 ## Understanding the Constant of Proportionality: Using Tables and Graphs

In the realm of mathematics, the Constant of Proportionality emerges as a key idea when examining proportional relationships. Essentially, it represents the constant ratio of two proportional quantities. In a scenario where one variable is always the product of the other and a constant, they share a proportional relationship. The constant here is referred to as the Constant of Proportionality.

## Step-by-Step Guide: Using Tables to Find the Constant of Proportionality

### Step 1: Understand the Scenario

The first action on the path to discovering the Constant of Proportionality is understanding the problem you are tackling. Are you considering a situation dealing with speed and time, or maybe a problem concerning cost and quantity?

### Step 2: Organize the Data in a Table

Subsequently, arrange the data presented in the problem in a tabular format. This table should encompass the two variables in question. For instance, if we are grappling with a scenario of speed and time, the table should capture these two quantities.

### Step 3: Identify the Ratio

With your data neatly tabulated, you can now discern the ratio of the two variables for each entry in the table.

### Step 4: Determine the Constant of Proportionality

If all these ratios coincide, then this shared value is the Constant of Proportionality.

## Step-by-Step Guide: Using Graphs to Find the Constant of Proportionality

### Step 1: Plot the Data

Given the data points, you need to plot these on a coordinate graph. If the data yields a straight line through the origin, the relationship between the variables is proportional.

### Step 2: Calculate the Slope

Calculate the slope of the line that the plotted points form. The slope can be determined by selecting two points on the line and computing the rise over the run (change in $$y$$ divided by the change in $$x$$).

### Step 3: Identify the Constant of Proportionality

In a proportional relationship, the slope of the line corresponds to the Constant of Proportionality.

And there you have it, math enthusiasts! You've now mastered the art of finding the Constant of Proportionality using both tables and graphs. Remember, practice is the key to mastering any concept, so don't hesitate to put these steps into action with various problems. Happy computing!

### Example

Scenario: A car is moving at a constant speed. The table below shows the distance the car traveled (in miles) for different amounts of time (in hours):

Hours (h) Distance (d)
1 50
2 100
3 150
4 200

1. Using a Table: To find the constant of proportionality, we can divide the distance by the time for each row in the table. This gives us:

For 1 hour, $$\frac{50}{1} = 50$$ mph,
For 2 hours, $$\frac{100}{2} = 50$$ mph,
For 3 hours, $$\frac{150}{3} = 50$$ mph,
For 4 hours, $$\frac{200}{4} = 50$$ mph.

As the ratios are the same, we can say that the constant of proportionality is 50 mph.

2. Using a Graph: If we plot the data from the table on a graph with time on the x-axis and distance on the y-axis, we would get a straight line passing through the origin. The slope of the line represents the constant of proportionality.

To calculate the slope, we can use the formula:
$$\text{slope} = \frac{\text{change in y}}{\text{change in x}} = \frac{\text{change in distance}}{\text{change in time}}$$

Selecting two points on the line, say (1, 50) and (2, 100), we can calculate the slope as:
$$\frac{100 - 50}{2 - 1} = 50$$

This matches our earlier result, confirming that the constant of proportionality is 50 mph.

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