Adil is looking at some function machines. 𝑥 times two times three equals 𝑦. Adil says, “If you know 𝑦, you can work out 𝑥 by either dividing by two first and then dividing by three or dividing by three first and then dividing by two.” Is Adil correct? You must give a reason for your answer.
According to Adil if we start with 𝑦, we could divide by two and then divide by three to get 𝑥. She also claims that this order doesn’t matter, that we could first divide by three and then divide by two. Why don’t we test it by plugging in some value for 𝑥? This function machine tells us that 𝑥 times two times three equals 𝑦. If 𝑥 equals one, one times two equals two, two times three equals 𝑦, six equals 𝑦. When 𝑥 equals one, 𝑦 equals six.
Now, we want to do the reverse 𝑦 divided by two divided by three should equal 𝑥. And according to Adil, 𝑦 divided by three divided by two should also equal 𝑥. In the first part, six divided by two divided by three: six divided by two equals three, three divided by three equals 𝑥, and 𝑥 equals one, which is correct. And what if we change the order of division? Is six divided by three divided by two also equal to the correct 𝑥-value? Six divided by three equals two, two divided by two equals 𝑥, and 𝑥 equals one.
We’ve shown that yes, Adil is correct. This is because divided by two and divided by three are the inverse operations of multiplied by two and multiplied by three, respectively and because the order of these operations does not matter. If the only operations we’re using are multiplication and division, then the order won’t matter.
Now, I’m gonna clear some space and consider part b), a second function machine.
Here is the second function machine. 𝑎 minus two multiplied by three equals 𝑏. Adil says, “If you know the value of 𝑏, you can work out the value of 𝑎 by first dividing 𝑏 by three and then adding two.” He says, “You cannot add two first and then divide by three.” Is Adil correct? You must give a reason for your answer.
Let’s map out what Adil has said. He says if we start with 𝑏, we first divide by three and then we add two to get 𝑎. First thing we notice is that dividing by three is the inverse of multiplying by three and that adding two is the inverse of subtracting two. We found the inverse operation. But now we need to consider the question, “does order matter here?”
We’re starting with 𝑎 minus two and we multiply that by three to give us 𝑏. Can we switch these two operations and still come up with the same answer? Is 𝑎 multiplied by three and subtract two the same value for 𝑏? Let’s try it if 𝑎 equals two. Here’s what we want to know: is two minus two multiplied by three the same thing as two times three minus two? Two minus two equals zero, zero times three equals zero, two times three equals six, six minus two equals four. Zero is not equal to four.
In this function machine, order matters And if order matters moving in the forward direction, then order will also matter moving in the reverse direction.
Adil is correct. Although divided by three and plus two are the inverse operations to multiply by three and subtract two, respectively, in this case, the order does matter. Given 𝑏, you cannot add two and then divide by three. You must divide by three and then add two to get 𝑎.