A home cleaning service takes more time to clean larger spaces. The table shows the number of minutes it takes for the service to clean a certain number of square feet within a home.
| | | | | |
| :-------------------- | :---- | :---- | :---- | :---- |
| Square Feet and Cleaning Time | | | | |
| Number of Square Feet, $x$ | 50 | 80 | 100 | 150 |
| Time in Minutes, $y$ | 20 | 32 | 40 | 60 |
Based on the table, how many minutes would it take for the service to clean 200 square feet?
A. 20 minutes
B. 80 minutes
C. 125 minutes
D. 400 minutes
Answer (2)
The cleaning service takes a constant rate of 0.4 minutes per square foot. Therefore, to clean 200 square feet, it would take 80 minutes. The correct answer is B. 80 minutes .
;
Calculate the cleaning rate for each given data point and observe that the rate is constant: 0.4 minutes/sq ft .
Recognize the linear relationship between square footage and cleaning time.
Multiply the square footage (200 sq ft) by the constant rate (0.4 minutes/sq ft) to find the cleaning time.
Conclude that it would take 80 minutes to clean 200 square feet: 80 minutes .
Explanation
Understanding the Problem We are given a table that shows the time it takes a cleaning service to clean different square footage. We need to determine how long it would take to clean 200 square feet.
Calculating Cleaning Rates First, let's calculate the rate of cleaning (minutes per square foot) for each data point in the table. This will help us determine if the relationship between square footage and cleaning time is constant (linear).
Determining Constant Rate The rates are:
For 50 square feet: 50 sq ft 20 minutes = 0.4 minutes/sq ft
For 80 square feet: 80 sq ft 32 minutes = 0.4 minutes/sq ft
For 100 square feet: 100 sq ft 40 minutes = 0.4 minutes/sq ft
For 150 square feet: 150 sq ft 60 minutes = 0.4 minutes/sq ft
Confirming Linear Relationship Since the rate is constant at 0.4 minutes per square foot, we can assume a linear relationship between the number of square feet and the cleaning time.
Calculating Time for 200 Square Feet To find the time it would take to clean 200 square feet, we multiply the square footage by the rate:
200 sq ft × 0.4 minutes/sq ft = 80 minutes
Final Answer Therefore, it would take 80 minutes to clean 200 square feet.
Examples
Understanding proportional relationships is useful in many real-life situations. For example, if you know the price of one item, you can calculate the price of multiple items if the price is directly proportional to the number of items. Similarly, in cooking, if you need to scale a recipe up or down, you use proportional relationships to adjust the amount of each ingredient. These concepts are also used in calculating distances on maps and understanding scale models.
