Problem 6 - Prealgebra

Question

Express each of the following rates as a ratio with the given units. Gallons/Minute A 225-gallon drum is filled in 3 minutes. What is the rate in gallons per minute?

Step-by-Step Solution

Answer

The rate is 75 gallons per minute.

Step 1: Identify the Given Values

We are provided with a 225-gallon drum and a time duration of 3 minutes for it to be completely filled.

Step 2: Set Up the Rate as a Division

To find the rate in gallons per minute, divide the total number of gallons by the total number of minutes.

Step 3: Perform the Calculation

Calculate the rate: \( \frac{225 \text{ gallons}}{3 \text{ minutes}} = 75 \text{ gallons per minute} \).

Step 4: Conclude with the Rate

The rate at which the drum is being filled is 75 gallons per minute.

Key Concepts

Rates

In mathematics, rates are a special kind of ratio that compare two different quantities with different units. Rates are found in many real-world contexts, such as speed (miles per hour), density (people per square mile), and our current example, flow rate (gallons per minute).
A rate helps us understand how one quantity changes in relation to another. For example, filling a 225-gallon drum in 3 minutes gives us a rate, because it shows how fast the process happens over time. The rate we calculate reflects this relationship.
When working with rates, we often use division to express one quantity per unit of the other. The resulting number tells us the average quantity of the first item per one unit of the second item. It's important to clearly specify these units to make sense of the rate.

Division

Division is a mathematical operation used to determine how many times one number is contained within another. When we express a rate, we often use division to distribute the values into a standard measurement.
In the example of the 225-gallon drum, we divide the total number of gallons by the total number of minutes. This calculation, \( \frac{225}{3} \), gives us the number of gallons per each minute of time: 75 gallons per minute.
Division helps break down larger quantities into smaller, manageable parts, allowing us to better understand the relationship between the quantities in a rate. By this operation, we assign a single quantity to one unit of another, simplifying broader concepts into easy-to-understand insights.

Units of Measurement

Units of measurement are essential for describing quantities in mathematical equations and real-world problems. They provide context for numbers and allow for precise communication about quantities. Common units of measurement include gallons for volume, minutes for time, and miles for distance.
When calculating rates, it's crucial to pay attention to the units involved. In our example, we are dealing with gallons and minutes. The final answer, 75 gallons per minute, is meaningful because it tells us how many gallons are filled in each minute.
Keeping units consistent and properly labeled ensures clarity, especially when comparing or converting rates. Understanding the role of units boosts comprehension and helps avoid mistakes in calculations.