The sum of the current ages of A and B is 77 years . Seven years ago A was 2.5 times as old as B . What is A' present age

Options

A. 48 years
B. 52 years
C. 50 years
D. 54 years

Answer

To solve the problem step by step, we will define the current ages of A and B, set up equations based on the information given, and solve for A's age.

### Step 1: Define the variables
Let:
- A's current age = A
- B's current age = B

### Step 2: Set up the first equation
From the problem, we know that the sum of their current ages is 77 years:
\[ A + B = 77 \] (Equation 1)

### Step 3: Set up the second equation
The problem states that seven years ago, A was 2.5 times as old as B. Seven years ago, A's age would be \( A - 7 \) and B's age would be \( B - 7 \). Therefore, we can write:
\[ A - 7 = 2.5 \times (B - 7) \] (Equation 2)

### Step 4: Simplify Equation 2
Expanding Equation 2:
\[ A - 7 = 2.5B - 17.5 \]
Now, rearranging gives:
\[ A = 2.5B - 10.5 \] (Equation 3)

### Step 5: Substitute Equation 3 into Equation 1
Now we will substitute Equation 3 into Equation 1:
\[ (2.5B - 10.5) + B = 77 \]
Combining like terms:
\[ 3.5B - 10.5 = 77 \]

### Step 6: Solve for B
Adding 10.5 to both sides:
\[ 3.5B = 87.5 \]
Now, divide by 3.5:
\[ B = \frac{87.5}{3.5} = 25 \]

### Step 7: Find A's age using B's age
Now that we have B's age, we can find A's age using Equation 1:
\[ A + 25 = 77 \]
Subtracting 25 from both sides:
\[ A = 77 - 25 = 52 \]

### Conclusion
A's present age is **52 years**.

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