The ratio of the present ages of A and B is 5:6. Eight years ago, the ratio of their ages was 4:5. What will be the ratio of the ages of A and B after 8 years from now?

Options

A. `9:11`
B. `8:9`
C. `6:7`
D. `7:8`

Answer

To solve the problem, let's follow these steps:

### Step 1: Define the Present Ages
Let the present ages of A and B be represented as:
- Age of A = 5x
- Age of B = 6x

### Step 2: Set Up the Equation for Ages 8 Years Ago
According to the problem, eight years ago, the ratio of their ages was 4:5. Therefore, we can express their ages eight years ago as:
- Age of A eight years ago = 5x - 8
- Age of B eight years ago = 6x - 8

We can set up the equation based on the given ratio:
\[ \frac{5x - 8}{6x - 8} = \frac{4}{5} \]

### Step 3: Cross-Multiply to Solve for x
Cross-multiplying gives us:
\[ 5(5x - 8) = 4(6x - 8) \]
Expanding both sides:
\[ 25x - 40 = 24x - 32 \]

### Step 4: Rearrange the Equation
Now, we will rearrange the equation to isolate x:
\[ 25x - 24x = -32 + 40 \]
This simplifies to:
\[ x = 8 \]

### Step 5: Calculate the Present Ages
Now that we have the value of x, we can find the present ages of A and B:
- Age of A = 5x = 5(8) = 40 years
- Age of B = 6x = 6(8) = 48 years

### Step 6: Calculate the Ages After 8 Years
To find their ages after 8 years:
- Age of A after 8 years = 40 + 8 = 48 years
- Age of B after 8 years = 48 + 8 = 56 years

### Step 7: Find the Ratio of Their Ages After 8 Years
Now, we can find the ratio of their ages after 8 years:
\[ \text{Ratio} = \frac{48}{56} = \frac{6}{7} \]

### Final Answer
The ratio of the ages of A and B after 8 years from now will be \(6:7\).

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