18 years ago the ratio of A's age to B's age was 8:13. Present ratio of their ages is 5:7. What is the present age of B?

Options

A. 90 years
B. 70 years
C. 60 years
D. 50 years

Explanation

To solve the problem, we will break it down step by step.

### Step 1: Define the variables
Let A's age 18 years ago be represented as \( 8x \) and B's age 18 years ago be represented as \( 13x \), based on the ratio given in the problem.

### Step 2: Write the present ages
Since the current age is 18 years later, we can express their present ages as:
- A's present age = \( 8x + 18 \)
- B's present age = \( 13x + 18 \)

### Step 3: Set up the equation for the current ratio
According to the problem, the present ratio of A's age to B's age is given as \( 5:7 \). Therefore, we can set up the equation:
\[ \frac{8x + 18}{13x + 18} = \frac{5}{7} \]

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

### Step 5: Expand both sides
Expanding both sides results in:
\[ 56x + 126 = 65x + 90 \]

### Step 6: Rearrange to isolate x
Now, let's rearrange the equation to isolate \( x \):
\[ 126 - 90 = 65x - 56x \]
\[ 36 = 9x \]

### Step 7: Solve for x
Dividing both sides by 9 gives:
\[ x = 4 \]

### Step 8: Find the present age of B
Now that we have \( x \), we can find B's present age:
\[ B's \, present \, age = 13x + 18 = 13(4) + 18 = 52 + 18 = 70 \]

### Final Answer
The present age of B is **70 years**.

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