How Much More Water is Needed to Fill a Bucket?

Imagine you have a bucket that is partially filled with water. You know that 4/5 of a certain amount of water fills 2/3 of the bucket. How do you calculate the total capacity of the bucket and determine how much more water you need to fill it to the top?

Understanding the Problem

The problem involves understanding how to scale a given amount of water to the full capacity of a bucket. Let's denote the total capacity of the bucket as B. We know that 4/5 of a certain quantity of water fills 2/3 of the bucket. This relationship can be expressed mathematically as:

[frac{4}{5} frac{2}{3} B]

Calculating the Bucket's Capacity

To find the total capacity B of the bucket, we need to solve for B. This can be done by isolating B on one side of the equation. Start by multiplying both sides of the equation by the reciprocal of 2/3:

[B frac{4}{5} times frac{3}{2}]

Next, perform the multiplication:

[B frac{4 times 3}{5 times 2} frac{12}{10} frac{6}{5}]

Therefore, the total capacity of the bucket is 6/5 units of water. This means the bucket can hold:

[B 1 frac{1}{5} 1.2]

The bucket's full capacity is 1.2 units of water.

Calculating the Remaining Water Needed

Now that we know the bucket's total capacity is 6/5 units of water, we can determine how much more water is needed to fill it. You already have 4/5 of a unit of water. Subtract this from the total capacity:

[text{Water needed} B - frac{4}{5}]

Substitute the value of B:

[text{Water needed} frac{6}{5} - frac{4}{5} frac{2}{5}]

Therefore, you need an additional 2/5 of a unit of water to fill the whole bucket. This amount can be expressed as 6 liters if the bucket's total capacity is 18 liters.

Generalizing the Solution

This calculation method can be generalized to other scenarios. For instance, if you have a jug that is 4/5 full and you want to fill a 5-gallon bucket, you would need:

[frac{5}{frac{4}{5}} 5 times frac{5}{4} frac{25}{4} 6.25]

This means you would need 6.25 jugs of 4/5 to fill the 5-gallon bucket.

Or, if you have 5 gallons of water and a 5-gallon bucket, the spill point is 10/3 gallons, or approximately 3.333 gallons. This means that to fill the bucket to 2/3, you need:

[text{2/3 of the bucket} text{2/3 of 5 gallons} frac{10}{3}text{ gallons} 3.333text{ gallons}]

Dividing this by 2/3, you need 1 2/3 gallons per 1/3 portion of the bucket to fill it completely.

In conclusion, understanding the relationship between the amount of water and the bucket capacity allows you to calculate how much more water is needed to fill the bucket completely, whether you are dealing with complex quantities or simpler scenarios.