Find the missing number in this input–output table.
In this problem, we’re given an input–output table. We can see that one row, the top row, is labeled in and another row, the bottom row, is labeled out. We can think of an input–output table as being a little bit like a number machine. A number goes into the machine. That’s the input. Something happens to that number. Maybe something is added to it or taken away or maybe it’s multiplied by something or divided by something. And then, a new number comes out. And this is the output. Now just by looking at the table, we’re not told what’s happening to each number, but we are told some inputs and some outputs. And we can use these to try and work out the rule to our input–output table.
So, if we look at each column of the table, we can see how a number changes. In the first column, we’ve got an input of three and an output of 18. In the second column, we have an input of six and the output is blank. This is the missing number that we’re being asked to find. In the third column, we have an input of seven and an output of 42. Then, an input of nine and an output of 54. And finally, an input of 12 and an output of 72. What rule are we using to change these numbers each time?
Let’s start by looking at our first pair of numbers. If these are smaller, they might be a bit easier to deal with. Our input, or the number that goes into our machine, is three. We know that the number that comes out is 18. So, what happens to three to get 18? Well, we know that 18 is three more than 15. So, one thing we could have done here is to add 15. Three plus 15 equals 18. Let’s see whether this rule works for one of our other pairs of numbers.
If we have an input of seven and then we add 15, what do we get? Well, five add 15 would be 20. So, if we add two more than that, the answer is going to be 22. This is not the same as the bottom number or the output in the third column. We’re told that the output, if we put in seven, is going to be 42 not 22. And if we quickly glance at the other two columns too, we can see, without working anything out, that this is not a plus-15 rule. So, if we’re not adding 15, what are we doing?
Let’s go back to our first two numbers and have another look at them. How could we get from three to 18? Well, what about multiplication? Three lots of six is 18. So, if we multiply three by six, we get 18. Perhaps this is the rule that we’re looking for. Let’s try our next pair of numbers again.
If we put seven into our number machine, this time we want to multiply by six. What is seven times six? Let’s count in sixes seven times to find the answer. Six, 12, 18, four sixes are 24, 30, six sixes are 36, and seven sixes are 42. This was the output we were looking for. Our rule seems to be, multiply the input by six. And if we check the last two columns, we know that nine sixes are 54. And if we have an input of 12, 12 times six equals 72. Now, we know that the rule is to multiply by six, we can find our missing number.
If we have an input of six, what is the output going to be? Well, here we can use our number line to help us. We know that seven sixes are 42. So, six lots of six is one lot of six less than 42. The answer is 36. So, we can complete our table with the missing number. We first found the answer by first identifying the rule for our input–output table. And then once we’ve done that, we could apply the rule to help us find the missing output. The missing number in our input–output table is 36.