Seven years ago, the ages ( in years ) of A and B were in the ratio `4:5` and 7 years hence, their ages will be in the ratio `5:6`. What will be the ratio of their ages 5 years from now ?

Options

A. `31:33`
B. `33:40`
C. `34:41`
D. `33:34`

Answer

To solve the problem step by step, we will follow the given information about the ages of A and B.

### Step 1: Define Present Ages
Let the present age of A be \( x \) years and the present age of B be \( y \) years.

### Step 2: Set Up the First Condition (Seven Years Ago)
Seven years ago, the ages of A and B were in the ratio \( 4:5 \). Therefore, we can write the equation:
\[ \frac{x - 7}{y - 7} = \frac{4}{5} \]
Cross-multiplying gives us:
\[ 5(x - 7) = 4(y - 7) \]
Expanding this, we get:
\[ 5x - 35 = 4y - 28 \]
Rearranging gives us the first equation:
\[ 5x - 4y = 7 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Condition (Seven Years Hence)
Seven years hence, the ages of A and B will be in the ratio \( 5:6 \). Thus, we can write:
\[ \frac{x + 7}{y + 7} = \frac{5}{6} \]
Cross-multiplying gives us:
\[ 6(x + 7) = 5(y + 7) \]
Expanding this, we get:
\[ 6x + 42 = 5y + 35 \]
Rearranging gives us the second equation:
\[ 6x - 5y = -7 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations Simultaneously
Now we have two equations:
1. \( 5x - 4y = 7 \)
2. \( 6x - 5y = -7 \)

To eliminate one variable, we can multiply Equation 1 by 5 and Equation 2 by 4:
\[ 25x - 20y = 35 \quad \text{(Equation 3)} \]
\[ 24x - 20y = -28 \quad \text{(Equation 4)} \]

### Step 5: Subtract Equation 4 from Equation 3
Now, subtract Equation 4 from Equation 3:
\[ (25x - 20y) - (24x - 20y) = 35 - (-28) \]
This simplifies to:
\[ x = 63 \]

### Step 6: Substitute Back to Find y
Now substitute \( x = 63 \) back into Equation 1 to find \( y \):
\[ 5(63) - 4y = 7 \]
\[ 315 - 4y = 7 \]
\[ 4y = 315 - 7 \]
\[ 4y = 308 \]
\[ y = 77 \]

### Step 7: Calculate Ages 5 Years from Now
Now we need to find the ratio of their ages 5 years from now:
- Age of A in 5 years: \( 63 + 5 = 68 \)
- Age of B in 5 years: \( 77 + 5 = 82 \)

### Step 8: Find the Ratio
The ratio of their ages 5 years from now is:
\[ \frac{68}{82} \]
This simplifies to:
\[ \frac{34}{41} \]

### Final Answer
Thus, the ratio of their ages 5 years from now is \( \frac{34}{41} \).

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