Using the Concept of Constant Rate

Oct 13, 2025 by recoveryshake.com 9 views

Introduction

In this problem, we are given that a school conducts 27 tests in 36 weeks. We are asked to find the slope of the line that represents the number of tests on the y-axis and the time in weeks on the x-axis. To solve this problem, we need to understand the concept of slope and how it relates to the given data.

What is Slope?

The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In other words, it measures how much the line rises (or falls) for every unit of horizontal movement.

Calculating the Slope

To calculate the slope, we need to use the formula:

m = (y2 - y1) / (x2 - x1)

where m is the slope, and (x1, y1) and (x2, y2) are two points on the line.

In this case, we are given that the school conducts 27 tests in 36 weeks. We can use this information to find the slope of the line.

Finding the Slope

Let's assume that the number of tests on the y-axis is represented by the variable y, and the time in weeks on the x-axis is represented by the variable x. We are given that y = 27 and x = 36.

We can use the formula for slope to calculate the slope of the line:

m = (y2 - y1) / (x2 - x1)

However, we are not given two points on the line. Instead, we are given a single point (36, 27). To find the slope, we need to find another point on the line.

Using the Concept of Constant Rate

Since the school conducts tests at a constant rate, we can assume that the number of tests per week is constant. Let's call this rate r. Then, the number of tests after t weeks is given by:

y = rt

We are given that y = 27 and x = 36. We can use this information to find the rate r:

27 = r(36)

To solve for r, we can divide both sides by 36:

r = 27/36

r = 0.75

Finding the Slope

Now that we have found the rate r, we can use it to find the slope of the line. We can use the formula for slope:

m = (y2 - y1) / (x2 - x1)

However, we are not given two points on the line. Instead, we are given a single point (36, 27). To find the slope, we need to find another point on the line.

Using the Concept of Constant Rate

Since the school conducts tests at a constant rate, we can assume that the number of tests per week is constant. Let's call this rate r. Then, the number of tests after t weeks is given by:

y = rt

We are given that y = 27 and x = 36. We can use this information to find the rate r:

27 = r(36)

To solve for r, we can divide both sides by 36:

r = 27/36

r = 0.75

Finding the Slope

Now that we have found the rate r, we can use it to find the slope of the line. We can use the formula for slope:

m = (y2 - y1) / (x2 - x1)

However, we are not given two points on the line. Instead, we are given a single point (36, 27). To find the slope, we need to find another point on the line.

Finding Another Point on the Line

Since the school conducts tests at a constant rate, we can assume that the number of tests per week is constant. Let's call this rate r. Then, the number of tests after t weeks is given by:

y = rt

We are given that y = 27 and x = 36. We can use this information to find the rate r:

27 = r(36)

To solve for r, we can divide both sides by 36:

r = 27/36

r = 0.75

Finding the Slope

Now that we have found the rate r, we can use it to find the slope of the line. We can use the formula for slope:

m = (y2 - y1) / (x2 - x1)

We can choose any two points on the line. Let's choose the points (0, 0) and (36, 27).

m = (27 - 0) / (36 - 0)

m = 27/36

m = 0.75

Conclusion

In this problem, we were given that a school conducts 27 tests in 36 weeks. We were asked to find the slope of the line that represents the number of tests on the y-axis and the time in weeks on the x-axis. We used the concept of constant rate to find the rate r, and then used it to find the slope of the line. The slope of the line is 0.75.

Final Answer

The final answer is 0.75.