Shivam's age is `4//5^(th)` of Avinash's age while age of Avinash after 5 years will be equal to twice the present age of Pradeep. If the sum of ages of all three is 37 years. Find the present age (in years) of Shivam.

Options

A. 15
B. 10
C. 12
D. 20

Answer

To solve the problem step by step, we will define the ages of Shivam, Avinash, and Pradeep using algebra.

### Step 1: Define the variables
Let Avinash's present age be \(5x\). According to the problem, Shivam's age is \( \frac{4}{5} \) of Avinash's age. Therefore, Shivam's age can be expressed as:
\[ \text{Shivam's age} = \frac{4}{5} \times (5x) = 4x \]

### Step 2: Express Pradeep's age
The problem states that Avinash's age after 5 years will be equal to twice Pradeep's present age. So, Avinash's age after 5 years is:
\[ \text{Avinash's age after 5 years} = 5x + 5 \]
Let Pradeep's present age be \(P\). According to the relationship given in the problem:
\[ 5x + 5 = 2P \]
From this, we can express Pradeep's age as:
\[ P = \frac{5x + 5}{2} \]

### Step 3: Set up the equation for the sum of ages
The sum of the ages of Shivam, Avinash, and Pradeep is given as 37 years. Therefore, we can write the equation:
\[ \text{Shivam's age} + \text{Avinash's age} + \text{Pradeep's age} = 37 \]
Substituting the expressions we have:
\[ 4x + 5x + \frac{5x + 5}{2} = 37 \]

### Step 4: Simplify the equation
To simplify the equation, first combine like terms:
\[ 9x + \frac{5x + 5}{2} = 37 \]
To eliminate the fraction, multiply the entire equation by 2:
\[ 2(9x) + (5x + 5) = 74 \]
This simplifies to:
\[ 18x + 5x + 5 = 74 \]
Combining the \(x\) terms gives:
\[ 23x + 5 = 74 \]

### Step 5: Solve for \(x\)
Now, isolate \(x\):
\[ 23x = 74 - 5 \]
\[ 23x = 69 \]
\[ x = \frac{69}{23} = 3 \]

### Step 6: Find Shivam's age
Now that we have \(x\), we can find Shivam's present age:
\[ \text{Shivam's age} = 4x = 4 \times 3 = 12 \]

### Conclusion
Thus, the present age of Shivam is \(12\) years.