The difference between the ages of two cousins is `10` years. `15` years ago, if the elder one was twice as old as the younger one, find their present ages.

Options

A. `15` and `25`
B. `25` and `35`
C. `20` and `30`
D. `10` and `20`

Answer

To solve the problem of finding the present ages of the two cousins, we can follow these steps:

### Step 1: Define the Variables
Let the age of the younger cousin be \( x \) years. Since the difference between their ages is 10 years, the age of the elder cousin will be \( x + 10 \) years.

### Step 2: Set Up the Equation for Ages 15 Years Ago
Fifteen years ago, the age of the younger cousin would have been \( x - 15 \) years, and the age of the elder cousin would have been \( (x + 10) - 15 = x - 5 \) years. According to the problem, 15 years ago, the elder cousin was twice as old as the younger cousin.

This gives us the equation:
\[ x - 5 = 2(x - 15) \]

### Step 3: Simplify the Equation
Now, we will simplify the equation:
\[ x - 5 = 2x - 30 \]

### Step 4: Rearrange the Equation
Next, we will rearrange the equation to isolate \( x \):
\[ x - 2x = -30 + 5 \]
\[ -x = -25 \]

### Step 5: Solve for \( x \)
Now, we can solve for \( x \):
\[ x = 25 \]

### Step 6: Find the Present Ages
Now that we have the age of the younger cousin, we can find the age of the elder cousin:
- Younger cousin's age: \( x = 25 \) years
- Elder cousin's age: \( x + 10 = 25 + 10 = 35 \) years

### Conclusion
The present ages of the cousins are:
- Younger cousin: 25 years
- Elder cousin: 35 years

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