On a Diwali day, komal purchases firecrackers and gave one-third to her sister Shurti and thrice the square root of the firecrackers to her brother Gobind. Still she had 27 firecrackers. Find the total number of firecrackers purchased by Komal.

Options

A. 81
B. 20
C. 21
D. 80

Answer

To solve the problem step-by-step, we will first define the variables and then set up the equation based on the information provided in the question.

### Step 1: Define the variable
Let \( X \) be the total number of firecrackers purchased by Komal.

### Step 2: Determine the amount given to Shruti
Komal gave one-third of her firecrackers to her sister Shruti. Therefore, the number of firecrackers given to Shruti is:
\[ \text{Firecrackers given to Shruti} = \frac{X}{3} \]

### Step 3: Determine the amount given to Gobind
Komal gave thrice the square root of the total firecrackers to her brother Gobind. Therefore, the number of firecrackers given to Gobind is:
\[ \text{Firecrackers given to Gobind} = 3\sqrt{X} \]

### Step 4: Set up the equation
After giving firecrackers to both Shruti and Gobind, Komal still had 27 firecrackers left. Thus, we can set up the equation:
\[ X - \left(\frac{X}{3} + 3\sqrt{X}\right) = 27 \]

### Step 5: Simplify the equation
First, we simplify the left side of the equation:
\[ X - \frac{X}{3} - 3\sqrt{X} = 27 \]
To combine the terms, we can find a common denominator for the \( X \) terms:
\[ \frac{3X}{3} - \frac{X}{3} - 3\sqrt{X} = 27 \]
This simplifies to:
\[ \frac{2X}{3} - 3\sqrt{X} = 27 \]

### Step 6: Eliminate the fraction
To eliminate the fraction, multiply the entire equation by 3:
\[ 2X - 9\sqrt{X} = 81 \]

### Step 7: Rearrange the equation
Rearranging gives us:
\[ 2X - 9\sqrt{X} - 81 = 0 \]

### Step 8: Substitute \( \sqrt{X} \)
Let \( \sqrt{X} = t \). Then \( X = t^2 \). Substitute \( t \) into the equation:
\[ 2(t^2) - 9t - 81 = 0 \]
This simplifies to:
\[ 2t^2 - 9t - 81 = 0 \]

### Step 9: Use the quadratic formula
To solve for \( t \), we will use the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 2 \), \( b = -9 \), and \( c = -81 \):
\[ t = \frac{9 \pm \sqrt{(-9)^2 - 4 \cdot 2 \cdot (-81)}}{2 \cdot 2} \]
Calculating the discriminant:
\[ t = \frac{9 \pm \sqrt{81 + 648}}{4} \]
\[ t = \frac{9 \pm \sqrt{729}}{4} \]
\[ t = \frac{9 \pm 27}{4} \]

### Step 10: Calculate the values of \( t \)
Calculating the two potential values for \( t \):
1. \( t = \frac{36}{4} = 9 \)
2. \( t = \frac{-18}{4} \) (this value will be rejected since \( t \) must be non-negative)

### Step 11: Find \( X \)
Since \( t = 9 \), we substitute back to find \( X \):
\[ \sqrt{X} = 9 \implies X = 9^2 = 81 \]

### Conclusion
The total number of firecrackers purchased by Komal is:
\[ \boxed{81} \]