Math Resources

When adding and subtracting fractions, we first need to make sure the fractions have a common denominator, or bottom number. Once the denominators are the same, we can simply add or subtract the numerators, or the numbers on the top of the fraction.

              

If the denominators are not the same, you need to convert the fractions to equivalent fractions with the same denominator. 

 

Here, we first need to find the least common multiple (LCM) of 3 and 5. A simple way of finding the LCM is to multiply the numbers, but oftentimes, this will not be the least common multiple. One strategy you can try is to list the multiples of each value until you reach one they have in common.

3: 3, 6, 9, 12, 15, 18, 21
5: 5, 10, 15, 20, 25, 30, 35

We can see that 3 and 5 both have 15 as a multiple, so that's the value we want in the denominator. To convert the fractions to equivalent fractions with 15 in the denominator, we need to multiply the top and bottom of each fraction by the same number. This ensures the value of the fraction is not changed.

To get from 3 to 15, we need to multiply by 5. So, we will multiply the numerator and denominator of the first fraction by 5 to get the equivalent fraction.

We repeat the same process for the second fraction. To get from 5 to 15, we multiply by 3.

Now that we have common denominators, we can add them as usual.

Applying this same process to a subtraction scenario:

Original:              Equivalent:             Answer:  

Adding & Subtracting Mixed Numbers

When we see a mixed number, we can think of the whole number and the fraction being added together. For example:

So, when adding or subtracting mixed numbers, we can work with whole numbers and fractions separately. First, we check for common denominators in the fractions. If they are not the same, then we convert them as we did above, ignoring the whole number.

Problem:             Equivalent:    

Now, we can add the whole numbers and add the fractions separately. Then, we recombine the parts to get the answer.

    and                 Sum:  

Should the fraction ever become an improper fraction, we can regroup to simplify the answer.

Problem:            Equivalent:            Answer:  

We can reconsider the answer in this way. We can rewrite the fractional portion as two fractions, one with the same numerator and denominator and the other being the remaining portion of the original fraction, such that if we added the two fractions, we would get the original fraction. We can then rethink of the first fraction as 1 since a number over itself is equal to 1. Now we can condense the like numbers to get a simplified answer that has been regrouped to eliminate the improper fraction.

 

The same processes can be applied when subtracting fractions. Consider the following example.

Problem:             Equivalent:      

    and               Answer:   

In the event that the second fraction is larger than the first fraction, we need to borrow from the whole number. This will look similar to when we regrouped to resolve the improper fraction when adding.

Problem:              Equivalent:  

We can't take 21 from 16, so we need to borrow from the whole number. That means we will be taking 1 away from the 5 and rewriting it as a fraction that we can then combine with the fractional portion to get an improper fraction.

Now we have a fraction that we can subtract 21 from.

 

Unlike addition and subtraction, multiplying and dividing fractions do not require a common denominator. When multiplying fractions, we can simply think about multiplying the numbers across the top (the numerators) and the bottom (the denominators). Consider the following example:

Multiplying across the top, we have 4 x 7 = 28. Multiplying across the bottom, we have 3 x 5 = 15. Since 28 over 15 is an improper fraction, we can regroup to convert it to a mixed number. 

Dividing fractions will be exactly like multiplying fractions with one small change. An easy way to divide fractions is to multiply by the reciprocal. That means we leave the first fraction the same, but we flip the second fraction over. Once the second fraction is flipped, we proceed with multiplying across the top and bottom. Here's an example:

   which becomes      then we multiply like above   

Cross-Cancelation

Sometimes we can make the math easier and save time reducing by using cross-cancelation. That means we look at the denominator in one fraction and the numerator in the other fraction and see if they cancel each other out. Let's explore this idea using an example.

Looking at the numerator in the first fraction (7) and the denominator in the second fraction (9), we cannot cancel because 7 does not go into 9 evenly. Looking at the denominator in the first fraction (3) and the numerator in the second fraction (30), we can cancel because 3 goes into 30 evenly. When canceling, we are essentially dividing both values by the same number. In this case, we're dividing both by 3. In the first fraction, 3 divided by 3 means our new denominator is 1. In the second fraction, 30 divided by 3 means the new numerator is 10. So, we can rewrite the equation and solve:

Note that if we multiplied the original fractions and then reduced the final answer, we would still get the same result. Canceling is not a required step, but it can help make the computations easier.

Working with Mixed Numbers

If either of the starting numbers is a mixed number, you first need to convert it to an improper fraction before you can multiply. Here are examples of what that would look like:

Multiplication (without cancelation):    

Multiplication (with cancelation):      

Division (without cancelation):     

Division (with cancelation):