Twelve years ago, the ratio of the ages of Anil and Bishu was 5:12. Eight years from now, the ratio of their ages will be 10:17. What is the ratio of the present ages of Anil and Bishu?

Options

A. `9:16 `
B. `8:15`
C. `5:8`
D. `7:13`

Answer

To solve the problem step by step, we need to set up equations based on the information given about the ages of Anil and Bishu.

### Step 1: Define the Variables
Let:
- A = Present age of Anil
- B = Present age of Bishu

### Step 2: Set Up the First Equation
According to the problem, twelve years ago, the ratio of their ages was 5:12. This can be expressed as:
\[ \frac{A - 12}{B - 12} = \frac{5}{12} \]
Cross-multiplying gives us:
\[ 12(A - 12) = 5(B - 12) \]
Expanding this, we get:
\[ 12A - 144 = 5B - 60 \]
Rearranging gives us the first equation:
\[ 12A - 5B = 84 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
The problem also states that eight years from now, the ratio of their ages will be 10:17. This can be expressed as:
\[ \frac{A + 8}{B + 8} = \frac{10}{17} \]
Cross-multiplying gives us:
\[ 17(A + 8) = 10(B + 8) \]
Expanding this, we get:
\[ 17A + 136 = 10B + 80 \]
Rearranging gives us the second equation:
\[ 17A - 10B = -56 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
We now have a system of two equations:
1. \(12A - 5B = 84\)
2. \(17A - 10B = -56\)

To eliminate one variable, we can multiply Equation 1 by 2:
\[ 24A - 10B = 168 \quad \text{(Equation 3)} \]

Now we can subtract Equation 2 from Equation 3:
\[ (24A - 10B) - (17A - 10B) = 168 - (-56) \]
This simplifies to:
\[ 7A = 224 \]
Dividing by 7 gives:
\[ A = 32 \]

### Step 5: Substitute to Find B
Now that we have A, we can substitute it back into Equation 1 to find B:
\[ 12(32) - 5B = 84 \]
This simplifies to:
\[ 384 - 5B = 84 \]
Rearranging gives:
\[ 5B = 300 \]
Dividing by 5 gives:
\[ B = 60 \]

### Step 6: Find the Ratio of Present Ages
Now we have the present ages:
- A = 32 (Anil's age)
- B = 60 (Bishu's age)

The ratio of their present ages is:
\[ \frac{A}{B} = \frac{32}{60} = \frac{8}{15} \]

### Final Answer
The ratio of the present ages of Anil and Bishu is \( \frac{8}{15} \).

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