12 yr ago the ratio between the ages of A and B was ` 3: 4` respectively. The present age of A is ` 3 3/4` times of C's present age. If C's present age is 10 yr, then what is B's present age? (in years)

Options

A. 48
B. 46
C. 36
D. 54

Explanation

To solve the problem step by step, we will follow the information provided in the question and use it to find B's present age.

### Step 1: Determine C's present age
Given that C's present age is 10 years.

### Step 2: Calculate A's present age
We know that A's present age is \(3 \frac{3}{4}\) times C's present age. We can convert \(3 \frac{3}{4}\) into an improper fraction:
\[ 3 \frac{3}{4} = \frac{15}{4} \]
Now, we can calculate A's present age:
\[ \text{A's present age} = \frac{15}{4} \times 10 = \frac{150}{4} = 37.5 \text{ years} \]

### Step 3: Determine A's age 12 years ago
To find A's age 12 years ago, we subtract 12 from A's present age:
\[ \text{A's age 12 years ago} = 37.5 - 12 = 25.5 \text{ years} \]

### Step 4: Set up the ratio of ages 12 years ago
According to the problem, the ratio of A's age to B's age 12 years ago was \(3:4\). We can express this as:
\[ \frac{\text{A's age 12 years ago}}{\text{B's age 12 years ago}} = \frac{3}{4} \]
Let B's age 12 years ago be \(B - 12\). Therefore, we can set up the equation:
\[ \frac{25.5}{B - 12} = \frac{3}{4} \]

### Step 5: Cross-multiply to solve for B
Cross-multiplying gives:
\[ 4 \times 25.5 = 3 \times (B - 12) \]
Calculating the left side:
\[ 102 = 3B - 36 \]

### Step 6: Solve for B
Now, we can isolate B:
\[ 3B = 102 + 36 \]
\[ 3B = 138 \]
\[ B = \frac{138}{3} = 46 \text{ years} \]

### Step 7: Calculate B's present age
Since we found B's age 12 years ago, we need to add 12 to find B's present age:
\[ \text{B's present age} = 46 + 12 = 58 \text{ years} \]

### Final Answer
B's present age is **58 years**.