The ratio of ages of A & B is at present 5:6. 6 years before this ratio was 4: 5. Find the present age of A & B.

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the variables
Let the present age of A be \(5x\) and the present age of B be \(6x\), where \(x\) is a common multiplier.

### Step 2: Set up the equation for ages 6 years ago
According to the problem, 6 years ago, the ages of A and B were in the ratio of 4:5. Therefore, we can express their ages 6 years ago as:
- Age of A 6 years ago: \(5x - 6\)
- Age of B 6 years ago: \(6x - 6\)

### Step 3: Write the equation based on the ratio 6 years ago
From the information given, we can set up the equation:
\[ \frac{5x - 6}{6x - 6} = \frac{4}{5} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 5(5x - 6) = 4(6x - 6) \]

### Step 5: Expand both sides of the equation
Expanding both sides results in:
\[ 25x - 30 = 24x - 24 \]

### Step 6: Rearrange the equation to solve for \(x\)
Now, we will rearrange the equation:
\[ 25x - 24x = -24 + 30 \]
\[ x = 6 \]

### Step 7: Calculate the present ages of A and B
Now that we have the value of \(x\), we can find the present ages:
- Present age of A: \(5x = 5 \times 6 = 30\) years
- Present age of B: \(6x = 6 \times 6 = 36\) years

### Final Answer
The present age of A is 30 years and the present age of B is 36 years.

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