Three brands of pens A, B and C are available in packets of 10, 12 and 24 respectively. If a shopkeeper wants to buy equal number of pens of each brand, what is the minimum number of packets of each brand, he should buy?

Question

Accepted Answer

To determine the minimum number of packets of each brand that the shopkeeper should buy to have an equal number of pens from each brand, we need to find the least common multiple (LCM) of the number of pens in each packet for brands A, B, and C.

1. **Brand A**: Packets contain 10 pens.
2. **Brand B**: Packets contain 12 pens.
3. **Brand C**: Packets contain 24 pens.

**Step 1: Calculate the LCM of 10, 12, and 24.**

- Prime factorization:
  - 10 = 2 × 5
  - 12 = 2² × 3
  - 24 = 2³ × 3

- LCM is found by taking the highest power of each prime number:
  - Highest power of 2: 2³
  - Highest power of 3: 3¹
  - Highest power of 5: 5¹

- LCM = 2³ × 3¹ × 5¹ = 8 × 3 × 5 = 120

The LCM of 10, 12, and 24 is 120. Therefore, the shopkeeper needs to buy enough packets so that the total number of pens from each brand equals 120.

**Step 2: Determine the number of packets needed for each brand.**

- **Brand A**: Each packet contains 10 pens.
  - Number of packets needed = 120 ÷ 10 = 12 packets

- **Brand B**: Each packet contains 12 pens.
  - Number of packets needed = 120 ÷ 12 = 10 packets

- **Brand C**: Each packet contains 24 pens.
  - Number of packets needed = 120 ÷ 24 = 5 packets

Thus, the shopkeeper should buy 12 packets of Brand A, 10 packets of Brand B, and 5 packets of Brand C to have an equal number of pens from each brand.

**Explanation of Options:**

- **Option 1 (A = 10, B = 12, C = 5)**: This option suggests buying 10 packets of Brand A, 12 packets of Brand B, and 5 packets of Brand C. This would result in 100 pens from Brand A, 144 pens from Brand B, and 120 pens from Brand C, which are not equal.

- **Option 2 (A = 5, B = 12, C = 10)**: This option suggests buying 5 packets of Brand A, 12 packets of Brand B, and 10 packets of Brand C. This would result in 50 pens from Brand A, 144 pens from Brand B, and 240 pens from Brand C, which are not equal.

- **Option 3 (A = 10, B = 5, C = 12)**: This option suggests buying 10 packets of Brand A, 5 packets of Brand B, and 12 packets of Brand C. This would result in 100 pens from Brand A, 60 pens from Brand B, and 288 pens from Brand C, which are not equal.

- **Option 4 (A = 12, B = 10, C = 5)**: This option correctly suggests buying 12 packets of Brand A, 10 packets of Brand B, and 5 packets of Brand C, resulting in 120 pens from each brand, which is equal.

Therefore, Option 4 is the correct answer as it ensures the shopkeeper buys an equal number of pens from each brand.