Ashish supplies bread and jams to a hospital and a school. Bread and jam are supplied in equal number of pieces. Bread comes in a bunch of 8 pieces and Jam comes in a pack of 6 pieces. On a particular day, Ashish has supplied x packets of bread and y packets of jam to the school. On the same day, Ashish has supplied 3x packets of bread along with sufficient packets of jam to hospital. It is known that the number of students in the school are between 500 and 550.
How many packets of jams are supplied in the hospital?

Options

A. 248 packets
B. 252 packets
C. 165 packets
D. 288 packets

Explanation

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step-by-Step Solution:

1. **Identify the Number of Students**:
We know that the number of students in the school is between 500 and 550. Since bread and jam are supplied in equal numbers, we need to find a common multiple of the pieces in a bunch of bread (8 pieces) and a pack of jam (6 pieces).

2. **Find the LCM of 8 and 6**:
The least common multiple (LCM) of 8 and 6 is calculated as follows:
- The prime factorization of 8 is \(2^3\).
- The prime factorization of 6 is \(2^1 \times 3^1\).
- The LCM is \(2^3 \times 3^1 = 24\).

3. **Determine Possible Student Numbers**:
We look for multiples of 24 between 500 and 550:
- The multiples of 24 in this range are 504 and 528.

4. **Calculate Packets of Bread and Jam**:
- If there are 504 students:
- The number of packets of bread \(x\) = \( \frac{504}{8} = 63\).
- The number of packets of jam \(y\) = \( \frac{504}{6} = 84\).
- If there are 528 students:
- The number of packets of bread \(x\) = \( \frac{528}{8} = 66\).
- The number of packets of jam \(y\) = \( \frac{528}{6} = 88\).

5. **Calculate Hospital Supply**:
- For the hospital, Ashish supplies 3 times the packets of bread supplied to the school:
- If \(x = 63\), then packets of bread supplied to the hospital = \(3x = 189\).
- If \(x = 66\), then packets of bread supplied to the hospital = \(3x = 198\).

6. **Equal Number of Bread and Jam**:
- Since the number of pieces of bread and jam supplied is equal, we need to find the number of packets of jam supplied to the hospital:
- For 189 packets of bread:
- Total pieces of bread = \(189 \times 8 = 1512\).
- Therefore, packets of jam = \( \frac{1512}{6} = 252\).
- For 198 packets of bread:
- Total pieces of bread = \(198 \times 8 = 1584\).
- Therefore, packets of jam = \( \frac{1584}{6} = 264\).

7. **Conclusion**:
Since the problem states that the number of students is between 500 and 550, we can conclude:
- If there are 504 students, the packets of jam supplied to the hospital is 252.
- If there are 528 students, the packets of jam supplied to the hospital is 264.

Given the options, the only valid answer is **252 packets of jam** supplied to the hospital.