The present age of A is 4 years less than twice the present age of B. B's present age is 6 years more than twice his age 15 years ago. Find the difference of their ages

Options

A. 30 years
B. 32 years
C. 20 years
D. 22 years

Explanation

To solve the problem, we need to set up equations based on the information given in the question. Let's break it down step by step.

### Step 1: Define the Variables
Let:
- A = Present age of A
- B = Present age of B

### Step 2: Set Up the Equations
From the question, we have two pieces of information:

1. The present age of A is 4 years less than twice the present age of B:
\[ A = 2B - 4 \quad \text{(Equation 1)} \]

2. B's present age is 6 years more than twice his age 15 years ago.
- B's age 15 years ago = B - 15
- Twice B's age 15 years ago = 2(B - 15) = 2B - 30
- Therefore, B's present age can be expressed as:
\[ B = 2(B - 15) + 6 \]
Simplifying this:
\[ B = 2B - 30 + 6 \]
\[ B = 2B - 24 \]
Rearranging gives us:
\[ 2B - B = 24 \]
\[ B = 24 \quad \text{(Equation 2)} \]

### Step 3: Substitute B's Age into Equation 1
Now that we have B's age, we can substitute it back into Equation 1 to find A's age:
\[ A = 2(24) - 4 \]
\[ A = 48 - 4 \]
\[ A = 44 \]

### Step 4: Find the Difference of Their Ages
Now that we have both A's and B's ages:
- A = 44
- B = 24

The difference in their ages is:
\[ \text{Difference} = A - B = 44 - 24 = 20 \]

### Final Answer
The difference of their ages is **20 years**.

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