12 years ago, the ratio of age of P to age of Q was 3:4. The present age of P is `3(3)/(5)` time of R's present age. If R's present age is 10 years then what is the Q's present age ?

Options

A. 32 years
B. 48 years
C. 44 years
D. 58 years

Explanation

To solve the problem step by step, we will follow the information given in the question and use algebraic expressions to find the present age of Q.

### Step 1: Define the variables
Let:
- P's present age = \( P \)
- Q's present age = \( Q \)
- R's present age = \( R \)

### Step 2: Use the information given
From the problem, we know:
- R's present age \( R = 10 \) years.
- The present age of P is \( \frac{18}{5} \) times R's present age.

### Step 3: Calculate P's present age
Using the information about R's age:
\[ P = \frac{18}{5} \times R = \frac{18}{5} \times 10 = \frac{180}{5} = 36 \text{ years} \]

### Step 4: Use the ratio of ages from 12 years ago
12 years ago, the ratio of the ages of P and Q was 3:4. Therefore, we can express this as:
\[ \frac{P - 12}{Q - 12} = \frac{3}{4} \]

### Step 5: Substitute P's present age into the equation
We already found that \( P = 36 \):
\[ \frac{36 - 12}{Q - 12} = \frac{3}{4} \]
This simplifies to:
\[ \frac{24}{Q - 12} = \frac{3}{4} \]

### Step 6: Cross-multiply to solve for Q
Cross-multiplying gives us:
\[ 24 \cdot 4 = 3 \cdot (Q - 12) \]
\[ 96 = 3Q - 36 \]

### Step 7: Solve for Q
Now, add 36 to both sides:
\[ 96 + 36 = 3Q \]
\[ 132 = 3Q \]
Now divide by 3:
\[ Q = \frac{132}{3} = 44 \text{ years} \]

### Conclusion
The present age of Q is \( 44 \) years.

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