 Mathematics

# Beginning Fractions Shawn Schafer's image for:
"Beginning Fractions"
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Image by: Okay, so first I'm going to define a term for 'normal' numbers. Normal numbers that do not involve fractions or decimals are often referred to as integers. They are also known as whole numbers.

So I'll jump into explaining fractions the traditional way. Lets say you have a pizza. The pizza is not cut yet. You could describe this pizza as one whole pizza, or for the purpose of this explanation, simply as 1. How do you describe the pizza if you cut it in half? You could call it 2 halves or 2/2 in fractions. If the whole pizza is '1' then each of these two very large pieces is '1/2' or one-half. If you cut each of these pieces in half again, you have 4 pieces. Since each of these pieces is only a fourth as large as the whole pizza, and the whole pizza can be described as '1', each piece is '1/4' or one-fourth. If you take two of the pieces and put them together, or add them, you have " 1/4 + 1/4= 2/4= 1/2" . This means that one fourth of a pizza and one fourth of a pizza make two fourths of a pizza, which could more simply be described as one half of a pizza. Similarly, if you eat one piece of this whole pizza that is cut into 4 pieces, you will only have three fourths of a pizza left or 3/4. This can be described mathematically as: " 1 - 1/4 = 3/4" or " 4/4 - 1/4= 3/4".

Now I will get away from analogies and get into mathematics, but I will keep it simple enough to follow. A fraction is a part of a number. The number on top is called the numerator, the one on the bottom is the denominator. "1/2" is half as large as "1" and is the same as dividing 1 by 2. The numerator describes how many parts there are, and the denominator describes how large these parts are. The larger the numerator, the larger the fraction, the larger the denominator, the smaller the fraction. Going back to the pizza thing, pieces of pizza are smaller when you cut them into, say, eighths; than if you cut them into halves. This means that "1/8" is smaller than "1/2"

Many fractions can and should be reduced to simpler fractions. For example, 2/4 is 1/2. The way you figure out if a fraction can be reduced is by figuring out if there is a number other than 1 that goes into both the numerator and the denominator. If there is such a number, you divide both the numerator and denominator by that number separately, Ideally, the number; also known as factor, they can both be divided by should be as large as possible. After reducing a fraction, check to see if the fraction you reduced it to can be reduced further. Reduce the fraction until it can't be reduced further.

Examples:

4/8, both 4 and 8 can be divided by 4. 4 divided by 4 is 1 and 8 divided by 4 is 2 so you get 1/2

5/8, there aren't any numbers other than 1 that go into both 5 and 8 so it is left as 5/8

6/15, 3 goes into both 6 and 15. 3 goes into 6 two times and 3 goes into 15 five times so it reduces to 2/5

There are also fractions that are larger than 1. For example, 3/2 is one half larger than 1. The way these fractions can be reduced is by finding how many whole numbers are in the fraction and then putting whatever fraction is left over next to the whole number

Examples:

4/2, 2/2 is 1 and 2/2 goes into 4/2 twice so 4/2=2

5/3, 3/3 is 1 and after converting the 3/3 of 5/3 into 1, there are still two-thirds left over so 5/3= 1 2/3

Of the four basic operations(multiplication, addition, division, subtraction), multiplication is, by far, the easiest to perform on fractions. You just multiply the numerator(top number) of the first fraction by the numerator of the second fraction and put the result on the top of your new fraction, and then multiply the bottom number(denominator) of the first fraction by the denominator of the second fraction, and put the result on the bottom of you new fraction.

Examples:

1/2 x 3/4 = 3/8

3/5 x 7/8 = 21/40

1/1 x 1/1 = 1/1

1/2 x 1/2 =1/4

Another way to think of multiplying fractions is to think of the multiplication symbol as the word 'of' . In this way, '1/2 x 1/2 = 1/4' is 'one half of one half is one fourth. When multiplying whole numbers by fractions, you should convert the whole number into a fraction by setting the whole number as a numerator and making the denominator 1. This is the same as dividing the number by one, so you aren't actually changing the number, you are only making it easier to work with.

Examples:

2 x 1/5 is 2/1 x 1/5= 2/5

3 x 4/9 is 3/1 x 4/9= 12/9= 1 3/9= 1 1/3

5 x 3/25 is 5/1 x 3/25= 15/25= 3/5

Dividing fractions is almost as easy. To divide fractions, you simply switch the numerator and denominator of the second fraction. I will be using the / character to describe division as well as to define fractions so just know that when there is a space between numbers and /, I'm dividing.

Examples:

2/4 / 3/5 = 2/4 x 5/3= 10/12= 5/6

1/2 / 1/2= 1/2 x 2/1= 2/2= 1

Subtracting and Adding fractions is slightly more complex. To add and subtract fractions, the denominators need to be the same. This means that if the denominators are different, you need to convert one or both the the fractions into a different fraction. To do this, you need to look at both of the denominators and figure out what the lowest number they both go into is. Once you find this number, you need to multiply the old numerators by what each original denominator had to be multiplied into to get the new denominator. You then add or subtract the new numerators and put it over the common denominator. I realize the explanation is kind of wordy so I'll give some examples:

2/4 + 2/4= 4/4=1 - Note that the denominators were already the same so I didn't have to alter either fraction before adding, I simply added the numerators and kept the denominator the same, then reduced.

2/5 + 3/10 = 4/10 + 3/10= 7/10- 5 and 10 are not equal so i had to find a denominator they both go into. The lowest number that both 10 and 5 go into is 10. 10 goes into 10 once so the 3 over the original 10 stays 3 because 3 x 1=3. 5 goes into 10 twice so the 2 over the 5 becomes 4 because 2 x 2= 4. I then added the new numerators and kept the new denominator the same

2/3 - 1/5 = 10/15 - 3/15= 7/15- 3 and 5 aren't equal and the lowest thing they both go into is 15. So I multiplied the two that is over the three by 5 because 3 goes into 15 five times, and I multiplied the 1 that is over the 5 by 3 because 5 goes into 15 three times. I then subtracted the first new numerator from the second new numerator and kept the new denominator the same.

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