- Use the formula \(T = 1,000 \times P_{1000} + 100 \times P_{100} + 10 \times P_{10} + 1 \times P_1\) to calculate the total pens for each site.
- Site A: \(1,000(2) + 100(16) = 3,600\) (Correct).
- Site B: \(1,000(3) + 100(5) + 10(10) = 3,600\) (Correct).
- Site C: \(1,000(3) + 100(4) + 10(10) = 3,500\) (Incorrect).
- Site D: \(1,000(3) + 10(60) = 3,600\) (Correct).
Explanation
- Problem Analysis and Formula Setup
We are tasked with verifying if four different sites (A, B, C, and D) correctly shipped exactly 3,600 pens each. Each site sends pens in packages of 1,000, 100, 10, and 1. The total number of pens shipped by a site can be calculated using the following formula:
\[\text{Total} = 1,000 \times (\text{# of 1,000-pack}) + 100 \times (\text{# of 100-pack}) + 10 \times (\text{# of 10-pack}) + 1 \times (\text{# of 1-pack})\]
We will calculate the total for each site and compare it to the target amount of 3,600.
- Evaluating Site A
For Site A:
- 1,000-pack: 2
- 100-pack: 16
- 10-pack: 0
- 1-pack: 0
Total shipped by Site A = \(1,000 \times 2 + 100 \times 16 + 10 \times 0 + 1 \times 0 = 2,000 + 1,600 + 0 + 0 = 3,600\).
Since 3,600 = 3,600, Site A shipped the correct amount.
- Evaluating Site B
For Site B:
- 1,000-pack: 3
- 100-pack: 5
- 10-pack: 10
- 1-pack: 0
Total shipped by Site B = \(1,000 \times 3 + 100 \times 5 + 10 \times 10 + 1 \times 0 = 3,000 + 500 + 100 + 0 = 3,600\).
Since 3,600 = 3,600, Site B shipped the correct amount.
- Evaluating Site C
For Site C:
- 1,000-pack: 3
- 100-pack: 4
- 10-pack: 10
- 1-pack: 0
Total shipped by Site C = \(1,000 \times 3 + 100 \times 4 + 10 \times 10 + 1 \times 0 = 3,000 + 400 + 100 + 0 = 3,500\).
Since 3,500 is not equal to 3,600, Site C did not ship the correct amount.
- Evaluating Site D
For Site D:
- 1,000-pack: 3
- 100-pack: 0
- 10-pack: 60
- 1-pack: 0
Total shipped by Site D = \(1,000 \times 3 + 100 \times 0 + 10 \times 60 + 1 \times 0 = 3,000 + 0 + 600 + 0 = 3,600\).
Since 3,600 = 3,600, Site D shipped the correct amount.
- Conclusion
Based on the calculations:
- Site A shipped 3,600 pens, so it shipped the correct amount.
- Site B shipped 3,600 pens, so it shipped the correct amount.
- Site C shipped 3,500 pens, so it did not ship the correct amount.
- Site D shipped 3,600 pens, so it shipped the correct amount.
Examples
This method of verifying counts is standard in inventory management and quality control. For instance, in a warehouse, a manager might receive a large order of items packed in different batch sizes and needs to quickly verify if the total quantity matches the purchase order. By systematically breaking down the shipments by package size, they can ensure accuracy and quickly identify any discrepancies in deliveries.