There are two buckets, one large and one small. The smaller bucket can hold only 3/5 of the water that the larger bucket can hold. If 6,000 buckets of the larger capacity are needed to fill a pond, how many buckets of the smaller capacity would be required to fill the same pond?

Introduction / Context:
This problem involves ratios and proportional reasoning. The capacities of two buckets are related by a simple fraction, and you are asked how many of the smaller buckets are needed when you already know how many larger buckets can fill the same pond.

Given Data / Assumptions:

• Capacity of smaller bucket = 3/5 of capacity of larger bucket.
• Number of larger buckets needed to fill the pond = 6000.
• We need to find the number of smaller buckets required.
• The volume of the pond remains the same in both scenarios.

Concept / Approach:
If capacity changes, the number of containers needed is inversely proportional to capacity. Since the small bucket holds only 3/5 as much water as the large bucket, more small buckets are required. Let capacity of large bucket = C units. Then capacity of small bucket = (3/5) * C. Total pond volume in terms of large buckets = 6000 * C. Number of small buckets needed = Total volume / small bucket capacity.

Step-by-Step Solution:
Step 1: Assume capacity of one large bucket is C units of water. Step 2: Then capacity of one small bucket = (3/5) * C. Step 3: Total volume of the pond = 6000 * C (because 6000 large buckets fill it). Step 4: Let N be the number of small buckets needed. Then N * (3/5) * C = 6000 * C. Step 5: Cancel C on both sides to get N * (3/5) = 6000. Step 6: Solve for N: N = 6000 * (5/3). Step 7: Compute 6000 * (5/3) = 6000 * 5 / 3 = 30,000 / 3 = 10,000.

Verification / Alternative check:
We can also think in terms of a unit volume like 1 pond. Larger bucket capacity = 1/6000 of the pond. Smaller bucket capacity is 3/5 of that = 3 / (5 * 6000) of the pond. To cover the full pond, we need: 1 divided by [3 / (5 * 6000)] = (5 * 6000) / 3 = 10,000 buckets. This matches the result from the algebraic approach.

Why Other Options Are Wrong:
8000: This assumes fewer small buckets than large ones, which is impossible since each small bucket holds less water. 12000 and 15000: These numbers are larger than necessary and result from miscalculating the ratio or inverting it incorrectly. 7500: This is again too low and does not satisfy the proportional relationship between capacities and counts.

Common Pitfalls:
A common mistake is to multiply by 3/5 instead of dividing by 3/5 when going from number of large buckets to number of small buckets. Students may also confuse direct proportion with inverse proportion. Remember that when capacity decreases, the number of buckets needed must increase in inverse proportion to that capacity change.

Final Answer:
To fill the same pond with the smaller bucket, 10,000 buckets are required.