Eighteen years ago, the ratio of A's age to B's age was `8 : 13`. Their present ratios are `5:7`. What is the present age of A ?

Options

A. 60 years
B. 70 years
C. 50 years
D. 40 years

Answer

To find the present age of A, we can follow these steps:

### Step 1: Set up the equations based on the given ratios.
Eighteen years ago, the ratio of A's age to B's age was 8:13. Let's denote A's age 18 years ago as \(8k\) and B's age as \(13k\).

### Step 2: Express their present ages.
Since 18 years have passed, we can express their present ages as:
- A's present age = \(8k + 18\)
- B's present age = \(13k + 18\)

### Step 3: Set up the equation using the current ratio.
The current ratio of their ages is given as 5:7. Therefore, we can set up the equation:
\[ \frac{8k + 18}{13k + 18} = \frac{5}{7} \]

### Step 4: Cross-multiply to eliminate the fraction.
Cross-multiplying gives us:
\[ 7(8k + 18) = 5(13k + 18) \]

### Step 5: Expand both sides of the equation.
Expanding both sides results in:
\[ 56k + 126 = 65k + 90 \]

### Step 6: Rearrange the equation to isolate \(k\).
Now, we will move all terms involving \(k\) to one side and constant terms to the other side:
\[ 56k - 65k = 90 - 126 \]
This simplifies to:
\[ -9k = -36 \]

### Step 7: Solve for \(k\).
Dividing both sides by -9 gives:
\[ k = 4 \]

### Step 8: Calculate A's present age.
Now that we have the value of \(k\), we can find A's present age:
\[ \text{A's present age} = 8k + 18 = 8(4) + 18 = 32 + 18 = 50 \]

### Final Answer:
The present age of A is **50 years**.

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