Nikhil's mother asks him to buy 100 pieces of sweets worth 100/-. The sweet shop has 3 kinds of sweets, kajubarfi, gulabjamun and sandesh. Kajubarfi costs 10/- per piece, gulabjamun costs 3/- per piece and sandesh costs 50 paise per piece. If Nikhil decides to buy at least one sweet of each type. How many gulabjamuns should he buy?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem step by step, we will define the variables and set up the equations based on the information given.

### Step 1: Define the Variables
Let:
- \( K \) = number of Kajubarfi pieces
- \( G \) = number of Gulabjamun pieces
- \( S \) = number of Sandesh pieces

### Step 2: Set Up the Equations
From the problem, we know:
1. The total number of sweets is 100:
\[ K + G + S = 100 \quad \text{(Equation 1)} \]
2. The total cost of the sweets is 100 rupees (or 10000 paise):
- Kajubarfi costs 10/- (or 1000 paise) per piece.
- Gulabjamun costs 3/- (or 300 paise) per piece.
- Sandesh costs 0.5/- (or 50 paise) per piece.

Therefore, the cost equation is:
\[ 10K + 3G + 0.5S = 100 \quad \text{(Equation 2)} \]

### Step 3: Eliminate the Fraction
To eliminate the fraction in Equation 2, multiply the entire equation by 2:
\[ 20K + 6G + S = 200 \quad \text{(Equation 3)} \]

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( K + G + S = 100 \) (Equation 1)
2. \( 20K + 6G + S = 200 \) (Equation 3)

We can subtract Equation 1 from Equation 3:
\[ (20K + 6G + S) - (K + G + S) = 200 - 100 \]
This simplifies to:
\[ 19K + 5G = 100 \quad \text{(Equation 4)} \]

### Step 5: Express G in Terms of K
From Equation 4, we can express \( G \):
\[ 5G = 100 - 19K \]
\[ G = \frac{100 - 19K}{5} \]

### Step 6: Determine Integer Values
Since \( G \) must be a non-negative integer, \( 100 - 19K \) must be divisible by 5. We can check the values of \( K \) to find valid integer solutions for \( G \).

### Step 7: Check Values of K
Let's check the values of \( K \):
- If \( K = 1 \):
\[ G = \frac{100 - 19(1)}{5} = \frac{81}{5} \quad \text{(not an integer)} \]
- If \( K = 2 \):
\[ G = \frac{100 - 19(2)}{5} = \frac{62}{5} \quad \text{(not an integer)} \]
- If \( K = 3 \):
\[ G = \frac{100 - 19(3)}{5} = \frac{43}{5} \quad \text{(not an integer)} \]
- If \( K = 4 \):
\[ G = \frac{100 - 19(4)}{5} = \frac{24}{5} \quad \text{(not an integer)} \]
- If \( K = 5 \):
\[ G = \frac{100 - 19(5)}{5} = \frac{5}{5} = 1 \quad \text{(integer)} \]

### Step 8: Conclusion
Thus, if \( K = 5 \), then \( G = 1 \). Since Nikhil must buy at least one of each type of sweet, the number of Gulabjamuns he should buy is:
\[ \boxed{1} \]