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

Options

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

Explanation

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

### Step 1: Define the present ages
Let the present age of A be \( 5x \) and the present age of B be \( 7x \) based on the given ratio of their present ages \( 5:7 \).

### Step 2: Define the ages 18 years ago
Eighteen years ago, A's age would be \( 5x - 18 \) and B's age would be \( 7x - 18 \).

### Step 3: Set up the equation based on the past ratio
According to the problem, the ratio of A's age to B's age 18 years ago was \( 8:13 \). Therefore, we can set up the equation:
\[ \frac{5x - 18}{7x - 18} = \frac{8}{13} \]

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

### Step 5: Expand both sides
Expanding both sides results in:
\[ 65x - 234 = 56x - 144 \]

### Step 6: Rearrange the equation
Now, we can rearrange the equation to isolate \( x \):
\[ 65x - 56x = 234 - 144 \]
\[ 9x = 90 \]

### Step 7: Solve for \( x \)
Dividing both sides by 9 gives:
\[ x = 10 \]

### Step 8: Calculate the present age of A
Now, substituting \( x \) back into the expression for A's present age:
\[ \text{Present age of A} = 5x = 5 \times 10 = 50 \]

Thus, the present age of A is **50 years**.

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