Team 5x5
#53535A
Fractions, Decimals, and Percentages
What are Fractions?
Fractions are a different way to express numbers. They can be used to represent values in between whole numbers, like maybe a number between 0 and 1 or between 3 and 4. Let's say we have two numbers, x and y. Fractions can be written as x/y or x
y
The top number is called the numerator, and the bottom number is called the denominator:
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x
y
Numerator
Denominator
Types of Fractions
There are 3 types of fractions, and they each have different names.
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Proper Fractions:
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Common fractions are fractions in which the numerator is a smaller number than the denominator. So, common fractions will always be a value between 0 and 1, unless they are negative, which we will cover later. Some examples of common fractions are 1/2, 3/4, and 6/7.
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Improper Fraction:
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Improper fractions are fractions in which the numerator is a bigger number than the denominator. These will always be numbers greater than 1, unless they are negative, which we will talk about soon. Some examples of improper fractions are 5/4, 8/5, and 3/2.
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Mixed Fractions:
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Also called Mixed Numbers, these are fractions that are converted from improper fractions. The from of a mixed fraction is a whole number first, and then a proper fraction after it. This from is a little weird, so here is a model of a mixed fraction:
Z
x
y
Converting Fractions
Mixed fractions are converted from improper fractions, or they could be converted to improper fractions. How? Well, to convert this mixed number to an improper fraction:
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We first multiply Z and Y, and then add X. This becomes the numerator of the improper fraction. The denominator doesn't change. So:
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Z
x
y
Z
x
y
y*z+x
y
Mixed Fraction form Improper Fraction Form
To convert improper fractions to mixed numbers, we think about it like division. Say we have improper fraction x/y, and remember, x is greater than y.
To convert this to a mixed number, do x divided by y. Leave your answer as a whole number and then a remainder. The whole number represents how many times y goes into x, and the remainder is how much is left over. Let's say the whole number is a, and the remainder is b. Then, converting to a mixed fraction, we have:
For example, lets say x is 11 and y is 4, so our fraction is 11/4. 11 divided by 4 is 2 remainder 3, because 4 goes 2 times into 11, and 3 is left over. So then, using what we have just learned, our mixed form is
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x
y
2
3
4
Improper Fraction Form Mixed Fraction Form
a
b
4
To make sure this works, let's test that when we convert this to an improper fraction, we get 11/4, because that is what we started with. 4 times 2 is 8, and 8 plus 3 is 11. So, we get 11/4. We did it! Now we have learned how to convert between two different fraction forms, even if they are the same fraction. Now, we will learn another way that two fractions can still be one fraction.
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Simplifying Fractions
Simplifying Fractions is making the numerator and denominator as small as possible, without changing the value of the fraction. We do this with only proper and improper fractions, so to simplify a mixed, first convert it to improper. So, we have ​our fraction x/y. We try to find a number that can go into both x and y, in other words, a factor of x and y, If there is no such number that is a factor of both x and y, besides 1, then x/y is already in it's simplest form. For example, the fraction 3/5. The only common factor of 3 and 5 is 1, which means that 3/5 is already in its simplest form.
However, say we have the fraction 4/8. We notice that 4 and 8 are both divisible by 2. Since they are both divisible by 2, we can divide the numerator and denominator by 2. This gives us 2/4. If you are curious, you can do 4 divided by 8 and 2 divided by 4 in a calculator, and you will get the same answer, which shows that we simplified this fraction without changing its value. However, the fraction is not completely simplified. 2/4 can be simplified more, because they are both divisible by 2. So, dividing the numerator and denominator by 2, we get 1/2. This cannot be simplified any more. However, we didn't have to go through the process of dividing by 2 twice. We can notice that when we have 4/8, both 4 and 8 are divisible by 4. So, this gives us 1/2, and it is quicker. So, when simplyfing a fraction, try to find the greatest common factor of the numerator and denominator, not just any factor. It still works when using another factor, but it will take longer.
This process of simplifying fractions also teaches something very important: when you have a fraction, and you multiply or divide the numerator and denominator by the same number, the value of the fraction will not change. However, when you add or subtract the same number from the numerator, most of the time the fractions value will change.
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x x*a
y y*a
=
x x+a
y y+a
=
Adding and Subtracting Fractions
Just like we add and subtract numbers, we can also add and subtract fractions. When we have two fractions that we want to add or subtract, we want to make the denominator the same for both fractions. We can change the denominator by multiplying the denominator and numerator by a number, like what we learnt earlier. Sometimes we have to multiply both fractions that we are adding, and other times it is just one. This process of getting the same denominator on both fractions is usually referred to as a common denominator. Once the denominator is the same, then all we have to do is add the numerators together. To find the common denominator, we just have to find the least common multiple of the two denominators. If we don't find the least common multiple, but we still find a multiple, then it will work, but then we will have to simplify at the end. So, it is faster to find the least common multiple. Let's practice. Here are two examples, one for addition and one for subtraction:
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3
4
+
47
3
4
-
47
We need to find the least common multiple of 4 and 7. The first few multiples of 7 are 7,14,21,28,35,41... Are any of these divisible by 4? We can see that 28 is, and none of the other ones are, so 28 is the least common multiple of 4 and 7, making it our common denominator. Now, we need to make these fractions have 28 in their denominator. Let's start with 3/4, which has a denominator of 4. 4*7 is 28, so we have to multiply the numerator and denominator by 7. 3*7=21, and 4*7=28, so 3/4=21/28. For 4/7, we need to multiply the numerator and denominator by 4. 4*4=16, 4*7=28. With these new fractions, our problem becomes:
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21
28
+
16
28
21
28
-
16
28
Now that we have a common denominator, all we have to do is add or subtract the two numerators, based on if it is addition or subtraction. For our addition example, 21+16=37. The denominator doesn't change, so our answer is 37/28. So, 3/4+4/7 =37/28. For our subtraction example, 21-16=5, so our answer is 5/28. Whenever we have a fraction as our answer, we want to simplify it completely. This is called the fractions "lowest terms". However, 37/28 and 5/28 both cannot be simplified any more, so that is our answer. ​
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Multiplying Fractions
Multiplying fractions is a bit more straightforward than adding of subtracting them, because there is no common denominator or anything. When multiplying fractions, we simply multiply the two numerators, which becomes the numerator of our answer. And we multiply the two denominators, which becomes the denominator of our answer.
a
b
x =
cd
a*c
b*d
Dividing Fractions
Dividing fractions has a unique way of being solved, and knowing how to multiply fractions comes in handy when dividing fractions. When we divide fractions, we multiply the first fraction by the another fraction, which is the second fraction but with its numerator and denominator flipped.
a
b
c
d
=
a
b
x
d
c
Once we turn our division problem into a multiplication problem, then all we have to do is solve it like we would in a multiplying fractions problem.