1 /4. (One-Half) (One-Quarter) (Three-Eighths)

Size: px

Start display at page:

Download "1 /4. (One-Half) (One-Quarter) (Three-Eighths)"

Alannah Potter
6 years ago
Views:

1 LESSON 4: Fractions: A fraction is a part of a whole. Slice a pizza, and you will have fractions: 1 /2 1 /4 3 /8 (One-Half) (One-Quarter) (Three-Eighths) The top number tells how many slices you have and the bottom number tells how many slices the pizza was cut into. Numerator / Denominator We call the top number the Numerator, it is the number of parts you have. We call the bottom number the Denominator, it is the number of parts the whole is divided into. NUMERATOR DENOMINATOR You just have to remember those names! (If you forget just think "Down"-ominator). Example: 3 / 4 means: We have 3 parts Each part is a quarter ( 1 / 4 ) of a whole 1.-Equivalent Fractions Equivalent Fractions have the same value, even though they may look different. Some fractions may look different, but are really the same, for example: 4 /8 2 /4 1 /2 (Four-Eighths) Two-Quarters) (One-Half)

2 It is usually best to show an answer using the simplest fraction ( 1/2 in this case ). That is called Simplifying, or Reducing the Fraction These fractions are really the same: Why are they the same? Because when you multiply or divide both the top and bottom by the same number, the fraction keeps it's value. The rule to remember is: What you do to the top of the fraction you must also do to the bottom of the fraction! So, here is why those fractions are really the same: And visually it looks like this: /2 2 /4 4 /8 Here are some more equivalent fractions, this time by dividing: If we keep dividing until we can't go any further, then we have simplified the fraction (made it as simple as possible).

3 Important: The top and bottom of the fraction must always be a whole number. You only multiply or divide, never add or subtract, to get an equivalent fraction. 2.-Three Types of Fractions There are three types of fraction: Proper Fractions: The numerator is less than the denominator Examples: 1 / 3, 3 / 4, 2 / 7 Improper Fractions: The numerator is greater than (or equal to) the denominator Examples: 4 / 3, 11 / 4, 7 / 7 Mixed Fractions: A whole number and proper fraction together a) Proper Fractions A Proper Fraction has a top number less than its bottom number 3 / 8 (Three-Eighths) Examples 3 / 8 1 / 4 14 / 15 4 / 5 See how the top number is smaller than the bottom number in each example. That makes it a Proper Fraction. So, a proper fraction is just a fraction where the numerator (the top number) is less than the denominator (the bottom number). Here are some examples of proper fractions:

4 1 / 2 1 / 4 3 / 8 (One-Half) (One-Quarter) (Three-Eighths) b) Improper Fractions 7 / 4 (seven-fourths or seven-quarters) An Improper fraction has a top number larger than (or equal to) the bottom number, It is "top-heavy" Examples 3 / 2 7 / 4 16 / / 5 See how the top number is bigger than the bottom number? That makes it an Improper Fraction, (but there is nothing wrong about Improper Fractions). Can be Equal What about when the numerator is equal to the denominator? For example 4 / 4? 4 / 4 Well, it is obviously the same as a whole, but it is written as a fraction, so most people agree that is a type of improper fraction.

$c) Mixed Fractions Also called "Mixed Numbers" A Mixed Fraction is a whole number and a proper fraction combined. 1 3 / 4 (one and three-quarters) such as 1 3 / 4.$

5 c) Mixed Fractions Also called "Mixed Numbers" A Mixed Fraction is a whole number and a proper fraction combined. 1 3 / 4 (one and three-quarters) such as 1 3 / 4. Examples 2 3 / / / / 5 See how each example is made up of a whole number and a proper fraction together? That is why it is called a "mixed" fraction (or mixed number). You can use either an improper fraction or a mixed fraction to show the same amount. For example 1 3 / 4 7 / 4, shown here: 1 3 / 4 7 / 4 Converting Improper Fractions to Mixed Fractions Divide the numerator by the denominator. Write down the whole number answer Then write down any remainder above the denominator. Example: Convert 11/4 to a mixed fraction. Divide: with a remainder of 3. Write down the 2 and then write down the remainder (3) above the denominator (4), like this: 2+3/4

6 Converting Mixed Fractions to Improper Fractions Multiply the whole number part by the fraction's denominator. Add that to the numerator Then write the result on top of the denominator. Example: Convert 3 2 / 5 to an improper fraction. Multiply the whole number by the denominator: Add the numerator to that: Then write that down above the denominator, like this: 17/5 3.-Simplifying Fractions To simplify a fraction, divide the top and bottom by the highest number thatcan divide into both numbers exactly. Simplifying (or reducing) fractions means to make the fraction as simple as possible. Why say four-eighths ( 4 / 8 ) when you really mean half ( 1 / 2 )? 4 /8 > 2 /4 > 1 /2 (Four-Eighths) (Two-Quarters) (One-Half) There are two ways to simplify a fraction: Method 1: Try dividing by the same number both the top and bottom of the fraction until you can't go any further (try dividing by 2,3,5,7,... etc). Example: Simplify the fraction 24 / 108 :

7 Method 2: Write the numerator and denominators as a product of prime factors. You have to take out the same factors in the numerator and in the denominator. So you get the most simple fraction. 4.-Comparing Fractions Sometimes we need to compare two fractions to discover which is larger or smaller. There are two easy ways to compare fractions: using decimals; or using the same denominator The Decimal Method of Comparing Fractions Just convert each fraction to decimals, and then compare the decimals. Which is bigger: 3 / 8 or 5 / 12? You need to convert each fraction to a decimal. You can do this using your calculator (3 8 and 5 12), or you can read about Converting Fractions to Decimals. Anyway, these are the answers I get: 3 / , and 5 / The Same Denominator Method So, 5 / 12 is bigger. If two fractions have the same denominator (the bottom number) then they are easy to compare. For example 4 / 9 is less than 5 / 9 (because 4 is less than 5) But if the denominators are not the same you need to make them the same (using Equivalent Fractions). Example: Which is larger: 3 / 8 or 5 / 12? If you multiply 8 3 you get 24, and if you multiply 12 2 you also get 24, so let's try that (important: what you do to the bottom, you must also do to the top): and, so it is now easy to see that 10 / 24 is bigger than 9 / 24, so 5 / 12 must be bigger.

$Then it is just a matter of changing each fraction to make it's denominator the Least Common Multiple. Example: Which is larger: 5 / 6 or 13 / 15? The Least Common Multiple of 6 and 15 is 30.$

8 How to Make the Denominators the Same The trick is to find the Least Common Multiple of the two denominators. In the previous example, the Least Common Multiple of 8 and 12 was 24. Then it is just a matter of changing each fraction to make it's denominator the Least Common Multiple. Example: Which is larger: 5 / 6 or 13 / 15? The Least Common Multiple of 6 and 15 is 30. So, let's do some multiplying to make each denominator equal to 30 : and, Now we can easily see that 26 / 30 is larger than 25 / 30, so 13 / 15 is the larger fraction. 5.-Adding Fractions You can add fractions easily if the bottom number (the denominator) is the same: 1 /4 + 1 /4 2 /4 1 /2 (One-Quarter) (One-Quarter) (Two-Quarters) (One-Half) 5 /8 + 1 /8 6 /8 3 /4

Adding Fractions with Different Denominators But what if the denominators are not the same? As in this example: 3 /8 + 1 /4? You must somehow make the denominators the same.

9 Adding Fractions with Different Denominators But what if the denominators are not the same? As in this example: 3 /8 + 1 /4? You must somehow make the denominators the same. In this case it is easy, because we know that 1/4 is the same as 2/8 : 3 /8 + 2 /8 5 /8 In that example it was easy to make the denominators the same, but it can be harder... so you may need to use the Least Common Denominator There are 3 Simple Steps to add fractions: Step 1: Make sure the bottom numbers (the denominators) are the same Step 2: Add the top numbers (the numerators). Put the answer over the same denominator. Step 3: Simplify the fraction (if needed). Example 1: ¼+¼ Step 1. The bottom numbers are already the same. Go straight to step 2. Step 2. Add the top numbers and put the answer over the same denominator: Step 3. Simplify the fraction:

10 Example 2: 1/3+1/6 Step 1: The bottom numbers are different. See how the slices are different sizes? We need to make them the same before we can continue, because we can't add them like this: 1 / / 6? To make the bottom numbers the same, multiply the top and bottom of the first fraction ( 1 / 3 ) by 2, like this: And now our question looks like this: 2 2 / / 6 The bottom numbers (the denominators) are the same, so we can go to step 2. Step 2: Add the top numbers and put them over the same denominator:

$2 1 2 + 1 3 + 6 6 6 6 In picture form it looks like this: 2 / 6 + 1 / 6 3 / 6 Step 3: Simplify the fraction: 3 1 6 2 In picture form the whole answer looks like this: 2 / 6 + 1 / 6 3 / 6 1$

11 In picture form it looks like this: 2 / / 6 3 / 6 Step 3: Simplify the fraction: In picture form the whole answer looks like this: 2 / / 6 3 / 6 1 / 2 Example 3: 1/3 + 1/5 Again, the bottom numbers are different (the slices are different sizes)! 1 / / 5? But let us try dividing them into smaller sizes that will each be the same:

12 5 / / 15 By multiplying the top and bottom of the first fraction by 5 we ended up with 5 / 15 : And by multiplying the top and bottom of the second fraction by 3 we ended up with 3 / 15 : The bottom numbers are now the same, so we can go ahead and add the top numbers: 3 5 / / 15 8 / 15

$6.-Subtracting Fractions There are 3 simple steps to subtract fractions Step 1.$ $Put the answer over the the same denominator. Step 3. Simplify the fraction. Example 1: 3/4-1/4 Step 1.$ $Simplify the fraction: 3 1 3 1 2 4 4 4 4 2 1 4 2 Example 2: ½ - 1/6 Step 1. The bottom numbers are different.$

13 6.-Subtracting Fractions There are 3 simple steps to subtract fractions Step 1. Make sure the bottom numbers (the denominators) are the same Step 2. Subtract the top numbers (the numerators). Put the answer over the the same denominator. Step 3. Simplify the fraction. Example 1: 3/4-1/4 Step 1. The bottom numbers are already the same. Go straight to step 2. Step 2. Subtract the top numbers and put the answer over the same denominator: Step 3. Simplify the fraction: Example 2: ½ - 1/6 Step 1. The bottom numbers are different. See how the slices are different sizes? We need to make them the same before we can continue, because we can't subtract them like this: 1 / 2-1 / 6?

14 To make the bottom numbers the same, multiply the top and bottom of the first fraction ( 1 / 2 ) by 3 like this: And now our question looks like this: 3 3 / 6-1 / 6 The bottom numbers (the denominators) are the same, so we can go to step 2. Step 2. Subtract the top numbers and put the answer over the same denominator: In picture form it looks like this: 3 / 6-1 / 6 2 / 6 Step 3. Simplify the fraction:

15 Multiplying Fractions Multiply the tops, multiply the bottoms. There are 3 simple steps to multiply fractions 1. Multiply the top numbers (the numerators). 2. Multiply the bottom numbers (the denominators). 3. Simplify the fraction if needed. Example 1: ½ x 2/5 Step 1. Multiply the top numbers: Step 2. Multiply the bottom numbers: Step 3. Simplify the fraction: Dividing Fractions Turn the the second fraction upside down, then just multiply. There are 3 Simple Steps to Divide Fractions:

16 Step 1. Turn the second fraction (the one you want to divide by) upside-down (this is now a reciprocal). Step 2. Multiply the first fraction by that reciprocal Step 3. Simplify the fraction (if needed) Example 1: ½ : 1/6 Step 1. Turn the second fraction upside-down (the reciprocal): Step 2. Multiply the first fraction by that reciprocal: Step 3. Simplify the fraction: Why Turn the Fraction Upside Down? Because division is the inverse (opposite) of multiplying. Let us see if it works... Multiply and divide are opposites, right? It works with simple numbers: Example: can be reversed by 50 / 5 10 So will the same work with fractions? Let us try: Example: Start with 100 and multiply by 3 / 4 So you divide by 4 then multiply by 3. So / 4 is 100 divided by 4 (25) then multiplied by 3 (75). Answer: / 4 75

17 Can we reverse that by dividing by 3 / 4? Example: 75 / ( 3 / 4 ) is also 75 ( 4 / 3 ), which is 75 divided by 3 (25) then multiplied by 4 (100). Answer: 75 / ( 3 / 4 ) Convert Decimals to Fractions To convert a Decimal to a Fraction follow these steps: Step 1: Write down the decimal divided by 1. Step 2: Multiply both top and bottom by 10 for every number after the decimal point. (For example, if there are two numbers after the decimal, then use 100, if there are three then use 1000, etc.) Step 3: Simplify (or reduce) the fraction Example 1: Express 0.75 as a fraction Step 1: Write down: 0.75 Step 2: Multiply both top and bottom by 100 (becasue there were 2 digits after the decimal place): Step 3: Simplify the fraction: 100 Do you see how it neatly turns the top number into a whole number? Answer 3/4

18 Note: 75/100 is called a decimal fraction and 3/4 is called a common fraction! Example 2: Express as a fraction Step 1: write down: Step 2: multiply both top and bottom by 1,000 (there were 3 digits after the decimal place so that is ,000) ,000 Step 3: Simplify the fraction (it took me two steps here): , Answer 5/8 10.-Convert Percent to Fraction To convert a Percent to a Fraction follow these steps: Step 1: Write down the percent divided by 100. Step 2: If the percent is not a whole number, then multiply both top and bottom by 10 for every number after the decimal point. (For example, if there is one number after the decimal, then use 10, if there are two then use 100, etc.) Step 3: Simplify (or reduce) the fraction Example 1: Express 75% as a fraction

19 Step 1: Write down: Step 2: The percent is a whole number, so no need for step 2. Step 3: Simplify the fraction: Answer 3/4 Note: 75/100 is called a decimal fraction and 3/4 is called a common fraction! Example 2: Express 62.5% as a fraction Step 1: Write down: 62.5 Step 2: Multiply both top and bottom by 10 (because there is 1 digit after the decimal place) , (See how this neatly makes the top a whole number?) Step 3: Simplify the fraction (this took me two steps, you may be able to do it one!) :

20 , Answer 5/8 Example 3: Express 150% as a fraction Step 1: Write down: Step 2: The percent is a whole number, so no need for step 2. Step 3: Simplify the fraction (I did it one step): Convert Fraction to Percent 50 Answer 3/2 Divide the top of the fraction by the bottom, multiply by 100 and add a "%" sign.

Improper Fractions. An Improper Fraction has a top number larger than (or equal to) the bottom number.

Improper Fractions. An Improper Fraction has a top number larger than (or equal to) the bottom number. Improper Fractions (seven-fourths or seven-quarters) 7 4 An Improper Fraction has a top number larger than (or equal to) the bottom number. It is "top-heavy" More Examples 3 7 16 15 99 2 3 15 15 5 See