Problem 59 - Algebra 2

Question

Find each product. Include the appropriate units with your answer. 5 gallons \(\left(\frac{4 \text { quarts }}{1 \text { gallon }}\right)\)

Step-by-Step Solution

Answer

20 quarts

Step 1: Identify the Units Given

We start with 5 gallons, and we need to convert this into quarts. The conversion factor given is 4 quarts per 1 gallon.

Step 2: Set Up the Conversion

To find the number of quarts, multiply the number of gallons by the conversion factor. Set up the equation as follows: \( 5 \text{ gallons} \times \frac{4 \text{ quarts}}{1 \text{ gallon}} \).

Step 3: Perform the Multiplication

Multiply the numbers. Here, 5 gallons is multiplied by \( \frac{4 \text{ quarts}}{1 \text{ gallon}} \): \( 5 \times 4 = 20 \text{ quarts} \).

Step 4: Confirm the Units

Ensure that the units cancel appropriately. Gallons cancel out, leaving us with quarts: \( 5 \text{ gallons} \times \frac{4 \text{ quarts}}{1 \text{ gallon}} = 20 \text{ quarts} \).

Key Concepts

Gallons to Quarts

Understanding how to convert gallons to quarts is a crucial skill in many real-world scenarios, such as cooking, scientific experiments, and mechanical tasks. A gallon is a larger unit of liquid measurement compared to a quart. Specifically, one gallon is equal to four quarts.
This conversion factor is essential to remember:

• 1 gallon = 4 quarts

By using this simple relationship, you'll be able to convert any gallon measurement into quarts, making it easier to understand and use in various applications.

Multiplication in Conversions

When converting units, especially when using a conversion factor like gallons to quarts, multiplication becomes your friend. The process involves multiplying your original quantity by the conversion factor.
For example:

• Your original quantity is 5 gallons.
• The conversion factor is 4 quarts per 1 gallon.

Thus, you perform the multiplication: \[ 5 \text{ gallons} \times \frac{4 \text{ quarts}}{1 \text{ gallon}} = 20 \text{ quarts} \] This arithmetic operation helps you in achieving the conversion from gallons to quarts smoothly and accurately.

Dimensional Analysis

Dimensional analysis is a method where you use conversion factors to move from one unit to another, ensuring that you're being consistent with your units. The key aspect of dimensional analysis is ensuring that units cancel out properly.
In our example:

• Start with 5 gallons.
• Multiply it by the conversion factor \( \frac{4 \text{ quarts}}{1 \text{ gallon}} \).
• "Gallons" in the numerator and denominator cancel each other out.

You’re then left with "quarts," which is the desired unit. This technique is invaluable because it ensures that the conversion is done correctly without unit errors.

Problem-Solving Steps

Solving a conversion problem involves clear and systematic steps that help ensure accuracy. Here's a simple guide:

• Step 1: Identify your starting and desired units. We started with gallons and wanted to end with quarts.
• Step 2: Set up the conversion using the appropriate conversion factor. In this case, \( \frac{4 \text{ quarts}}{1 \text{ gallon}} \).
• Step 3: Perform the multiplication to calculate the result.
• Step 4: Confirm that the units cancel out, and you're left with the desired units.

By following these steps, even the most complex unit conversions can be simplified and made efficiently manageable.