Answered

Katrina drinks 0.5 gallons of water per day. Which expression shows how to find the number of cups of water she drinks in a week?



There are 16 cups in a gallon.


A$\frac{0.5 \text{ gallons}}{1 \text{ day}} \times \frac{16 \text{ cups}}{1 \text{ gallon}} \times \frac{1 \text{ week}}{7 \text{ days}}$
B$\frac{0.5 \text{ gallons}}{1 \text{ day}} \times \frac{1 \text{ gallon}}{16 \text{ cups}} \times \frac{7 \text{ days}}{1 \text{ week}}$
C$\frac{0.5 \text{ gallons}}{1 \text{ day}} \times \frac{1 \text{ gallon}}{16 \text{ cups}} \times \frac{1 \text{ week}}{7 \text{ days}}$
D$\frac{0.5 \text{ gallons}}{1 \text{ day}} \times \frac{16 \text{ cups}}{1 \text{ gallon}} \times \frac{7 \text{ days}}{1 \text{ week}}$

Verified Answer :

  • Start with the rate of \(0.5 \text{ gallons/day}\).
  • Convert gallons to cups by multiplying by the ratio \(\frac{16 \text{ cups}}{1 \text{ gallon}}\).
  • Convert days to weeks by multiplying by the ratio \(\frac{7 \text{ days}}{1 \text{ week}}\).
  • The final expression is \(\frac{0.5 \text{ gallons}}{1 \text{ day}} \times \frac{16 \text{ cups}}{1 \text{ gallon}} \times \frac{7 \text{ days}}{1 \text{ week}}\), which simplifies to \(\boxed{\frac{0.5 \text{ gallons}}{1 \text{ day}} \times \frac{16 \text{ cups}}{1 \text{ gallon}} \times \frac{7 \text{ days}}{1 \text{ week}}}\).

Explanation

  1. Problem Analysis and Data Preparation

To determine the amount of water Katrina consumes in a week, we need to convert her daily consumption rate into a weekly volume measured in cups. We are given:

  • Daily consumption rate: \(0.5\) gallons per day.
  • Conversion factor: \(16\) cups per \(1\) gallon.
  • Time conversion: \(1\) week is \(7\) days.

Our goal is to reach a unit of "cups per week" by using the factor-label method (dimensional analysis), which ensures that units cancel out correctly to leave us with the desired final units.

  1. Step-by-Step Conversion and Selection

We start with the given rate:
\[\frac{0.5 \text{ gallons}}{1 \text{ day}}\]
To convert "gallons" into "cups", we multiply by the ratio \(\frac{16 \text{ cups}}{1 \text{ gallon}}\). This places "gallons" in the denominator, allowing the unit to cancel out:
\[\frac{0.5 \text{ gallons}}{1 \text{ day}} \times \frac{16 \text{ cups}}{1 \text{ gallon}} = \frac{0.5 \times 16 \text{ cups}}{1 \text{ day}} = \frac{8 \text{ cups}}{1 \text{ day}}\]
Now, we have the rate in cups per day. To find the amount consumed in a week, we need to multiply by the conversion factor relating days to weeks. Since there are \(7\) days in \(1\) week, we multiply by \(\frac{7 \text{ days}}{1 \text{ week}}\):
\[\frac{8 \text{ cups}}{1 \text{ day}} \times \frac{7 \text{ days}}{1 \text{ week}} = \frac{8 \times 7 \text{ cups}}{1 \text{ week}} = \frac{56 \text{ cups}}{1 \text{ week}}\]
Comparing this process to the given options, we identify the expression that matches our step-by-step conversion.

  1. Conclusion and Selection of the Correct Option

The expression that correctly represents the conversion from "gallons per day" to "cups per week" is:
\[\frac{0.5 \text{ gallons}}{1 \text{ day}} \times \frac{16 \text{ cups}}{1 \text{ gallon}} \times \frac{7 \text{ days}}{1 \text{ week}}\]
This expression correctly cancels the units to result in cups per week: \(\frac{\text{gallons}}{\text{day}} \times \frac{\text{cups}}{\text{gallon}} \times \frac{\text{days}}{\text{week}} = \frac{\text{cups}}{\text{week}}\). The correct option is the fourth one.

Examples

Understanding unit conversion is essential in many daily scenarios, such as cooking, where you might need to convert fluid ounces to tablespoons or cups to milliliters. Similarly, when traveling in different countries, you might need to convert between miles and kilometers or gallons and liters to estimate fuel efficiency or travel distance. Mastering dimensional analysis allows you to accurately scale measurements and perform these types of conversions with confidence.

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