Isabella is learning a strategy to subtract. She creates calculations with the same answer. Whatever you add to one number, you also have to add to the other. So, 17 take away eight equals 19 take away 10. Use her strategy to find the missing numbers. 93 take away 78 equals what take away 80? 317 take away 196 equals what take away 200? And 873 take away 382 equals what take away 400?
In this problem, we’re introduced to a strategy that’s really helpful if we are subtracting two numbers. The aim of the strategy is to create an easier calculation that has the same answer. And this is what Isabella tries to do. In her explanation, Isabella says the following, whatever you add to one number, you also have to add to the other. And we can see that in the example we’re given, Isabella is trying to work out the answer to 17 take away eight.
What does she do to make this calculation easier to work out? Well, she adds two to both numbers in the subtraction. Adding two to 17 gives her an answer of 19. And adding two to eight gives her an answer of 10. So, the calculation turns from 17 take away eight into 19 take away 10. Can you see how this is now an easier calculation to work out? She’s taking away a multiple of 10, a nice round number.
We know that taking away two numbers is the same as finding the difference between them. So, if we add the same amount to both numbers, the difference between them stays the same. If we imagine two friends running a race. One of them runs five meters towards the finishing line, and so does the other. The distance between them stays the same. So, this can be a good strategy to use to convert a subtraction into something a little easier to work out the same answer.
We’re asked to use Isabella’s strategy to find the missing numbers. And in each of the three calculations we’re given, we can see there’s a missing number. And it’s always the first number in the easier subtraction. Let’s look at the first calculation, 93 take away 78 equals what take away 80? Well, by looking at the arrows, we can see how the numbers have changed. We’ve added two to both numbers in the subtraction to make the calculation easier.
In particular, look how 78 changes when we add two to it, it becomes a multiple of 10. And as we’ve said already, it’s easier to subtract multiples of 10 often. And if we add two to the second number, we’ve also got to add two to the first number. So, our missing number is going to have a value of 93 plus two, which we know is 95. Our missing number is 95. Now, it’s a lot easier to work out the answer to 95 take away 80, or eight tens, than it is to work out in 93 take away 78.
Our second subtraction is 317 take away 196. And again, our missing number is the first number of the second subtraction, what take away 200. How have our numbers changed? What have we added to them to create our new subtraction? Well, only one of the arrows is labeled this time. And we can see that we’ve added four to 196 to make 200. This time, as well as being a multiple of 10, it’s actually multiple of 100. This is going to be an easier number to subtract than 196.
Now, remember Isabella’s rule, whatever we add to one number, we have to add to the other. We’ve added four to the second number. We know that because it’s moved from 196 to 200, so we also need to add four to 317. 17 plus four is 21, so we know 317 plus four is 321. And again, 321 take away 200 is a lot easier than the first subtraction.
Onto the last calculation, 873 take away 382 equals what take away 400? Now, the arrows aren’t labeled in this diagram, but we can see that 382 has changed into 400. What do we add to 382 to get to 400? Well, to get from 82 to 90, we need to add eight. And then, to get to the next hundred, we need to add another 10. So, that’s 18 we’ve added all together. So, to get from 382 to 400, we’ve added 18. And to apply Isabella’s rule, we need to also add 18 to the first number. Then, the difference between both numbers will stay the same.
So, what is 873 add 18? Well, we know that 18 is two less than 20. If we had 20 to 873, we get 893. But we’ve added two too many here, so we need to take away two from this number. 873 plus 18 then must be 891. And as we can see, 891 take away 400 is a lot easier to work out than 873 take away 382.
We’ve made each subtraction simpler by adding the same number to both numbers in the subtraction. 93 take away 78 equals 95 take away 80. 317 take away 196 equals 321 take away 200. And 873 take away 382 equals 891 take away 400. Our three missing numbers are 95, 321, and 891.