# Math Study Sheet

### Adding and Subtracting Fractions

## Subtracting Regular Fractions

1/2 - 2/5 = ?

To answer this question you will need to first find a common denominator or the lowest common multiple. To find a common denominator you multiply the two bottom numbers:

2x5=10 so therefore your common denominator is 10. There is a well known rule that you must follow and it is that whatever you do to the bottom number you must do to the top. Lets start with 1/2, we multiplied 2x5 to get 10 so now we have to do exactly what we did with the bottom number to the top number which is multiply by 5 so we have to multiply the top number (1) by 5 so 1x5=5. Now we have to do the exact same thing to the other fraction: 2x2=4. Now we have two fractions with the same denominator and the equation is much easier to solve : 5/10 - 4/10 = 3/10. If both the denominator and numerator are even numbers that means that your final answer is not in its lowest form and you will have to reduce it.

## Subtracting Improper fractions

Subtracting Improper fractions are very similar to subtracting simple fractions find a common denominator or the lowest common multiple. To find a common denominator you multiply the two bottom numbers:

10x20=200 so therefore your common denominator is 200. There is a well known rule that you must follow and it is that whatever you do to the bottom number you must do to the top. Lets start with 14/10, we multiplied 10x20 to get 280 so now we have to do exactly what we did with the bottom number to the top number which is multiply by 20 so we have to multiply the top number (14) by 20 so 14x20=280. Now we have to do the exact same thing to the other fraction: 22x10=220. Now we have two fractions with the same denominator and the equation is much easier to solve : 280/200 - 220/200 = 240/200. If after subtracting your final answer is still an improper fraction you will have to convert it into a mixed number: 1 40/200 and then since both numbers are even you have to keep reducing it until one or both numbers are odd: 1 5/25 is the final answer.

## Subtracting Mixed Fractions

To subtract a mixed fraction I find it easier to convert it into an improper fraction and then subtracting it. So improper fractions of these examples would be 14/10-12/20. You have to find a common denominator or the lowest common multiple. To find a common denominator you multiply the two bottom numbers:

10x20=200 so therefore your common denominator is 200. There is a well known rule that you must follow and it is that whatever you do to the bottom number you must do to the top. Lets start with 14/10, we multiplied 10x20 to get 280 so now we have to do exactly what we did with the bottom number to the top number which is multiply by 20 so we have to multiply the top number (14) by 20 so 14x20=280. Now we have to do the exact same thing to the other fraction: 22x10=220. Now we have two fractions with the same denominator and the equation is much easier to solve : 280/200 - 220/200 = 240/200. If after subtracting your final answer is still an improper fraction you will have to convert it back into a mixed number: 1 40/200 and then since both numbers are even you have to keep reducing it until one or both numbers are odd: 1 5/25 is the final answer.

## Adding Fractions

## Adding Mixed Fractions

The equation:

1 5/10 + 3 5/20

To add the fractions together we first would have to find the common denominator.

To find the common denominator we are going to look at the multiples of 10 and 20

10: 10 **20** 30 40

20: **20** 40 60 80

The common multiple from the two denominators is 20. Next we are going to multiple the denominator 10 by 2, since thats how we get to the denominator 20. There is a rule that is always applied to adding,subtracting,multiplying and dividing fractions which is, what you do to the bottom you have to do to the top. So we would have to multiply 5 by 2, since we multiplied 10 by 2.

10 x 2 = 20 < the new denominator

5 x 2 = 10 < whatever you do to the bottom you have to do to the top

The new fraction is 1 10/20. You also keep the whole numbers regular and dont do anything to them.

The fraction 3 5/20 stays the same since the denominators are the same. But if it was different then we would have to change it.

So now that we have found our common denominators we can add the fractions together.

1 10/20 + 3 5/20 = 4 15/20

When adding fractions, you only add the numerators together and the whole numbers together. You always keep the denominator. And at some points the numerators can be larger than the denominator.

## Video about adding different types of fractions

## Adding Regular Fractions

If we have the equation 2/4 + 2/3

The first thing to do is find the common denominator. To find the common denominator is to look at the multiples

3: 3 6 9 **12**

4: 4 8 **12** 16

Or you could just multiply the 2 denominators together, which is the simplest way to do it.

Now we are going to change the denominators:

2/4:

4 x 3 = 12 < **multiplying by 3 to get the denominator of 12**

2 x 3 = 6 <** whatever you do to the bottom you have to do to the top**

Our new fraction is now:** 6/12**

2/3:

3 x 4 = 12 < **multiplying by 4 to get the denominator of 12**

2 x 4 = 8 < **whatever you do to the bottom you have to do to the top**

Our new fraction is now: **8/12**

Now that both of our fractions have a common denominator we are able to add.

**6/12 + 8/12 = 14/12**

Our answer is 14/12.

In this answer 14/12, the numerator is larger than the denominator. This is called an improper fraction.

To make it a proper fraction we would have to change it into a mixed number. To change it into a mix number the first thing we would have to do, is see how many times the numerator is able to go into the denominator.

how many times can 14 go into 12.

14 can go into 12, 1 time. With the 1 thats going to be the whole in the fraction. We have leftover numbers which is the number 2 because 14 - 12 = 2. And since 14 can only go into 12 once we have a left over of 2. The 2 is now going to be our numerator. And the denominator stays the same.

So our answers would be:

**14/12 < improper fraction**

**1 2/12 < changing it into a mixed fraction**

## Adding Improper Fractions

If we have the equation:

8/5 + 5/3

We already know we have to find the common denominator.

5: 5 10 **15** 20

3: 3 6 9 12 **15**

The common denominator is 15.

**8/5 **

5 x 3 = 15 **< multiplying by 3 to get the common denominator **

8 x 3 = 24 **< whatever you do to the bottom, you have to do to the top**

Our new fraction is 24/15

**5/3**

3 x 5 = 15 **< multiplying by 5 to get the common denominator**

5 x 5 = 25 **< whatever you do to the bottom you have to do to the top**

Our new fraction is 25/15

Now we can add the improper fractions together:

24/15 + 24/15 = 49/15

Our answer is now 49/15. We can change this answer into a mixed number since its an improper fraction, by looking at how many times 49 goes into 15 to get the whole. And looking at the left over numbers as the numerator.

49 goes into 15 3 times.

So our whole number is 3.

The left over number is 4 which is our numerator.

And we keep the denominator the same.

or we can leave the fraction how it is.

Answer:

**Improper: 49/15**

**Mixed: 3 4/15**

## Reducing your fractions

The equation:

5/20 + 5/20 = 10/20

We can make 10/20 into a smaller fraction by simplifying it. To reduce/simplify the fraction all your doing is dividing the numerator and the denominator by 2.

10 divided by 2 = 5

20 divided by 2 = 10

So the reduced new fraction answer would be: 5/10.

It is possible to divide the fraction even further but it depends if you have a prime number. If you have a prime number you can divide it even further. But if you don't have a prime number you can divide it even further.