5 years ago, age of A is 3 times of the age of B at that time. Ratio of age of B, 5 year hence, and age of C, 5 year ago, is 3:5. If difference between age of Cand A is 10 years(C is younger than A), then find the age of B.

Options

A. A)20 years
B. B)15 years
C. C)18 years
D. D)25 years

Answer

To solve the problem step by step, we will define the ages of A, B, and C based on the information given in the question.

### Step 1: Define the variables
Let the present age of B be \( k \) years. According to the problem, 5 years ago, the age of A was 3 times the age of B.

### Step 2: Express A's age in terms of B's age
Five years ago, the age of A was \( 3 \times \) (age of B 5 years ago). Therefore:
- Age of B 5 years ago = \( k - 5 \)
- Age of A 5 years ago = \( 3(k - 5) \)

### Step 3: Express A's current age
The current age of A can be expressed as:
\[ \text{Age of A} = 3(k - 5) + 5 = 3k - 15 + 5 = 3k - 10 \]

### Step 4: Express C's age in terms of A's age
We know that C is 10 years younger than A:
\[ \text{Age of C} = \text{Age of A} - 10 = (3k - 10) - 10 = 3k - 20 \]

### Step 5: Set up the ratio of B's future age and C's past age
The problem states that the ratio of the age of B 5 years hence to the age of C 5 years ago is 3:5. Therefore:
- Age of B 5 years hence = \( k + 5 \)
- Age of C 5 years ago = \( 3k - 20 - 5 = 3k - 25 \)

Setting up the ratio:
\[ \frac{k + 5}{3k - 25} = \frac{3}{5} \]

### Step 6: Cross-multiply to solve for k
Cross-multiplying gives us:
\[ 5(k + 5) = 3(3k - 25) \]
Expanding both sides:
\[ 5k + 25 = 9k - 75 \]

### Step 7: Rearranging the equation
Rearranging the equation to isolate \( k \):
\[ 25 + 75 = 9k - 5k \]
\[ 100 = 4k \]
\[ k = \frac{100}{4} = 25 \]

### Step 8: Find the age of B
The present age of B is:
\[ \text{Age of B} = k = 25 \text{ years} \]

### Conclusion
The age of B is **25 years**.