James cuts on average 15 logs in 10 minutes. Write an equation for the number of logs 𝑙 he cuts in 𝑚 minutes.
We’re told that during a period of 10 minutes, James can cut 15 logs, where 𝑙 is the number of logs he cuts and 𝑚 is the number of minutes. Now, since this is an average, we can assume that if James had 20 minutes, he would cut double the amount of logs. If he had 30 minutes, he’d cut triple the amount of logs, and so on. So we make an assumption that 𝑙 and 𝑚, the variables, are directly proportional to one another.
This symbol here, which isn’t in fact the Greek letter 𝛼, is the directly proportional to symbol. This means 𝑙 is directly proportional to 𝑚. As the number of minutes increases, the number of logs cut also increases at the same factor.
Now, suppose we have a pair of variables 𝑦 and 𝑥. If those variables are directly proportional to one another, then we say that their ratio is constant; 𝑦 divided by 𝑥 equals 𝑘. So in this case, we can say that 𝑙 divided by 𝑚 equals 𝑘. The number of logs cut per minute will be equal to some constant. Since when 𝑙 equals 15, 𝑚 equals 10, we can find 𝑘 by dividing 15 by 10. 15 divided by 10 is 1.5. So 𝑘, which we call our constant of proportionality or constant of variation, must be equal to 1.5.
And before we use this to find an equation for the number of logs cut in 𝑚 minutes, let’s define the units. 15 logs are cut in 10 minutes. Since we divided the number of logs by the number of minutes, we’re basically saying that 1.5 logs are cut per minute. Let’s replace 𝑘 with 1.5 in our earlier equation. So 𝑙 over 𝑚 equals 1.5.
We’re going to make 𝑙 the subject since we’re forming an equation for 𝑙 in terms of 𝑚. And to do so, we multiply both sides of this equation by 𝑚. On the left-hand side, we get 𝑙. And on the right, we get 1.5𝑚. And so the equation for the number of logs 𝑙 James cuts in 𝑚 minutes is 𝑙 equals 1.5𝑚.