Problem 36 - Prealgebra

Question

Convert each rate using dimensional analysis. $$4.8 \mathrm{qt} / \mathrm{min}=\square \mathrm{gal} / \mathrm{h}$$

Step-by-Step Solution

Answer

4.8 qt/min is equivalent to 72 gal/h.

Step 1: Understanding the question

We need to convert a given rate of 4.8 quarts per minute to gallons per hour. This involves converting both the volume unit (from quarts to gallons) and the time unit (from minutes to hours).

Step 2: Conversion factors

Identify the conversion factors: 1 gallon = 4 quarts and 1 hour = 60 minutes. We will use these to convert the units in our rate.

Step 3: Convert quarts to gallons

Since 1 gallon = 4 quarts, we convert quarts to gallons by dividing the number of quarts by 4: \[ 4.8 ext{ qt/min} \times \frac{1 ext{ gal}}{4 ext{ qt}} = 1.2 ext{ gal/min} \]

Step 4: Convert minutes to hours

We need to convert the time unit from minutes to hours like so: \[ 1.2 ext{ gal/min} \times 60 ext{ min/h} = 72 ext{ gal/h} \]

Step 5: Final Answer

The converted rate is 72 gallons per hour.

Key Concepts

Unit Conversion

Unit conversion is the process of changing a measurement from one unit to another. This is particularly important in science and engineering where measurements using different units need to be standardized. In the context of our exercise, we want to convert quarts to gallons, which are both units of volume. The relationship between these units is essential to correctly change the values. For instance, 1 gallon is equivalent to 4 quarts. By knowing this, you can easily perform the conversion by dividing the number of quarts by four to get the number of gallons:

• The formula for conversion: \[ \frac{\text{gallons}}{\text{quart}} = \frac{1}{4} \]

This concept aids in ensuring consistency and eliminates confusion in various calculations. When performing conversions, writing the conversion factor as a fraction helps to keep track of the units you are converting from and to.

Rate Conversion

Rate conversion takes unit conversion a step further by involving change over time or a specific amount. It often involves converting one rate to another by modifying both the top and bottom units. In our exercise, we are converting from quarts per minute to gallons per hour.The two main steps include changing the volume unit and the time unit:

• Volume unit conversion: From quarts to gallons, as explained before. The conversion factor is 1 gallon = 4 quarts.
• Time unit conversion: From minutes to hours where 1 hour = 60 minutes. To convert the rate, multiply by the conversion factor for time: \[ \frac{\text{minutes}}{\text{hour}} = 60 \]

Executing both conversions together is crucial to ensure that your final answer reflects the correct rate in the desired units of measurement.

Mathematical Problem Solving

Mathematical problem solving involves breaking down a problem into smaller, manageable parts to find a solution effectively. This approach is essential in problems like unit and rate conversions. Here is how you can apply it efficiently:

• Identify the Given Information: Understanding the units you're starting with and the units you want to end with is crucial. In this exercise, we begin with quarts per minute and want gallons per hour.
• Determine Conversion Factors: Recognize the relationships between units (e.g., 1 gallon = 4 quarts and 1 hour = 60 minutes). This step allows the structured conversion of each component of the original rate.
• Step-by-Step Conversion: Convert one variable at a time—first the volume and then the time. Calculate the conversions and multiply them to shift from the initial rate to the desired rate.

By systematically working through each part of a problem, one can efficiently solve complex mathematical problems, minimizing errors and ensuring accuracy.