's present age is 10 years more than the present age of A, then the present age (in years) of C is:
52
Let's break down this age problem step by step to find the present age of C.
Understanding the Age Ratio Problem
We are given the ratio of the present ages of two individuals, A and B. We are also told how this ratio changes after a certain number of years. Finally, we are given a relationship between the present age of A and the present age of C. Our goal is to use this information to calculate C's present age.
Setting Up the Equations
Let the present ages of A and B be represented by \(7x\) and \(8x\) years, respectively, based on the given ratio of 7:8.
- Present age of A = \(7x\) years
- Present age of B = \(8x\) years
After 6 years from now, their ages will be:
- Age of A after 6 years = \(7x + 6\) years
- Age of B after 6 years = \(8x + 6\) years
The problem states that the ratio of their ages after 6 years will be 8:9. We can write this as an equation:
\(\frac{7x + 6}{8x + 6} = \frac{8}{9}\)
Solving for the Unknown Variable (x)
Now, we solve this equation for \(x\). We can do this by cross-multiplication:
\(9(7x + 6) = 8(8x + 6)\)
\(63x + 54 = 64x + 48\)
To isolate \(x\), we can subtract \(63x\) from both sides and subtract 48 from both sides:
\(54 - 48 = 64x - 63x\)
\(6 = x\)
So, the value of \(x\) is 6.
Calculating Present Ages
Now that we have the value of \(x\), we can find the present ages of A and B:
- Present age of A = \(7x = 7 \times 6 = 42\) years
- Present age of B = \(8x = 8 \times 6 = 48\) years
We can quickly verify the ratio after 6 years:
- Age of A after 6 years = \(42 + 6 = 48\) years
- Age of B after 6 years = \(48 + 6 = 54\) years
- Ratio after 6 years = \(\frac{48}{54} = \frac{8}{9}\). This matches the problem statement.
Finding C's Present Age
The problem states that C's present age is 10 years more than the present age of A.
- Present age of A = 42 years
- Present age of C = Present age of A + 10 years
- Present age of C = \(42 + 10 = 52\) years
Therefore, the present age of C is 52 years.
Here is a summary of the ages:
| Person | Present Age | Age after 6 years |
|---|---|---|
| A | 42 years | 48 years |
| B | 48 years | 54 years |
| C | 52 years | - |
Revision Table: Key Steps in Solving Age Problems
| Step | Description |
|---|---|
| 1 | Represent unknown ages using variables and given ratios. |
| 2 | Formulate equations based on how ages change over time (adding or subtracting years). |
| 3 | Set up an equation using the ratio of ages at a future or past point in time. |
| 4 | Solve the equation to find the value of the variable. |
| 5 | Substitute the variable's value back into the expressions for the ages to find the actual ages. |
| 6 | Use any additional information (like C's age relative to A's) to find the required age. |
Additional Information: Ratio Concepts in Math
A ratio is a comparison of two quantities. In this age problem, the ratio 7:8 for A and B's present ages means that for every 7 years of A's age, B is 8 years old. This relationship holds true as long as both ages are multiplied by the same factor (our variable \(x\)).
When dealing with age problems involving ratios and time changes:
- The difference between the ages of two people remains constant throughout their lives. In this case, B is always \(8x - 7x = x\) years older than A. Since \(x=6\), B is always 6 years older than A.
- Present: B (48) - A (42) = 6 years
- After 6 years: B (54) - A (48) = 6 years
- Adding or subtracting years affects both individuals' ages equally. This is why we added 6 to both \(7x\) and \(8x\).
Solving ratio-based age problems often involves algebra to find the unknown multiplier that relates the ratio parts to the actual ages.
Important Questions from Problem on Age
Three years ago, the ratio of the age of father to that of his son was 8 ∶ 3. After 4 years, their ages will be in the ratio 11 ∶ 5. What is the present age (in years) of the father?
At present, A is younger than B by 8 years. If 4 years ago, their ages were in the ratio 1 ∶ 2, then what is the present age of B (in years)?
Eight years ago, the ratio of ages of A and B was 5 ∶ 4. The ratio of their present ages is 6 ∶ 5. What will be the sum (in years) of the ages of A and B after 7 years from now?
Ratio of the present age of a mother to that of the daughter is 7 ∶ 1. After 5 years the ratio will become 4 ∶ 1. What is the difference (in years) in their present ages?
The sum of the presents age of a father and son is 52 years Four years hence, the son's age will be 1/4 that of the father. What will be the ratio of the age of the son and father, 10 years from now?
