We saw in our last video that if two variables, like the number of hours worked and money earned, are directly proportional to each other, then as one of them increases, the other one increases proportionally. So in this case, the more hours you work, the more money you'll earn. In this video though, we're going to cover how to write an equation to express this sort of relationship. So for this example, if we were to say that E is the money earned and H is equal to the hours worked, then the equation that describes the relationship between money earned and hours worked would be E equals 12H. So if you worked for 3 hours, E would equal 12 times 3, which is 36. So you'd have earned £36 for those 3 hours.
In exams though, you'll often have to figure these equations out for yourself without using a graph. So to see how we do that, let's jump straight in with a typical question. So the first thing we're told here is that the time in seconds that it takes to boil some water is directly proportional to the mass of the water, measured in grams, that's in the kettle. So basically the time taken, which we represent with the letter T, is directly proportional to the mass of water that we have, which we represent with the letter M.
or if we wanted to put it in algebra form, we could say that t is proportional to m. Next, we're told that when t equals 600, m equals 200, or in other words, it takes 600 seconds to boil 200 grams of water. And finally, the question is asking us to find t when m equals 450, so to find the time it would take to boil 450 grams of water. Now, the idea with this type of question is to first of all use the values that we're given to help us write an expression to describe the relationship, in this case the relationship between time and mass. And then once we have that, we can plug in something like the mass of 450 to help us find out the corresponding value of t.
So if we go back to our expression so far, which says that t is proportional to m, The first thing we have to do is change the proportional sign to an equal sign, but when we do that we also have to add a constant of proportionality, which we normally represent like this using the letter k. This constant has an actual value though, like 0.2, 3, or 50, and to find it we need to rewrite the equation using these values from the question, which in this case would give us the time of 600 equals k times the mass of 200. So then we can divide both sides by 200 to find that 3 equals k. And now that we know that k is 3, we can rewrite our equation as t equals 3m. So we now have an algebraic equation that links time and mass, which we can use to convert between the two of them. So to find the time it would take to heat 450 grams of water, like it asks in the question.
All we have to do is plug the 450 into our equation, so you get t equals 3 times 450, or t equals 1350. So it would take 1350 seconds to heat up 450 grams of water. Let's have a go at one more. The length of a piece of water is directly proportional to its diameter. A piece of wire that's 25cm long has a diameter of 2cm. Write an expression for the relationship between length and diameter, and calculate the diameter of a piece of wire that's 40cm long.
So from this first part, we know that length is directly proportional to diameter, so L is proportional to D. Then if we add our constant of proportionality, we get L equals kd. And now we can use this second sentence to figure out what k is.
So if we plug in our length of 25 and our diameter of 2, we get 25 equals k times 2. And then we can divide both sides by 2 to find that 12.5 equals k, which means that our equation, that length, length, and diameter, is l equals 12.5d. Then, moving on to the last part of the question. We can now use our equation to find the diameter of a 40cm piece of wire. So we plug in 40 in the place of L to get 40 equals 12.5d, and then divide both sides by 12.5 to find out 3.2 equals d. So the 40cm wire would have a diameter of 3.2cm.
Anyway, that's the end of this video, so hope you found it useful! And we'll see you again soon!