The ratio of age of A & B at present is 6.5 & the sum of their ages is 54 years after 5 years. What will be the ratio of their ages after 8 years ?

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the Given Information
We know that the present age ratio of A and B is 6:5, and the sum of their ages after 5 years will be 54 years.

### Step 2: Set Up the Present Ages
Let the present age of A be \(6x\) and the present age of B be \(5x\), where \(x\) is a common multiplier.

### Step 3: Calculate the Sum of Their Ages After 5 Years
In 5 years, A's age will be \(6x + 5\) and B's age will be \(5x + 5\). The sum of their ages after 5 years can be expressed as:
\[ (6x + 5) + (5x + 5) = 54 \]

### Step 4: Simplify the Equation
Combine like terms:
\[ 6x + 5 + 5x + 5 = 54 \]
\[ 11x + 10 = 54 \]

### Step 5: Solve for \(x\)
Subtract 10 from both sides:
\[ 11x = 44 \]
Now divide by 11:
\[ x = 4 \]

### Step 6: Find the Present Ages of A and B
Now substitute \(x\) back to find the present ages:
- Age of A: \(6x = 6 \times 4 = 24\) years
- Age of B: \(5x = 5 \times 4 = 20\) years

### Step 7: Calculate Their Ages After 8 Years
After 8 years, their ages will be:
- Age of A: \(24 + 8 = 32\) years
- Age of B: \(20 + 8 = 28\) years

### Step 8: Find the Ratio of Their Ages After 8 Years
The ratio of their ages after 8 years will be:
\[ \text{Ratio} = \frac{32}{28} = \frac{8}{7} \]

### Final Answer
The ratio of their ages after 8 years is \(8:7\).

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