Determining Total Tourists

Let the total number of tourists be denoted by $T$.

According to the problem statement:

• Tourists visiting Qutub Minar = $\frac{1}{3}T$
• Tourists visiting Red Fort = $4\sqrt{T}$
• Tourists visiting Kartavya Path = 18

The sum of tourists at all locations must equal the total number of tourists:

$ \frac{T}{3} + 4\sqrt{T} + 18 = T $

Solving the Tourist Equation

To find the total number of tourists $T$, we solve the equation:

1. Rearrange the equation:

   $ 4\sqrt{T} + 18 = T - \frac{T}{3} $

   $ 4\sqrt{T} + 18 = \frac{2T}{3} $

2. Multiply the entire equation by 3 to eliminate the fraction:

   $ 3 \times (4\sqrt{T} + 18) = 3 \times \frac{2T}{3} $

   $ 12\sqrt{T} + 54 = 2T $

3. Let $x = \sqrt{T}$. Then $T = x^2$. Substitute this into the equation:

   $ 12x + 54 = 2x^2 $

4. Rearrange into a standard quadratic equation ($ax^2 + bx + c = 0$):

   $ 2x^2 - 12x - 54 = 0 $

5. Divide by 2 to simplify:

   $ x^2 - 6x - 27 = 0 $

6. Factor the quadratic equation:

   $ (x - 9)(x + 3) = 0 $

7. The possible values for $x$ are $x = 9$ or $x = -3$. Since $x = \sqrt{T}$, it must be a positive value. Therefore, $x = 9$.
8. Substitute back $T = x^2$:

   $ T = 9^2 = 81 $

Verification of the Result

Let's check if $T = 81$ satisfies the conditions:

• Qutub Minar visitors: $\frac{1}{3} \times 81 = 27$
• Red Fort visitors: $4\sqrt{81} = 4 \times 9 = 36$
• Kartavya Path visitors: 18
• Total tourists: $27 + 36 + 18 = 81$. This matches our calculated total.

The total number of tourists is 81.