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` times 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

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the given information
We know that:
- 12 years ago, the ratio of the ages of P to Q was 3:4.
- The present age of P is \( \frac{3 \frac{3}{5}}{1} \) times the present age of R.
- R's present age is 10 years.

### Step 2: Calculate the present age of P
First, we need to convert \( 3 \frac{3}{5} \) into an improper fraction:
\[ 3 \frac{3}{5} = \frac{3 \times 5 + 3}{5} = \frac{15 + 3}{5} = \frac{18}{5} \]

Now, we can find P's present age using R's present age:
\[ \text{Present age of P} = \frac{18}{5} \times 10 = \frac{180}{5} = 36 \text{ years} \]

### Step 3: Set up the equation for Q's age
Let Q's present age be \( Q \). According to the problem, 12 years ago the ages of P and Q were:
- Age of P 12 years ago = \( 36 - 12 = 24 \)
- Age of Q 12 years ago = \( Q - 12 \)

Using the ratio of their ages 12 years ago:
\[ \frac{24}{Q - 12} = \frac{3}{4} \]

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

### Step 5: 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
Thus, Q's present age is **44 years**.

---