Fill in the blank. 93 times 96 times 43 equals what times 43 times 93.
In this problem, we have two multiplication expressions. And the first thing we can say about both expressions is that they both contain three numbers. In the first expression, we have 93 multiplied by 96 multiplied by 43. And in the second expression, we have an unknown number, this is the blank that we need to fill in, multiplied by 43 multiplied by 93.
What else can we say about these expressions? Well, the second thing we can say is that both multiplications are worth the same. We can see an equals sign in between them. The final thing that we can notice about both expressions is that they contain similar numbers. They both contain the number 93. They also contain the number 43. Although the numbers are in different orders in each expression.
So, what could our missing number be? To answer the question, we need to think about one of the rules of multiplication. This rule of multiplication tells us that it doesn’t matter how numbers in a multiplication are grouped. The answer stays the same. And if we understand this rule, we don’t need to do any calculating at all here. We’ll know what the missing number is.
Let’s remind ourselves how this rule works. Instead of using large numbers like 93, 96, and 43, we’ll use some smaller numbers. What about two, three, and four? We could think of this calculation as representing two lots of three times four. Three times four is 12. And so, two lots of three times four or two times three times four equals 24.
Now, what if we write the numbers in a different order? Three times four times two. We could think of this as three lots of four times two. Four times two is eight, and so three lots of four times two is the same as three lots of eight, which is also 24. So, it doesn’t matter which way around we write our three numbers, the answer is the same.
Now, let’s go back to our original calculation. On the left, we have 93 times 96 times 43. And so, because we know the expression on the right is worth the same as the one on the left, and because we can also see that it contains two of the numbers in the first expression, our missing number must be the third number in the first expression. It must be 96. We’ve used that law of multiplication that we talked about, the associative law, to juggle the numbers around without changing their product. 93 times 96 times 43 is the same as 96 times 43 times 93. The missing number that we needed to fill in is 96.