Fill in the blank. Nine times 684 equals nine times 600 plus what plus four.
In this problem, we can see two expressions that are equal to each other. On the left-hand side, we have a multiplication, nine lots of 684. And we can see that this is equal to nine multiplied by the answer to an addition, because we can see a pair of parentheses or brackets. And what these mean is that we need to work out whatever is inside of them first.
Once we’ve worked out whatever is inside of them, we then multiply nine by that number. And what’s inside of the parentheses is an addition. We have 600 plus something plus four. And the problem is asking us to find out what this missing number is. If we look at the multiplication on the left-hand side, we could model this by drawing a rectangle. Our rectangle could have a width or a height of nine and a length of 684. Now, we know the area of a rectangle is its length multiplied by its width. Well, in this case, we could switch those two letters around. And so we can think of the rectangle is showing us nine multiplied by 684.
In our second multiplication, we can see that we’re multiplying nine by another number. But the number has to equal 684 because we know both expressions are equal. So if we know that the expression inside the parentheses is equal to 684, then we can think of this addition as being a way to partition 684. How can we split up this number into an addition?
We’re told the first part of the addition is 600. And the last part is four. 600 is the number of hundreds in 684. And four is the number of ones. What are we missing? We’re missing the number of tens. 684 has eight tens, which have a value then of 80. Nine multiplied by 600 plus nine multiplied by 80 plus nine multiplied by four is exactly the same as nine multiplied by 684. Nine multiplied by 684 is equal to nine multiplied by 600 plus 80 plus four.
Our missing number is 80.