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INTRODUCTION TO FRACTION

SIMPLE FRACTION IN OUR TOYS

Fraction plays an important role in our daily life. 

When we were kids we used to play with these toys without knowing the hidden facts behind it.

Let's recall back our childhood memory by seeing and investigating this picture.

Multiplying and Dividing Fractions

Multiplying Fractions 

Three simple steps are required to multiply two fractions:

• Step 1: Multiply the numerators from each fraction by each other (the numbers on top). The result is the numerator of the answer.
• Step 2: Multiply the denominators of each fraction by each other (the numbers on the bottom). The result is the denominator of the answer.
• Step 3: Simplify or reduce the answer.

Examples of multiplying fractions: 

 
In the first example you can see that we multiply the numerators 2 x 6 to get the numerator for the answer, 12. We also multiply the denominators 5 x 7 to get the denominator for the answer, 35. 

In the second example we use the same method. In this problem the answer we get is 2/12 which can be further reduced to 1/6. 

Multiplying Different Types of Fractions 

The examples above multiplied proper fractions. The same process is used to multiply improper fractions and mixed numbers. There are a couple of things to watch out for with these other types of fractions. 

Improper fractions - With improper fractions (where the numerator is greater than the denominator) you may need to change the answer into a mixed number. For example, if the answer you get is 17/4, your teacher may want you to change this to the mixed number 4 ¼. 

Mixed numbers - Mixed numbers are numbers that have a whole number and a fraction, like 2 ½. When multiplying mixed numbers you need to change the mixed number into an improper fraction before you multiply. For example, if the number is 2 1/3, you will need to change this to 7/3 before you multiply. 

You may also need to change the answer back to a mixed number when you are done multiplying. 

Example: 

 
In this example we had to change 1 ¾ to the fraction 7/4 and 2 ½ to the fraction 5/2. We also had to convert the multiplied answer to a mixed number at the end. 

Dividing Fractions 

Dividing fractions is very similar to multiplying fractions, you even use multiplication. The one change is that you have to take the reciprocal of the divisor. Then you proceed with the problem just as if you were multiplying.

• Step 1: Take the reciprocal of the divisor.
• Step 2: Multiply the numerators.
• Step 3: Multiply the denominators.
• Step 4: Simplify the answer.

Taking the reciprocal: To get the reciprocal, invert the fraction. This is the same as taking 1 divided by the fraction. For example, if the fraction is 2/3 then the reciprocal is 3/2. 

Examples: 

Adding and Subtracting Fractions

Adding and subtracting fractions may seem tricky at first, but if you follow a few simple steps and work a lot of practice problems, you will have the hang of it in no time. 

Here are some steps to follow:

• Check to see if the fractions have the same denominator.
• If they don't have the same denominator, then convert them to equivalent fractions with the same denominator.
• Once they have the same denominator, add or subtract the numbers in the numerator.
• Write your answer with the new numerator over the denominator.

Note: The denominator may have changed when you converted the fractions to the same common denominator. 

Simple Example 

A simple example is when the denominators are already the same: 

 

Since the denominators are the same in each question, you just add or subtract the numerators to get the answers. 

Harder Example 

Here we will try a problem where the denominators are not the same. 

 

As you can see, these fractions do not have the same denominator. Before we can add the fractions together, we must first create equivalent fractions that have common denominators. 

Find the Common Denominator 

To find a common denominator, we must multiply each fraction by the other fraction's denominator (the one the bottom). If we multiply both the top and the bottom of the fraction by the same number, its just like multiplying it by 1, so the value of the fraction stays the same. See the example below: 

 

Add the Numerators 

 

Now that the denominators are the same, you can add the numerators and put the answer over the same denominator. 

Subtracting Fractions Example 

Here is an example of subtracting fractions where only one denominator needs to be changed: 

 

Reduce Your Final Answer 

Sometimes the answer will need to be reduced. Here is an example: 

 

The initial answer after adding the numerators was 10/15, however this fraction can be further reduced to 2/3 as shown in the last step. 

Tips for Adding and Subtracting Fractions

• Always make sure that the denominators are the same before you add or subtract.
• If you multiply the top and the bottom of a fraction by the same number, the value stays the same.
• Be sure to practice converting fractions to common denominators. This is the hardest part of adding and subtracting fractions.
• You may need to simplify your answer after you are done adding and subtracting. Sometimes the answer can be reduced even though the original fractions could not be reduced.
• The same process is used for both adding and subtracting, if you can add fractions, you can subtract them.
• If there are mixed numbers that you are adding or subtracting be sure to convert them to improper fractions before you start the process.

Equivalent Fractions

When fractions have different numbers in them, but have the same value, they are called equivalent fractions. 

Let's take a look at a simple example of equivalent fractions: the fractions ½ and 2/4. These fractions have the same value, but use different numbers. You can see from the picture below that they both have the same value. 

 

How can you find equivalent fractions? 

Equivalent fractions can be found by multiplying or dividing both the numerator and the denominator by the same number. 

How does this work? 

We know from multiplication and division that when you multiply or divide a number by 1 you get the same number. We also know that when you have the same numerator and denominator in a fraction, it always equals 1. For example: 

 

So as long as we multiply or divide both the top and the bottom of a fraction by the same number, it's just the same as multiplying or dividing by 1 and we won't change the value of the fraction. 

Multiplication example: 

 

Since we multiplied the fraction by 1 or 2/2, the value doesn't change. The two fractions have the same value and are equivalent. 

Division example: 

 

You can also divide the top and bottom by the same number to create an equivalent fraction as shown above. 

Cross Multiply 

There is a formula you can use to determine if two fractions are equivalent. It's called the cross multiply rule. The rule is shown below: 

 

This formula says that if the numerator of one fraction times the denominator of the other fraction equals the denominator of the first fraction times the numerator of the second fraction, then the fractions are equivalent. It's a bit confusing when written out, but you can see from the formula that it's fairly simple to work out the math. 

If you get confused on what to do, just remember the name of the formula: "cross multiply". You are multiplying across the two fractions like the pink "X" shown in the example below. 

 

 

Comparing Fractions 

How can you tell if one fraction is bigger than another? 

In some cases it's pretty easy to tell. For example, after working with fractions for a while, you probably know that ½ is bigger than ¼. It's also easy to tell if the denominators are the same. Then the fraction with the larger numerator is bigger. 

However, sometimes it's difficult to tell which is bigger just by looking at two fractions. In these cases you can use cross multiplication to compare the two fractions. Here is the basic formula: 

 

Here is an example: 

 

Key Things to Remember

• Equivalent fractions may look different, but they have the same value.
• You can multiply or divide to find an equivalent fraction.
• Adding or subtracting does not work for finding an equivalent fraction.
• If you multiply or divide by the top of the fraction, you must do the same to the bottom.
• Use cross multiplication to determine if two fractions are equivalent.