A comparison of ages of A, B, C, D and E are as follows
I. B' s age is half the age of A
II. B' s age is `1 ½` times the age of C
III. D' s age is 12 years less than C .
IV. D' s age is `1 ½` times than age of E .
V. The age of E is 12 years
With the given data what will be the difference in the ages of A and C ?

Options

A. 64
B. 60
C. 40
D. 36

Answer

To solve the problem step by step, we will analyze the given information about the ages of A, B, C, D, and E.

### Step 1: Define the ages based on the information provided.
Let:
- Age of A = A
- Age of B = B
- Age of C = C
- Age of D = D
- Age of E = E

### Step 2: Write equations based on the comparisons given.
1. From the first statement, "B's age is half the age of A":
\[ B = \frac{1}{2}A \quad \text{(Equation 1)} \]

2. From the second statement, "B's age is \(1 \frac{1}{2}\) times the age of C":
\[ B = \frac{3}{2}C \quad \text{(Equation 2)} \]

3. From the third statement, "D's age is 12 years less than C":
\[ D = C - 12 \quad \text{(Equation 3)} \]

4. From the fourth statement, "D's age is \(1 \frac{1}{2}\) times the age of E":
\[ D = \frac{3}{2}E \quad \text{(Equation 4)} \]

5. From the fifth statement, "The age of E is 12 years":
\[ E = 12 \quad \text{(Equation 5)} \]

### Step 3: Substitute the value of E into Equation 4.
Using Equation 5 in Equation 4:
\[ D = \frac{3}{2} \times 12 = 18 \quad \text{(Equation 6)} \]

### Step 4: Substitute the value of D into Equation 3 to find C.
Using Equation 6 in Equation 3:
\[ 18 = C - 12 \\ C = 18 + 12 = 30 \quad \text{(Equation 7)} \]

### Step 5: Substitute the value of C into Equation 2 to find B.
Using Equation 7 in Equation 2:
\[ B = \frac{3}{2} \times 30 = 45 \quad \text{(Equation 8)} \]

### Step 6: Substitute the value of B into Equation 1 to find A.
Using Equation 8 in Equation 1:
\[ 45 = \frac{1}{2}A \\ A = 45 \times 2 = 90 \quad \text{(Equation 9)} \]

### Step 7: Calculate the difference in ages of A and C.
Now we have:
- Age of A = 90
- Age of C = 30

The difference in ages of A and C is:
\[ \text{Difference} = A - C = 90 - 30 = 60 \]

### Final Answer:
The difference in the ages of A and C is **60 years**.

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