Bucket Capacity Reduction and Its Impact on Filling a Tank

When dealing with tasks involving measuring and filling large volumes of liquid, understanding how the capacity of the measuring tool affects the overall task can be crucial. In this article, we will explore the relationship between the bucket capacity and the number of buckets needed to fill a tank, focusing on a specific scenario where the bucket capacity is reduced to two-fifths of its original size.

Understanding the Task

Let's consider a practical scenario: To fill a tank, 25 buckets of water are required when each bucket has a certain capacity. Now, if the capacity of the bucket is reduced to two-fifths of its present capacity, what would be the new number of buckets required to fill the same tank?

Mathematical Derivation

1. **Initial Setup**: The current setup tells us that the tank requires 25 buckets when each bucket has a capacity of (C).

2. **Total Tank Capacity**: Therefore, the total capacity of the tank is (25C).

3. **New Bucket Capacity**: If the capacity of the bucket is reduced to two-fifths of its present capacity, the new capacity of one bucket becomes (frac{2}{5}C).

4. **Calculation of New Buckets Required**: To find the new number of buckets required, we use the formula:

(text{Total number of buckets needed} frac{text{Total capacity of the tank}}{text{New capacity of one bucket}})

Substituting the values:

(text{Total number of buckets needed} frac{25C}{frac{2}{5}C} 25C times frac{5}{2C} 25 times frac{5}{2} frac{125}{2} 62.5)

Since it is impossible to have a fraction of a bucket, we would need 63 buckets to completely fill the tank.

Example Calculation

Let's take a numerical example to illustrate this concept:

Let the initial capacity of the bucket be 100 litres. Therefore, the total volume of water required to fill the tank is 25 buckets × 100 litres per bucket  2500 litres. If the bucket capacity is reduced to two-fifths of its original size, the new capacity of the bucket is 100 litres × frac{2}{5}  40 litres. To calculate the number of buckets needed, we use the formula:

The new number of buckets required to fill the tank would be:

(frac{2500 text{ litres}}{40 text{ litres per bucket}} 62.5) buckets.

Since we cannot have 0.5 of a bucket, we would need 63 buckets.

General Formula and Application

The relationship between the number of buckets and the bucket capacity can be described by the following formula:

(N_1C_1 N_2C_2)

Where:

(N_1) is the initial number of buckets, (C_1) is the initial bucket capacity, (N_2) is the number of new buckets, (C_2) is the new bucket capacity.

Using this formula, if the initial capacity of the bucket is 100 litres and the number of buckets required is 20, and the new capacity of the bucket is 0.4 of the original capacity, the new number of buckets required would be:

(20 div 0.4 50) buckets.

Conclusion

In conclusion, the relationship between the bucket capacity and the number of buckets needed to fill a tank is inversely proportional. When the bucket capacity is reduced, the number of buckets required increases, and vice versa. This understanding is crucial in various fields, from farming to industrial applications, where precise measurement and calculation of liquid volume are essential.