The ages of A and B are in the ratio 5:7. Five years ago, their ages were in the ratio 5:8. The respective present ages(in years) are:
A और B की उम्र 5 : 7 के अनुपात में है | पांच वर्ष पहले उनकी उम्र 5 : 8 के अनुपात में थी | उनकी वर्तमान आयु (क्रमशः ) है -

Options

A. 10,14
B. 20,28
C. 25,40
D. 15,21

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the present ages of A and B
Let the present age of A be \( 5x \) and the present age of B be \( 7x \), where \( x \) is a common multiplier.

### Step 2: Set up the equation for their ages 5 years ago
Five years ago, the ages of A and B would be:
- Age of A = \( 5x - 5 \)
- Age of B = \( 7x - 5 \)

According to the problem, five years ago their ages were in the ratio \( 5:8 \). Therefore, we can set up the equation:
\[ \frac{5x - 5}{7x - 5} = \frac{5}{8} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 8(5x - 5) = 5(7x - 5) \]

### Step 4: Expand both sides of the equation
Expanding both sides results in:
\[ 40x - 40 = 35x - 25 \]

### Step 5: Rearrange the equation to isolate \( x \)
Now, we will move all terms involving \( x \) to one side and constant terms to the other side:
\[ 40x - 35x = -25 + 40 \]
\[ 5x = 15 \]

### Step 6: Solve for \( x \)
Dividing both sides by 5 gives:
\[ x = 3 \]

### Step 7: Calculate the present ages of A and B
Now that we have \( x \), we can find the present ages:
- Age of A = \( 5x = 5 \times 3 = 15 \) years
- Age of B = \( 7x = 7 \times 3 = 21 \) years

### Final Answer
The respective present ages of A and B are 15 years and 21 years.

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