The difference between the ages of two persons is 10 years. 15 years ago the elder one was twice as old as the younger one. The present age of the elder person is

Options

A. 25 years
B. 35 years
C. 45 years
D. 55 years

Answer

To solve the problem step by step, we will define the ages of the two persons and set up equations based on the information given.

### Step 1: Define the Variables
Let:
- A = present age of the elder person
- B = present age of the younger person

### Step 2: Set Up the First Equation
According to the problem, the difference between their ages is 10 years. Therefore, we can write the first equation as:
\[ A - B = 10 \]
(Equation 1)

### Step 3: Set Up the Second Equation
The problem states that 15 years ago, the elder one was twice as old as the younger one. Therefore, we can express their ages 15 years ago as:
- Elder's age 15 years ago = A - 15
- Younger’s age 15 years ago = B - 15

From the information given, we can write the second equation as:
\[ A - 15 = 2(B - 15) \]
Expanding this gives:
\[ A - 15 = 2B - 30 \]
Rearranging this, we get:
\[ A - 2B = -15 \]
(Equation 2)

### Step 4: Solve the Equations
Now we have two equations:
1. \( A - B = 10 \) (Equation 1)
2. \( A - 2B = -15 \) (Equation 2)

We can solve these equations simultaneously. From Equation 1, we can express A in terms of B:
\[ A = B + 10 \]

Now, substitute this expression for A into Equation 2:
\[ (B + 10) - 2B = -15 \]
This simplifies to:
\[ 10 - B = -15 \]
Now, isolate B:
\[ -B = -15 - 10 \]
\[ -B = -25 \]
\[ B = 25 \]

### Step 5: Find the Present Age of the Elder Person
Now that we have B, we can find A using Equation 1:
\[ A = B + 10 \]
Substituting the value of B:
\[ A = 25 + 10 = 35 \]

### Conclusion
The present age of the elder person is:
\[ \boxed{35} \]