The rule for this input–output table is divide by four. The input numbers follow the pattern add four. What number is 𝑥? The output numbers will also follow a pattern. What is it? Using the numbers in the table, find two ways to calculate 𝑦. 20 multiplied by four and four plus two, 20 divided by four and four plus one, or 20 plus four and three plus two.
This problem is based on an input–output table. Remember that with an input–output table, numbers come in, something happens to them, and then numbers come out. The first column of our table shows the inputs. These are 12, 16, 20, 𝑥, which stands for a number we don’t know, and then 28. And then the second column, that’s the right-hand column, shows us the outputs. As we’ve said already, something happens to the inputs to get these outputs. And we can see at the top of the table, and also in the question, that the rule for the input–output table is to divide by four.
We can see that our output column is only partially completed. We have three, four, 𝑦, which is a letter that represents another unknown number, and then the last two outputs haven’t been completed yet. But still, this is enough information for us to see how the table works. For example, if we start with 12, then divide by four, 12 divided by four is three. The input’s 12 and the output’s three. 16 divided by four gives an output of four, and so on. Now that we understand how the input–output table works, let’s go through the three different parts to our problem, and they’re all about finding patterns in the inputs and outputs.
The first part tells us that the input numbers follow the pattern add four. What number is 𝑥? You may have noticed that the input numbers follow the pattern add four as we read them together. 12, 16, 20, these are all multiples of four. And if we look to the left of the input column, we can see the arrows that show us this pattern plus four every time. 12 plus four gives us 16. 16 plus four gives us 20. And so to find the value of 𝑥, we need to calculate 20 plus four. The answer is 24. So, the value of 𝑥 equals 24.
We’re told that the output numbers will also follow a pattern. What is it? At the moment, we can only see two actual output numbers. That’s three followed by four. Now, this looks like it might be a pattern of adding one. But when we only have one piece of information like this, it’s very hard to show a pattern. For example, the pattern could be plus one plus two plus three plus four. It doesn’t have to be plus one plus one plus one. But we can use reasoning to help us answer this question.
If we take a moment to look back at the inputs, as we’ve said already, this column shows multiples of four. But importantly, it shows multiples of four in order. Three times four is 12, and then we have four times four is 16. Then, five times four, six times four, seven times four. So, when we divide each of these multiples of four by four, we can see that they will be a pattern that goes up in ones each time. As we’ve just said, there are three fours in 12, four fours in 16, five fours in 20, six fours in 24, and seven fours in 28. So, we can say that the pattern will be to add one each time.
Finally, we’re asked to use the numbers in the table to find two ways to calculate 𝑦. The first possible method is different for each answer. We have multiply, divide, and add. But the second possible method contains three additions. This must be to do with the output column. So, let’s start by looking at the output column. The first possible answer suggests that we start on four and add two. But we know if we start at four, we only have to add one to get to 𝑦. The first answer is not correct.
The second answer could be correct. As we’ve just said, we start at four and then add one to get to 𝑦. But also the final answer could be correct. If we start at the top of our output column, at the number three, we have two jumps of plus one to get to 𝑦, that’s three plus two. Of course, we know that 𝑦 equals five. Both these give a possible answer of five. So, which is the correct answer?
The first part of each answer talks about the input. And, of course, we know that if we have an input of 20, the rule is divide by four. Direct ways to calculate 𝑦 are 20 divided by four and four plus one. We’ve used the information inside our input–output table and the rule divide by four to help us answer the questions. The input numbers follow the pattern add four, and so we know that 𝑥 equals 24. The output numbers also follow a pattern, which is to add one each time. And our two ways to calculate 𝑦 are 20 divided by four and four plus one.