Find the missing number in this input-output table.
This input-output table is a little bit like a number machine. We have a row of numbers that go in and then something happens to them. And we have a row of numbers that goes out. But one of our numbers is missing. To help us to find this missing number, we first need to work out what the rule is for change as an input to an output. Let’s start by looking at the first two numbers in the table.
We can see that the input is 31. That’s three 10s and one one. Our output is 10m which is simply one 10. So to get from three 10s and one one down to a one 10, we need to lose one one and two 10s. In other words, we need to subtract 21. To check that this is the rule for the whole table, let’s try another pair of numbers.
What if our input is 72? 72 is made up of seven tens and two ones. Now, if we subtract 21, that means again we need to take away two 10s and one one. So instead of seven 10s, we now have five 10s. And instead of two ones, we now have one one. Our output number is 51 and this matches the output in the table. So this means the whole table has a one-step rule that we need to follow. We simply subtract 21 from the input to find the output.
Now that we know the rule, let’s have a look at the column where our missing number is. This is interesting because we can see that our missing number isn’t an output. We know what the output is. It’s 27. But we don’t know what the input was. In other words, what number do we take 21 from to give us 27? To find the answer, we can apply the rule backwards or the inverse operation. In other words, we can start with 27. And then, instead of subtracting 21, we add 21. And this should find us our input. 27 is equal to two 10s and seven ones.
Now, if we want to add 21 to this number, we need to add another two 10s, which takes us to four 10s and also another one. We now have eight ones. And the number we have as an answer is 48. 27 plus 21 equals 48. This means our missing number must be 48. Let’s subtract 21 from 48 just to check whether the input turns into the output. 48 has four 10s. And if we subtract two 10s, this leaves us with two 10s. And 48 has eight ones. And if we subtract one one, we end with seven ones. This is exactly the opposite of what we’ve just done. 48 is an input. We know we’re going to get the output 27.
The one-step rule for our input-output table was to take away 21 each time. To find our missing input, we needed to work backwards and use the inverse operation. So instead of taking away 21, we added 21. The missing number in our input-output table is 48.