The rule for this input–output table is divide by three. Jacob has shown that the pattern for the input numbers is subtract nine. Describe the pattern that the output numbers will follow.
So, in this problem, we’re given an input–output table. This is a little bit like a number machine. On the left-hand side, that’s the left-hand column in the table, we have the numbers that are the inputs. These are the numbers going into the machine, and these are 42, 33, 24, 15, and six. And then on the right-hand side, we have the output column. This is the column of numbers after they’ve had something done to them. And the thing that they have done to them is the rule is to divide by three.
The first row has been completed as an example. 42 is the input. It’s then being divided by three. Because 42 divided by three is 14, the output is the number 14. Now, Jacob is clearly being looking closely at this table because he can see a pattern in all the numbers that are going in. In other words, he’s looked at 42, 33, 24, 15, and six and said to himself, I can see a pattern there. And the pattern he spotted is to subtract nine each time. 42 take away nine gives him his next input of 33. 33 take away nine gives him his next input of 24, and so on.
Now, we’re asked to describe the pattern that the output numbers will follow. The inputs are decreasing by nine each time. Remember, we’re dividing them by three. There are two ways that we could use to find the pattern that the output numbers will follow. The first method is to complete the table, and once we’ve completed the table, to look at all the numbers in the output column, and to see if we can spot a pattern.
The second method that we could use doesn’t involve doing any working out of outputs at all. We just need to use reasoning. But this method is a little bit more complicated, so let’s use the first method. Perhaps we’ll go over the second one at the end quickly.
So, our first row’s been completed for us, 42 and 14. And then, let’s look at our second row. 33 is our input, but what’s our output? What’s 33 divided by three? Well, we know that 30 divided by three is 10, so 33 is one more three than this. It’s 11 threes. If 33 is the input, the output is 11. How many threes are there in 24? We know eight times three is 24. So, 24 divided by three leaves us with eight. What about if 15 is the input? 15 divided by three equals five. There are five threes in 15. And the nice one to end with, how many threes are there in six? There are two threes in six.
The numbers in the output column are 14, 11, eight, five, and two. What pattern can we see? Well, the difference between 14 and 11 is three. If we carry on going on the output column, we can see if we subtract three each time, we find the next output. The pattern that the output numbers will follow is subtracting three each time. Now, we did say there was another method we could use here without working out any of the outputs. If you’re interested, we’ll go over it now.
Let’s imagine we haven’t worked out any of the outputs yet. We know that the first output is 14 because there are 14 threes in 42. And what we do know is that the input is going to decrease by nine. How many threes has the input decreased by if it’s decreased by nine? Well, there are three threes in nine. So, the number of threes in our next answer is going to have decreased by three. And then, again, if we take nine of the input, what we really do is taking away three lots of three. So, the output, which shows us the number of threes in the number, is going to decrease by three each time.
There we go. That’s a way to find the answer without doing any calculations at all. But the method we used was to find all those outputs. All the outputs are 14, 11, eight, three, and two. And the pattern that we noticed the output numbers will follow is subtracting three every time.