Multiplication is a topic that can be understood best using whole numbers. Multiplying the whole numbers 6 × 4 can be represented by an array containing 6 columns and 4 rows.
There are a total of (6 × 4 =) 24 ovals in the diagram. Multiplication of whole numbers can be represented by rows and columns. Now consider multiplying by a fraction instead of a whole number. The problem 2 × 6 can be represented by 2 rows that each contain 6 ovals while the problem × 6 is represented by half of a single row of 6 ovals.
When you take a number and multiply it by another whole number your answer becomes larger than what you started with. An unusual aspect of multiplying by a fraction is that is instead of increasing the number you started with, the fraction actually takes a part of the original number. Your resulting quotient is smaller than your original number when you multiply it by a fraction.
The majority of students learn best when they can gain a visual understanding of a mathematical concept. Multiplying fractions can be represented visually using a box containing overlapping rows and columns.
The problem can be stated
When rows and columns overlap, the result is boxes. There are 4 rows and 3 columns here, resulting in a total of 12 boxes. The answer will be the number of boxes where the rows and columns overlap divided by the total number of boxes. There are 6 boxes that result from the fractions overlapping, so the answer is the fraction 6/12, which reduces to ½. The answer is sensible here because two thirds of any number will be less than the original number. Our result of one-half makes sense here.
The diagram above is shown to help you understand the thinking involved in multiplying fractions, but in reality the process is quite simple: the answer is simply the product of the numerators divided by the product of the denominators.
Beginning Problems: Just multiply
Many of these problems are quite simple. Take these as an example of how to multiply two fractions.
Each problem here can be done by simply multiplying the two numerators on top and the two denominators on the bottom.
Intermediate Problems: Multiply, then reduce the answer
If either of your numerators have a factor in common with either of your denominators, then you will need to reduce your answer to lowest terms.
Each of the problems was done by simply finding the answer, then reducing the numerator and denominator by a common factor to change the fraction into lowest terms. Both worked out perfectly after reducing them a single time. If you reduce the fractions and it still isn’t in lowest terms then go ahead and reduce it again until the fraction is in lowest terms.
In the problem above it should be obvious that 42/140 is not in lowest terms. Reducing both by a factor of 7 simplifies the fraction to 6/50 but does not put it into lowest terms. Reducing a second time yields and answer of 3/25 and this fraction is your answer in lowest terms. It is OK to reduce two, three, or even four separate times. Just keep going until the result is in lowest terms.
When the fractions become larger or there are 3 or more fractions being multiplied, you will often have a lot to reduce. In this case, you can consider the problem to be “bigger” and can reduce the fractions first, then multiply.
“Bigger” Problems: Reduce first, then multiply
Students who are comfortable with multiplying fractions generally find it easier to reduce the fractions first, then multiply them. Here is an example of how it works when 5 fractions are being multiplied.
Each factor is reduced on the top and bottom. In this example, each of the individual reductions is represented with a different color. A red slash shows the 2’s canceling out, an orange slash shows the 3’s canceling out, and so on. The result is simply all 1’s on the top and four 1’s and a six on the bottom. Multiplying each of the remaining number yields our answer of 1/6. This same process can be used when multiplying any fractions that can be reduced.
It is important to take a moment to discuss how fractions can be simplified in multiplication problems. The basic thought is that any numerator can reduce with any denominator. The numbers being reduced can be on top of each other or can be diagonal with one on top and one on the bottom.
It is a mathematical delight to find easy ways to do hard problems. The final two example problems here are mathematical delights because they can be greatly reduced before multiplying. The actual multiplications can be done in your head once the fractions are reduced completely.