Some students will be familiar with complex fractions by grade six from their individual math tutoring lessons. If not, complex fractions can be tricky, but this is nothing new for math students in the sixth grade.
While the topic of complex fractions is difficult at first, it is an essential mathematical concept if middle school math students are to fully understand more complicated topics as they progress through middle school, high school, and beyond, to SAT exams.
Let’s not get ahead of ourselves yet. Let’s come back to sixth grade and look at a topic that can be considered a friendly introduction to complex fractions: comparing fractions.
We know that a fraction is a number that represents a part of a whole. Fractions are usually written as one number – the numerator – over another number, known as the denominator. There are many different types of fractions, such as improper fractions, reciprocals, and equivalent fractions. There are also many different ways that we can write the same fraction, and this is the area of fractions we are interested in today.
Sometimes we need to identify which of two (or more) fractions is bigger or smaller. We can easily do this when they are expressed in the same way as one another – for example, we know that ¾ is greater than ¼.
This task becomes more difficult, however, when our fractions are not expressed in the same way as each other. Imagine we have the question, which of the following numbers is bigger: ⅔ or ¾?
Without a calculator, if we tried to solve this problem, we may not know off the top of our heads which fraction is smaller. To find out, we can take one of two approaches, the decimal method, or the same denominator method.
The Decimal Method
Of course, we can use a calculator to find the answer to our previous problem. 2 divided by 3 is equal to 0.6666… whereas ¾ is equal to 0.75. Therefore, we can see that, since 0.66 is less than 0.75, then similarly ⅔ is less than ¾.
This is essentially the decimal method. It involves rewriting your fractions as decimals and comparing them to see which is smaller and which is larger. Without a calculator, it can be difficult, but it isn’t impossible.
To convert a fraction into a decimal, we can follow this process:
- Find a number you can multiply by the bottom of the fraction to make it 10, 100, or 1000 or any 1 followed by 0s.
- Step 2: Multiply both the numerator and the denominator by that number.
- Step 3. Write down just the numerator, putting the decimal point one space from the right-hand side for each zero in the denominator.
For example, if we have ⅘:
- We can multiply both the numerator and the denominator by 20.
- Our number is now 80/100, so we write 80.
- Since we have two zeros in our denominator, we have to put the decimal point 2 spaces from the right.
- Our 80 becomes 0.80, which equals ⅘.
Luckily, students will be able to use calculators in most cases and will not need to know this method, but it is extremely valuable and will help students become significantly faster performing complex mental arithmetic in future math lessons.
There is another popular method to compare fractions, known as the same denominator method, which students will be expected to know, understand, and use.
The Same Denominator Method
The basic idea behind the same denominator method of comparing fractions is to get each fraction written in the same format writing them in terms with the same denominator.
For example, if we try to find out which is larger, 7/12 or 1/8.
If we multiply 7/12 by 2, we get 14/24.
Similarly, if we multiply 1/8 by 3, we get 3/24.
Now that we have two numbers with the same denominator, we can clearly see that the first number (7/12) is larger than our second number (1/8).
Converting fractions and making them have a common denominator can be done in two ways: the lowest common denominator method (which we used above), and the common denominator method, where you simply multiply each fraction by the denominator of the other, and simplify your answer.
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