At our training Institute the number of boys is same as that of the girls, Last week, except 2/3rd of the girls all the students, went to picnic,where they bought some samosas but later on they found exactly one dozen samosas were not fresh so those 12 samosas had been throuwn away. After it the samosas were divided equally between boys and girls. Further when boys dealtout the samosas equally among themselves 39 samosas left undistributed, but when the girls dealtout the same number of samosas equally among themselves 12 samosas were still left undistributed. The number of students at our training institute is:

Options

A. 1. 60
B. 2. 156
C. 3. 162
D. 4. can't be determined

Answer

To solve the problem step by step, we can follow this approach:

### Step 1: Understanding the Problem
We know that the number of boys is equal to the number of girls at the training institute. Let's denote the number of boys (B) and the number of girls (G) as follows:
- B = G

### Step 2: Determine the Number of Students Who Went to the Picnic
According to the problem, except for 2/3 of the girls, all students went to the picnic. This means:
- The number of girls who went to the picnic = 1/3 of G = (1/3)G
- The total number of students who went to the picnic = B + (1/3)G = G + (1/3)G = (4/3)G

### Step 3: Calculate the Number of Samosas
After the picnic, they bought some samosas, but 12 samosas were found to be spoiled and were thrown away. Therefore, the number of samosas available for distribution is:
- Total samosas = S - 12 (where S is the total number of samosas bought)

### Step 4: Distribution of Samosas
The remaining samosas were divided equally between boys and girls. Let’s denote the number of samosas each group received as:
- Samosas per group = (S - 12) / 2

### Step 5: Analyze the Leftover Samosas
From the problem, we know:
- When the boys distributed the samosas among themselves, 39 samosas were left undistributed.
- When the girls distributed the same number of samosas, 12 samosas were left undistributed.

Let’s denote the number of boys (B) and girls (G) as equal to N (since B = G = N).
- For boys: (S - 12) / 2 = k * N + 39 (where k is the number of samosas each boy received)
- For girls: (S - 12) / 2 = m * N + 12 (where m is the number of samosas each girl received)

### Step 6: Set Up the Equations
From the above equations, we can derive:
1. (S - 12) / 2 = kN + 39
2. (S - 12) / 2 = mN + 12

### Step 7: Equate the Two Expressions
Since both expressions equal (S - 12) / 2, we can set them equal to each other:
- kN + 39 = mN + 12

### Step 8: Solve for N
Rearranging gives:
- (k - m)N = 12 - 39
- (k - m)N = -27

### Step 9: Find the Number of Girls Who Went to the Picnic
From the problem, we know that the number of girls who went to the picnic is:
- Girls who went to picnic = 1/3 of G = 27 (from the video explanation)
Thus, G = 3 * 27 = 81.

### Step 10: Calculate the Total Number of Students
Since the number of boys is equal to the number of girls:
- B = G = 81
- Total students = B + G = 81 + 81 = 162.

### Final Answer
The total number of students at the training institute is **162**.

---