The ratio of the present ages of A and B is 5 : 8. If after 8 years the ratio of ages of A and B will be 2 : 3, then what is the present age (in years) of A?

Options

A. a)64
B. b)40
C. c)48
D. d)72

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the present ages of A and B
Let the present ages of A and B be represented as:
- Age of A = 5X
- Age of B = 8X

### Step 2: Set up the equation for their ages after 8 years
After 8 years, the ages of A and B will be:
- Age of A after 8 years = 5X + 8
- Age of B after 8 years = 8X + 8

According to the problem, the ratio of their ages after 8 years will be 2:3. Therefore, we can set up the equation:
\[ \frac{5X + 8}{8X + 8} = \frac{2}{3} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 3(5X + 8) = 2(8X + 8) \]

### Step 4: Expand both sides
Expanding both sides results in:
\[ 15X + 24 = 16X + 16 \]

### Step 5: Rearrange the equation to solve for X
Now, we will rearrange the equation to isolate X:
\[ 15X + 24 - 16X = 16 \]
\[ -1X + 24 = 16 \]
\[ -1X = 16 - 24 \]
\[ -1X = -8 \]
\[ X = 8 \]

### Step 6: Calculate the present age of A
Now that we have the value of X, we can find the present age of A:
\[ \text{Present age of A} = 5X = 5 \times 8 = 40 \]

### Final Answer
The present age of A is **40 years**.

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