Is 𝐚 multiplied by 𝐛 divided by 𝐜 in parentheses equal to 𝐚𝐛 divided by 𝐚𝐜?
This question is easier to see than is to hear. It’s quite tricky to read aloud. So let’s go through what it’s asking us. Firstly, we can see that there are two statements either side of an equal sign. And the question is asking us are the two statements the same. You may recognize the statements or think you’ve seen them somewhere before. Well, they’re very similar to what we call the distributive law of multiplication. This is the law that tells us that if we multiply the total of two numbers by another number — so in other words, 𝐚 multiplied by the total of 𝐛 plus 𝐜 — this is the same as multiplying both the numbers separately and then adding them together.
So, for example, you might have something like 10 multiplied by five plus two. Well, we’d add five plus two to start with to get seven and then multiply it by 10. The answer would be 70. This is exactly the same as multiplying the two numbers separately and then adding them. 10 times five is 50 and 10 times two is 20. And if we add 50 and 20 together, we get 70. Now we know that the distributive law of multiplication works. But what’s the difference between this law, which is written in green, and our question? Well, the only difference is that in our question instead of an addition sign, we have a division symbol.
So what this question is really asking us is, Does the distributive law of multiplication work when you divide instead of add? To find the answer, let’s replace the letters 𝐚, 𝐛, and 𝐜 with some numbers. We’ll pick some numbers that are going to give us straightforward numbers to work with. What if we say 𝐚 is equal to five, 𝐛 is equal to 10? And we need 𝐜 to be a number that we can divide 10 by. So we’ll have two. Let’s check the left-hand side of our statement first of all. We have 𝐚, 𝐛 divided by 𝐜, where the term 𝐛 divided by 𝐜 is written inside parentheses.
Notice how the letter 𝐚 is written right up close next to the parentheses. There’s no symbol to add or divide. Remember when we do this with a letter in algebra, it means that we need to multiply. So really, what this is saying is 𝐚 multiplied by 𝐛 divided by 𝐜. And the fact 𝐛 divided by 𝐜 is inside parentheses means that we need to work this out first. Now that we’ve given our letters some numbers, let’s replace those letters for those numbers.
So the left-hand side of our statement now reads five multiplied by 10 divided by two. We need to work out the parentheses first. 10 divided by two equals five. So, really, what our calculation is asking us to do is multiply five by five. The value of the left-hand side of our statement is 25. But what about the right-hand side of our statement? Does this equal 25 too? Can we multiply both numbers in the division by five and still get the answer 25? The right-hand side of our statement reads 𝐚𝐛 divided by 𝐚𝐜.
And as we’ve said already, when we write two letters next to each other, we mean that we need to multiply them. So using the numbers that we have already, 𝐚𝐛 becomes five times 10 and 𝐚𝐜 becomes five times two. We’ve written both these parts in parentheses to show that we need to work them out first. So let’s do that. Five multiplied by 10 equals 50 and five multiplied by two equals 10. So our calculation becomes 50 divided by 10. And we can see already it’s not going to be 25. 50 divided by 10 equals five and 25 is not equal to five.
By putting numbers of our own into the calculation, we’ve shown the distributive law of multiplication doesn’t work when you divide instead of add. So in an answer to our question is 𝐚 multiplied by 𝐛 divided by 𝐜 equal to 𝐚𝐛 divided by 𝐚𝐜, no, it’s not. The distributive law of multiplication doesn’t work when both parts are divided.