Decimals and Fractions 101

In order to use fractions to add, subtract, multiply, and divide you will need to know how to find factors.

Factors are numbers we can multiply together to get another number. Some numbers only have 1 pair of factors. We call these prime numbers. 2,3,5,7, and 11 are examples of prime numbers. Other numbers, called composite, have several factor pairs. For example, the number 12 has three factor pairs: (1x12), (2x6), and (3x4).

The best way to find all the factors of a number is to start with number 1 and keep going up until you repeat a factor. Consider the number 12 again. We know 1 is paired with 12. Now we move to the number 2. Does 2 times anything equal 12? Yes, 2x6=12. Now we move to number 3. Does 3 times anything equal 12? Yes, 3x4=12. Now we move to number 4, but since we have already used it (3x4=12), we know that we have found all the factors. We have found that the factors of 12 are 1,2,3,4,6, and 12.

We also need to be able to find something called GCF, which stands for Greatest Common Factor. To find the common factor of 2 or more numbers, you must find all the factors of each number and pick the largest number that they all have in common. Let's find the GCF of 12 and 20...

Here are all the factors of 12 and 20:

12: 1,2,3,4,6,12

20: 1,2,4,5,10,20

The numbers that are the same in each row are: 1, 2, and 4. Since 4 is the largest of these factors, it is the GCF.

Before we begin working with fractions, we need to learn one more skill. We must learn how to find the LCM. LCM stands for Least Common Multiple.

To find the LCM, we must be able to find multiples of a number. Let's use 3 as an example: 3, 6, 9, 12, 15, 18, 21. As you can see, all you need to do to find multiples is to skip count. Now let's try the number 4. We find: 4, 8, 12, 16, 20, 24, 28. We now have two lists of multiples. We must find the smallest number that is in both lists:

3 6 9 12 15 18 21

4 8 12 16 20 24 28

The smallest number that is in both lists is 12.

There is no rule to determine how many multiples you should write down. However, you should start with at least 5 multiples and then keep going if you haven't found the LCM. Let's look at another example. This time we will find the multiples of 5 and 7:

5 10 15 20 25

7 14 21 28 35

As you can see, none of the numbers match. We need to keep adding multiples until we have one in common from both rows:

5 10 15 20 25 30 35

7 14 21 28 35 42 49

Now we have found our LCM of 35. Sometimes you need to find more than 5 multiples for each number. Keep going until you find the LCM.

To pass level 4, you will need to learn how to multiply fractions in two different ways:

1. Overlapping Arrays
2. Multiplying Across

Overlapping Arrays - In this method we use arrays to represent two fractions. The portion of the arrays that overlaps will be our answer. This method takes much longer, but provides an excellent visual model for multiplying fractions.

We will use 7/8 times 1/3 as our example...

We start with a simple square that will represent 7/8:

Next we divide our square into 8 sections:

Then we mark 7 of the 8 sections to represent 7/8:

Now we must add an array for 1/3 by splitting our image into 3 sections:

Now we mark 1 of the 3 sections to represent 1/3:

Finally, we highlight the intersecting arrays:

To find the numerator, you simply count the highlighted sections. You can see there are 7 sections. To find the denominator, you count the total number of sections, including the highlighted sections. If you count all the sections, you find a denominator of 24. So the answer to 7/8 times 1/3 is 7/24.

Multiply Across - In this method we simply need to multiply each numerator and then multiply each denominator. Let's look at the same problem we did earlier...

• To find our new numerator we multiply 7 x 1 = 7
• To find our new denominator we multiply 8 x 3 = 24

Level 5 - Division

Dividing fractions is not as simple as multiplication. With multiplication, you can find an exact answer by using arrays or mathematics. With division, only mathematics will give us an exact answer. We can visualize a problem by using images, but it will not provide a precise solution. Let me show you two examples to illustrate this point:

1. A turtle needs to travel 1/2 of a mile to reach a pond. The turtle can travel 1/6 of a mile in one hour. How long will it take the turtle to reach the pond?

This question can be written as 1/2 ÷ 1/6 = ?

We can draw 1/2 and 1/6 to get an idea of the answer:

It appears that 1/6 will fit into 1/2 three times. In this case we were able to use a picture to find our answer of 3. However, in our next example we will not be so lucky:

2. After reaching the pond, the turtle discovered that he forget his wallet and he needs to walk back. However, for this trip our turtle will wear roller skates that allow him to travel at 1/3 of a mile per hour. How long will the 1/2 mile trip take him this time?

This question can be written as 1/2 ÷ 1/3 = ?

Again let's try to use a diagram to find the answer:

This diagram shows that our answer lies somewhere between 1 and 2, which is correct. However, it does not tell us the exact answer.

So how do we find the "real" answer? Well, we need to Keep Change Flip...

1/2 ÷ 1/3 = ?

1/2 (keep) ÷ (change) 1/3 (flip) so...

1/2 x 3/1 = 3/2

We keep the fraction that we are splitting up (dividing) the same, we change ÷ to x, and we flip the other fraction upside down ( 1/3 becomes 3/1)

Let's check one more example before you watch the video:

3/4 ÷ 2/5 = ?

3/4 x 5/2 = 15/8

That's it. Don't worry that the numerator is bigger than the denominator. I will teach you how to fix that in level 9.

Level 6 - Addition

Adding fractions can be tricky, but since we have already learned how to find the LCM, it should be pretty easy for us. We will learn two different methods for adding fractions:

1. Common Denominators - In this method we must find the LCM to help us add fractions. The nice thing about this method is that you should not have to simplify your answer. (We will learn how to simplify our answers in lesson 9.)
2. Criss Cross Under Sauce - This method is much faster than finding common denominators, however you often need to simplify your answer before you are finished.

In the first method we will use common denominators by finding the LCM of the denominators of each fraction. Let's break down the process step by step with the example 2/6 + 3/8:

Step 1: Find the LCM of 6 and 8 (the denominators) to create a new denominator.

6 12 18 24 30

8 16 24 32 40

Step 2: Now that we have a new denominator of 24 we need to change the numerators:

Step 3: The hard part is over now we add the numerators, but not the denominators:

You see that we added 8 + 9 to get 17 as the numerator, however our denominator does not change. This makes sense because the denominator represents the "whole". The whole does not change if you add two fractions together. This can be a bit confusing, hopefully the video clears things up for you.

*** If the denominators are already the same, you don't need to go through all these steps. Simply add the numerators and do not change the denominator. For example: 2/7 + 3/7 = 5/7.

In the second method we will be using a criss cross under sauce method. I'm sure this method has another, more professional name, but I like criss cross under sauce. Let's break this method down step by step by using 3/4 + 5/7:

Step 1: Criss

3 x 7 = 21 This is our first numerator, we will have two numerators.

Step 2: Cross

4 x 5 = 20 This is our second numerator.

Step 3: Under Sauce

4 x 7 = 28 This will be our denominator.

Step 4: Add the numerators:

Our answer of 41/28 is called an improper fraction because the numerator is larger than the denominator. In level 8 we will learn how to change these into a mixed number. For now, don't worry if the numerator is larger than the denominator.

The video below will show both methods and how to represent your answers as a picture or diagram.

Level 8 - Simplifying Fractions

In this level we will learn how to simplify our answers after we have added, subtracted, multiplied, or divided fractions. Back in level 2 we learned about factors. We will need to apply that knowledge again on this level in order to move forward.

When we simplify (also called reducing) the fraction, we are creating an equal fraction that is easier to work with. In order to simplify, we divide both the numerator and the denominator by a common factor. Things go faster if we divide by the greatest common factor. Let's look at 12/20 as an example:

Step 1: Find the factors

12: 1, 2, 3, 4, 6, 12

20: 1, 2, 4, 5, 10, 20

Step 2: Find the GCF

12: 1, 2, 3, 4, 6, 12

20: 1, 2, 4, 5, 10, 20

Step 3: Divide the numerator and denominator by the GCF

Numerator: 12 ÷ 4 = 3

Denominator: 20 ÷ 4 = 5

Step 4: Check for other common factors (the factor 1 does not count)

3: 1, 3

5: 1, 5

Since there are no other common factors, 3/5 is our simplified answer.

Level 9 - Improper Fractions

In this section we will learn how to change improper fraction into mixed numbers. Improper Fractions are fractions where the numerator is larger than the denominator. Improper fractions are a problem because they don't really make sense. Think about this situation. Jim orders a pizza. The pizza is cut into 8 pieces. The largest amount of pizza Jim can eat is 8 pieces out of 8 pieces or 8/8. Yet, often times when we add and divide we end up with something like 11/8. In our situation, it is impossible for Jim to eat 11 pieces of pizza because there are only 8 pieces... unless there were another whole pizza. Then Jim could have 8 pieces of the first pizza and 3 pieces of the second pizza. That situation represents a Mixed Number. A mixed number contains a whole number and a fraction. Our mixed number would look like:

Where 1 represents our first whole pizza and 3/8 represents the fraction of our second pizza. You could also show this situation as a diagram or picture:

Now we will go through the steps involved in changing an improper fraction into a mixed number using the example:

Step 1: Divide

Step 2: The whole number is the quotient (the blue 2)

Step 3: The numerator of our fraction is our remainder (the red 2)

Step 4: The denominator of our fraction is the devisor (the black 5)

Students typically find that the video in this level is easier to understand than the written instructions. Please check it out. Plus, it gives you a lot more examples.

Multiplying decimals is far different than adding or subtracting. However we still follow three steps:

1. Multiply like normal (ignore the decimals)
2. Count the digits to the right of the decimal
3. Insert the decimal

Check out the example below:

Step 1:

Step 2:

Step 3:

When we need to divide numbers with decimals we set up a normal long division problem, then we follow another 3 step strategy:

1. Divide like normal (ignore the decimals)
2. If there is a decimal in the divisor (the number on the left) then you need to move the decimal in the dividend. If there is no decimal in the divisor you can ignore this step.
3. Raise the decimal into your answer

Check out the example below:

Step 1:

Step 2:

Step 3:

We know these steps may seem a little complex at first. Don't get discouraged. If you haven't watched the video yet, please give it a try. The video has a few more examples where we break this method down step by step. Good Luck!

Adding decimals is just like adding whole numbers. However, you need to be very careful that you line up the place values correctly. Let's look at an example: 1.31 + 2.3

A common mistake would be to set up and solve the problem as shown below:

This example is incorrect because the place values are not lined up correctly. The easiest way to check your place values is to make sure the decimals are lined up. Look at the correct method below:

As you can see in the example, you can add a zero into any blank space that is caused by lining up the correct place values. Then all you need to do is add and bring the decimal point straight down.

Since you have passed level 2, you will have no problem with level 3! All the steps are the same, except you subtract at the end instead of adding. Take a look at 2.3 - 1.31

There are only 3 simple steps:

1. Line up the decimals
2. Add zeros to any empty spaces (You can see the red zero above)
3. Subtract