Cross-Multiplication:

Cross-multiplication is one of the most useful and practical tools in math, both in solving math problems in school and in real life. Unfortunately, it is not familiar to a lot of people, even though it could make their lives a lot easier.

We will go into the theory behind this concept, but before we do, let us look at a practical example and the way to solve it:

John and Julie are making sushi for a party. If they make 24 pieces of sushi in 10 minutes, how many are they going to make in 45 minutes?

Cross-multiplication is a short cut, allowing you to bypass a step. In the above example, if we know that it takes John and Julie 10 minutes to make 24 pieces of sushi and we need to calculate how many pieces they will make in 45 minutes, we should first calculate that if in 10 minutes, John and Julie make 24 sushi pieces, in 1 minute, they will make 10 times less:

In 1 minute, John and Julie will make 2.4 pieces of sushi and therefore in 45 minutes, they will make:

In 45 minutes, John and Julie will make 108 pieces of sushi.

Cross-multiplication allows us to take a faster and simpler method to solve problems of direct proportion. This is how we structure the problem:

The rule of cross-multiplication is that we always multiply B by C and divide the result by A to give us D. In other words: D = BC/A.

Our calculation is therefore going to look like this, with S the number of sushi in 45 min:

We start with our values of 24, 45 and 10 and can immediately simplify the expression, since both 45 and 10 are divisible by 5 and can further simplify the expression, since
both 24 and 2 are divisible by 2.

We get the same answer of 108 in one step!

In the above example, to get the correct answer, we need to multiply 45 x 24 and then to divide by 10, If you look at the beginning of this section, where we solved this problem by finding the “unit measure”, we have done exactly the same thing.

Cross-multiplication allows you to skip the step of calculating the “unit measure” or in the above example, the number of sushi made per 1 minute, before going on to find the number of sushi in 45 minutes.

Remember: It is essential to have the units or products or elements identified correctly and have the above formula “filled in” with the numbers in the right places.

The above example is pretty simple, but look at the following one:

If it takes 1/4 of an hour to empty 3/8 of the closet, how long will it take to empty the entire closet?

Our cross-multiplication query will read:

Remember - the rule is that:

D = BC/A

Therefore:

Since both 4 and 8 are divisible by 4, our expression is simplified to 2/3.
Cross-multiplication can be extremely helpful when we deal with fractions.

How does cross-multiplication work?
The principle is that:

If (A) .............................(B)
Then (C) ......................(D)

where D is the number we are looking for and we said that:
The rule of cross-multiplication is that we always multiply B by C and divide the result by A to give us D.
Let us go back to our sushi example. Think about what happens with each additional minute in terms of how many sushi are made - the number of sushi made will be increasing proportionately to the number of minutes.

Therefore if we divide 45 : 10 to find out how many 10 minute “intervals” there are in 45 minutes and multiply our 24 sushi by that number (which is 4.5), we get 108.

In terms of the formula we have given you:

d = bc/a

d = 45 x 24 : 10 = 108

Reminder: Always make sure that you are lining up the right units with the right units!