Proportional Relationships Explained: Discover 'k'

Let's explore proportional relationships using a car traveling at a constant speed.When a car travels at a constant speed of 2 miles per hour, the distance it travels is proportional to the time spent driving.Let's collect some data about the car's journey and organize it in a table.For each hour of travel, we'll record the distance covered and calculate the ratio between distance and time.When we plot these points on a coordinate plane, we can see a pattern emerge.Each point represents a time and its corresponding distance. Notice how they form a straight line passing through the origin.A key characteristic of proportional relationships is that their graphs always pass through the origin, where both quantities are zero.In a proportional relationship, the ratio between the two quantities stays constant. In this case, the car travels 2 miles for every 1 hour, consistently.This constant ratio is what makes the relationship proportional, and it's why the graph forms a straight line through the origin.The constant of proportionality, k, is the value that connects two quantities in a proportional relationship.We find k by dividing any y-value by its corresponding x-value in the relationship.Let's use a real example with cookies and their cost. Here's our first point: 4 cookies cost 2 dollars.To find k, we divide the cost, 2 dollars, by the number of cookies, 4.This gives us k equals zero point five dollars per cookie. This means each cookie costs fifty cents.Let's verify this relationship holds true for other points. For 2 cookies, the cost is 1 dollar.And for 6 cookies, the cost is 3 dollars.When we connect these points, we get a straight line. This is because k remains constant throughout the relationship.This gives us our equation: y equals k times x, or in this case, y equals zero point five x.Now that we understand k, let's see how to use it to solve real-world problems.Let's return to our cookie store example. Four cookies cost two dollars.To find k, we divide the total price by the number of cookies.This gives us k equals fifty cents per cookie.Once we know k, we can find the price for any number of cookies using the formula y equals k x.Let's solve some examples. For seven cookies:For three cookies:And for ten cookies:When we plot these points, they all fall on a straight line because this is a proportional relationship.This relationship scales perfectly: if you double the number of cookies, the price doubles too!