Arjit's age is 3 years more than 1.5 times the age of Heera, whose age in tum is 10 years more than the age of Deepika. If 5 times the age of Deepika is equal to 3 times the age of Arjit, what is the sum of the present ages of the trio?

Options

A. 99 years
B. 95 years
C. 100 years
D. 97 years

Answer

To solve the problem, we will define the ages of Arjit, Heera, and Deepika using algebraic expressions based on the information provided in the question.

Let's denote:
- Deepika's age as \( D \)
- Heera's age as \( H \)
- Arjit's age as \( A \)

### Step 1: Set up the equations based on the problem statement.

1. According to the problem, Arjit's age is 3 years more than 1.5 times Heera's age:
\[ A = 1.5H + 3 \]

2. Heera's age is 10 years more than Deepika's age:
\[ H = D + 10 \]

3. It is also given that 5 times Deepika's age is equal to 3 times Arjit's age:
\[ 5D = 3A \]

### Step 2: Substitute Heera's age into Arjit's age equation.

From the second equation, we can substitute \( H \) in the first equation:
\[ A = 1.5(D + 10) + 3 \]
Expanding this gives:
\[ A = 1.5D + 15 + 3 = 1.5D + 18 \]

### Step 3: Substitute Arjit's age into the third equation.

Now we can substitute \( A \) in the third equation:
\[ 5D = 3(1.5D + 18) \]
Expanding this gives:
\[ 5D = 4.5D + 54 \]

### Step 4: Solve for Deepika's age.

Now, let's isolate \( D \):
\[ 5D - 4.5D = 54 \]
\[ 0.5D = 54 \]
\[ D = \frac{54}{0.5} = 108 \]

### Step 5: Find Heera's age.

Now that we have Deepika's age, we can find Heera's age using the second equation:
\[ H = D + 10 = 108 + 10 = 118 \]

### Step 6: Find Arjit's age.

Now we can find Arjit's age using the first equation:
\[ A = 1.5H + 3 = 1.5(118) + 3 = 177 + 3 = 180 \]

### Step 7: Calculate the sum of their ages.

Now we can find the sum of their ages:
\[ \text{Sum} = A + H + D = 180 + 118 + 108 = 406 \]

### Final Answer:
The sum of the present ages of Arjit, Heera, and Deepika is \( 406 \).

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