## Subjects

# Proportionality

Cake sales are described by the line $y=kx$, where x is the number of days that have passed since the start. Maria's $k$ value is twice as large as Michael's. This means that...

## Proportionality

If you haven't learned about the coordinate system yet, go through that first, because there are things that you need to know before you would be able to understand lines in the coordinate system. Two points in a coordinate system can be connected by a line. We can use this to show how two things are connected to each other, for example, how some value changed over time. The number of cakes Maria sold can be represented by a point in the coordinate system. Let the Y axis represent the total number of cakes sold.

And the X axis represents time, in days. Now, we can see that by the end of the first day she sold 1 cake, by the end of the second day she sold 2 cakes, and so on. We see that Maria sells 1 cake a day. Mikael also sells cakes. Here are his results.

Mikael's line isn't as steep as Maria's line. It tells us that Mikael sells at a slower rate than Maria. Now, Monika also starts to sell cakes. And she sells a lot. Her line is steeper than two previous lines.

We call this the line's slope, or if we want to sound particularly elegant, a gradient. If the X axis represents time, the slope will represent the rate of change - how quickly something changes over time. We can say that the results of the three cake sellers are directly proportional to time. For every day that passes, the sales increase by the same amount. We can write this as an equation, like this: Maria's total sales are equal to 1 cake times the number of days that have passed.

The number of days are on the X axis. So we write 1 times X. And since it's precisely one X, we don't need to write the number at all. Total sales are represented by the Y axis, so we write Y there. The function Y equals X describes Maria's sales.

Monika sells two cakes every day. Her sales are also directly proportional to time, but she sells twice as much per day. Her total sales, Y, are equal to 2 times the number of days. Y equals two X. Mikael sells one cake every other day, or half a cake a day on average.

If we call the total sales Y and the days X here as well, we get the function Y equals zero-comma-five X. These are three equations that describe how something - cake sales - changes as a function of another item - time. We write the function as Y equals K X, where K is the slope. K tells us how much Y changes every time we increase X by one. In the coordinate system equation Y equals K X will always result in a straight line passing through the origin.