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

Transcription

1 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 See how the top number is bigger than (or equal to) the bottom number? That makes it an Improper Fraction, (but there is nothing wrong about Improper Fractions ). Three Types of Fractions There are three types of fraction:

2 Fractions A Fraction (such as 7 /4) has two numbers: Numerator Denominator The top number (the Numerator) is the number of parts we have. The bottom number (the Denominator) is the number of parts the whole is divided into. Example: 7 /4 means: We have 7 parts Each part is a quarter ( 1 /4) of a whole So we can define the three types of fractions like this: 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

3 Examples: 1 1 / 3, 2 1 / 4, 16 2 / 5 Improper Fraction So an improper fraction is a fraction where the top number (numerator) is greater than or equal to the bottom number (denominator): it is top-heavy. 4 /4 Can be Equal What about when the numerator is equal to the denominator? For example 4 /4? Well it is the same as a whole, but it is written as a fraction, so most people agree it is a type of improper fraction. Improper Fractions or Mixed Fractions We 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

4 = Converting Improper Fractions to Mixed Fractions To convert an improper fraction to a mixed fraction, follow these steps: 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: 11 4 = 2 with a remainder of 3 Write down the 2 and then write down the remainder (3) above the denominator (4), like this: Converting Mixed Fractions to Improper Fractions To convert a mixed fraction to an improper fraction, follow these steps:

5 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: 3 5 = 15 Add the numerator to that: = 17 Then write that down above the denominator, like this: 17 5 Are Improper Fractions Bad? NO, they aren't bad! For mathematics they are actually better than mixed fractions. Because mixed fractions can be confusing when we write them in a formula: should the two parts be added or multiplied? Mixed Fraction: What is: / 4? Is it: / 4 = 3 1 / 4? Or is it: / 4 = 1 1 / 2? Improper Fraction: What is: / 4? It is: 4 / / 4 = 13 / 4

6 But, for everyday use, people understand mixed fractions better. Example: It is easier to say "I ate 2 1 /4 sausages", than "I ate 9 /4 sausages" Equivalent Fractions Equivalent Fractions have the same value, even though they may look different. These fractions are really the same: 1 2 = 2 4 = 4 8 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: "Change the bottom using multiply or divide, And the same to the top must be applied" So, here is why those fractions are really the same: = = And visually it looks like this: /2 2 /4 4 /8

7 = = See the Animation See Fractions on the Number Line... it shows you many equivalent fractions We also have a Chart of Fractions with many examples of equivalent fractions. Dividing Here are some more equivalent fractions, this time by dividing: = = Choose the number you divide by carefully, so that the results (both top and bottom) stay whole numbers.

8 If we keep dividing until we can't go any further, then we have simplified the fraction (made it as simple as possible). Summary: You can make equivalent fractions by multiplying or dividing both top and bottom by the same amount. You only multiply or divide, never add or subtract, to get an equivalent fraction. Only divide when the top and bottom would still be whole numbers. Greatest Common Factor The highest number that divides exactly into two or more numbers. It is the "greatest" thing for simplifying fractions! Let's start with an Example... Greatest Common Factor of 12 and 16 Find all the Factors of each number, Circle the Common factors, Choose the Greatest of those So... what is a "Factor"?

9 Factors are the numbers you multiply together to get another number: A number can have many factors: Factors of 12 are 1, 2, 3, 4, 6 and because 2 6 = 12, or 4 3 = 12, or 1 12 = 12. (Read how to find All the Factors of a Number. In our case we don't need the negative ones.) What is a "Common Factor"? Let us say you have worked out the factors of two numbers: Example: Factors of 12 and 30 Factors of 12 are 1, 2, 3, 4, 6 and 12 Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30 Then the common factors are those that are found in both lists: Notice that 1, 2, 3 and 6 appear in both lists? So, the common factors of 12 and 30 are: 1, 2, 3 and 6 It is a common factor when it is a factor of two or more numbers. (It is then "common to" those numbers.) Here is another example with three numbers: Example: The common factors of 15, 30 and 105

10 Factors of 15 are 1, 3, 5, and 15 Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30 Factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105 The factors that are common to all three numbers are 1, 3, 5 and 15 In other words, the common factors of 15, 30 and 105 are 1, 3, 5 and 15 What is the "Greatest Common Factor"? It is simply the largest of the common factors. In our previous example, the largest of the common factors is 15, so the Greatest Common Factor of 15, 30 and 105 is 15 The "Greatest Common Factor" is the largest of the common factors (of two or more numbers) Why is this Useful? One of the most useful things is when we want to simplify a fraction: Example: How could we simplify 12 /30? Earlier we found that the Common Factors of 12 and 30 were 1, 2, 3 and 6, and so the Greatest Common Factor is 6. So the largest number we can divide both 12 and 30 evenly by is 6, like this: 6

11 12 2 = The Greatest Common Factor of 12 and 30 is 6. And so 12 /30 can be simplified to 2 /5 Finding the Greatest Common Factor Here are three ways: 1. You can: find all factors of both numbers (I have an All Factors Calculator to help you), then select the ones that are common to both, and then choose the greatest. Example: Two Numbers Factors Common Factors Greatest Common Factor Example Simplified Fraction 9 and 12 9: 1,3,9 12: 1,2,3,4,6,12 1,3 3 9 / 12 = 3 / 4 And another example: Two Numbers Factors Common Factors Greatest Common Factor Example Simplified Fraction

12 6 and 18 6: 1,2,3,6 18: 1,2,3,6,9,18 1,2,3,6 6 6 / 18 = 1 / 3 2. You can find the prime factors and combine the common ones together: Two Numbers Thinking... Greatest Common Factor Example Simplified Fraction 24 and = 24, and = = / 108 = 2 / 9 3. And sometimes you can just play around with the factors until you discover it: Two Numbers Thinking... Greatest Common Factor Example Simplified Fraction 9 and = 9 and 3 4 = / 12 = 3 / 4 But in that case you had better be careful you have found the greatest common factor. Greatest Common Factor Calculator There is another easy method, you can use our Greatest Common Factor Calculator to find it automatically. Other Names

13 The "Greatest Common Factor" is often abbreviated to "GCF", and is also known as: the "Greatest Common Divisor (GCD)", or the "Highest Common Factor (HCF)" Least Common Multiple The smallest positive number that is a multiple of two or more numbers. Let's start with an Example... Least Common Multiple of 3 and 5: List the Multiples of each number, Find the first Common (same) value The Least Common Multiple of 3 and 5 is is a common multiple of 3 and 5, and is the smallest (least) common multiple So... what is a "Multiple"? We get a multiple of a number when we multiply it by another number. Such as multiplying by 1, 2, 3, 4, 5, etc, but not zero. Just like the multiplication table. Here are some examples:

14 The multiples of 4 are: 4,8,12,16,20,24,28,32,36,40,44,... The multiples of 5 are: 5,10,15,20,25,30,35,40,45,50,... What is a "Common Multiple"? Say we have listed the first few multiples of 4 and 5: the common multiples are those that are found in both lists: The multiples of 4 are: 4,8,12,16,20,24,28,32,36,40,44,... The multiples of 5 are: 5,10,15,20,25,30,35,40,45,50,... Notice that 20 and 40 appear in both lists? So, the common multiples of 4 and 5 are: 20, 40, (and 60, 80, etc..., too) What is the "Least Common Multiple"? It is simply the smallest of the common multiples. In our previous example, the smallest of the common multiples is so the Least Common Multiple of 4 and 5 is 20. Finding the Least Common Multiple List the multiples of the numbers until we get our first match. Example: Find the least common multiple of 4 and 10:

15 The multiples of 4 are: 4, 8, 12, 16, 20,... and the multiples of 10 are: 10, 20,... Aha! there is a match at 20. It looks like this: So the least common multiple of 4 and 10 is 20 Example: Find the least common multiple of 6 and 15: The multiples of 6 are: 6, 12, 18, 24, 30,... and the multiples of 15 are: 15, 30,... There is a match at 30 So the least common multiple of 6 and 15 is 30 More than 2 Numbers We can also find the least common multiple of three (or more) numbers. Example: Find the least common multiple of 4, 6, and 8 Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36,... Multiples of 6 are: 6, 12, 18, 24, 30, 36,... Multiples of 8 are: 8, 16, 24, 32, 40,... So 24 is the least common multiple (I can't find a smaller one!) Hint: You can have smaller lists for the bigger numbers.

16 Least Common Denominator... is the Least Common Multiple of the denominators... What is a Denominator? The denominator is the bottom number in a fraction. It shows how many equal parts the item is divided into Fractions with Different Denominators You can't add fractions with different denominators: 1 /3 + 1 /6 =? So what do you do? How can they be added? Answer: You need to make the denominators the same. Common Denominator

17 But what should the new denominator be? One simple answer is to multiply the current denominators together: 3 6 = 18 So instead of having 3 or 6 slices, we will make both of them have 18 slices. The pizzas now look like this (I will show calculations later): 6 / /18 = 9 /18 (Read more about Common Denominators.) Least Common Denominator That is all fine, but 18 is a lot of slices... can you do it with fewer slices? Here is how to find out: 1 /3 List the multiples of 3: 3, 6, 9, 12, 15, 18, 21,... 1 /6 List the multiples 6: 6, 12, 18, 24,... Then find the smallest number that is the same multiples of 3: 3, 6, 9, 12, 15, 18, 21,... multiples 6: 6, 12, 18, 24,... The answer is 6, and that is the Least Common Denominator.

18 So let us try using it! We want both to have 6 slices. When we multiply top and bottom of 1/3 by 2 we get 2/6 1/6 already has a denominator of 6 And our question now looks like: 2 /6 + 1 /6 = 3 /6 One last step is to simplify the fraction (if possible). In this case 3/6 is simpler as 1/2: 2 /6 + 1 /6 = 3 /6 = 1 /2 And that is what the Least Common Denominator is all about. It lets you add (or subtract) fractions using the least number of slices. What Did We Do? The trick was to list the multiples of each denominator, then find the Least Common Multiple

19 In the previous example the Least Common Multiple of 3 and 6 was 6. In other words the Least Common Denominator of 1 /3 and 1 /6 is 6. Here are the steps to follow: Find the Least Common Multiple of the denominators (which is called the Least Common Denominator). Change each fraction (using equivalent fractions) to make their denominators the same as the least common denominator Then add (or subtract) the fractions, as you wish! Example: What is 1 /6 + 7 /15? The Denominators are 6 and 15: multiples of 6: 6, 12, 18, 24, 30, 36,... multiples 15: 15, 30, 45, 60,... So the Least Common Multiple of 6 and 15 is 30. Now let's try to make the denominators the same. Note: what you do to the bottom of the fraction, you must also do to the top When you multiply 6 5 you get 30, and when you multiply 15 2 you also get 30: 5 2 and 1 = 5 7 = 14

20 Now we can do the addition by adding the top numbers: 5 / /30 = 19 /30 The fraction is already as simple as it can be, so that is the answer. Least Common Multiple Tool To find the least common denominator automatically, you can use our Least Common Multiple Tool- just put in the denominators, press the button, and the least common denominator is shown for you. One More Example Example: What is 3 /8 + 5 /12? List the multiples of 8 and 12 multiples of 8: 8, 16, 24, 32, 40,... multiples 12: 12, 24, 36, 48,... The Least Common Multiple is 24 Let's try to make the denominators the same... when you multiply 8 3 you get 24, and when you multiply 12 2 you also get 24. So, let's use that: 3 and 2

21 3 9 = = Now we can do the addition: 9 / /24 = 19 /24 The fraction is already as simple as it can be, so that is the answer. Adding Fractions 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 thedenominator Step 3: Simplify the fraction (if needed) Example 1: Step 1. The bottom numbers (the denominators) are already the same. Go straight to step 2.

22 Step 2. Add the top numbers and put the answer over the same denominator: = = Step 3. Simplify the fraction: 2 1 = 4 2 In picture form it looks like this: 1 /4 + 1 /4 = 2 /4 = 1 /2... and do you see how 2 /4 is simpler as 1 /2? (see Equivalent Fractions.) Example 2: Step 1: The bottom numbers are different. See how the slices are different sizes? 1 /3 + 1 /6 =?

23 We need to make them the same before we can continue, because we can't add them like that. The number "6" is twice as big as "3", so to make the bottom numbers the same we can multiply the top and bottom of the first fraction by 2, like this: = Important: you multiply both top and bottom by the same amount, to keep the value of the fraction the same Now the fractions have the same bottom number ("6"), and our question looks like this: 2 /6 + 1 /6 The bottom numbers are now the same, so we can go to step 2.

24 Step 2: Add the top numbers and put them over the same denominator: = = 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 /2

25 With Pen and Paper And here is how to do it with a pen and paper (press the play button): Play with it! Try the Adding Fractions Animation. A Rhyme To Help You Remember "If adding or subtracting is your aim, The bottom numbers must be the same! "Change the bottom using multiply or divide, But the same to the top must be applied, "And don't forget to simplify, Before its time to say good bye" Example 3: Again, the bottom numbers are different (the slices are different sizes)! 1 /3 + 1 /5 =?

26 But let us try dividing them into smaller sizes that will each be the same: 5 / /15 The first fraction: by multiplying the top and bottom by 5 we ended up with 5 /15 : = The second fraction: by multiplying the top and bottom by 3 we ended up with 3 /15 : = 5 15

27 3 The bottom numbers are now the same, so we can go ahead and add the top numbers: 5 / /15 = 8 /15 The result is already as simple as it can be, so that is the answer: 8 /15 Making the Denominators the Same In the previous example how did we know to cut them into 1 /15ths to make the denominators the same? Read how to do this using either one of these methods: Common Denominator Method, or the Least Common Denominator Method They both work, use which one you prefer! Example: Cupcakes You want to make and sell cupcakes:

28 A friend can supply the ingredients, if you give them 1 / 3 of sales And a market stall costs 1 / 4 of sales How much is that altogether? We need to add 1 / 3 and 1 / 4 1 1? + = 3 4? First make the bottom numbers (the denominators) the same. Multiply top and bottom of 1 / 3 by 4: 1 4 1? + = 3 4 4? And multiply top and bottom of 1 / 4 by 3: ? + = ? Now do the calculations: = = Answer: 7 / 12 of sales go in ingredients and market costs. Subtracting Fractions You might like to read Adding Fractions first. There are 3 simple steps to subtract fractions

29 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 same denominator. Step 3. Simplify the fraction (if needed). Example 1: 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: 2 1 = 4 2 (If you are unsure of the last step see Equivalent Fractions.) Example 2: 1 1

30 2 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 =? To make the bottom numbers the same, multiply the top and bottom of the first fraction ( 1 /2) by3 like this: = And now our question looks like this: 3 /6 1 /6 The bottom numbers (the denominators) are the same, so we can go to step 2.

31 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: 2 1 = 6 3 With Pen and Paper And here is how to do it with a pen and paper (press the play button): Subtracting Mixed Fractions I have a special page on Adding and Subtracting Mixed Fractions. Making the Denominators the Same

32 In the previous example it was easy to make the denominators the same, but it can be harder... so you may need to use either the Common Denominator Method, or the Least Common Denominator Method They both work, use which one you prefer! Example: Cupcakes You want to sell cupcakes at a market: You get paid 2 / 5 of total sales But you have to pay 1 / 4 of total sales for the market stall How much do you get? We need to subtract 1 / 4 from 2 / 5 2 1? = 5 4? First make the bottom numbers (the denominators) the same. Multiply top and bottom of 2 / 5 by 4: 2 4 1? = 5 4 4? And multiply top and bottom of 1 / 4 by 5:

33 ? = ? Now do the calculations: = = Answer: you get to keep 3 / 20 of total sales. Adding and Subtracting Mixed Fractions 1 3 /4 (one and three-quarters) Quick Definition: A Mixed Fraction is a whole number and a fraction combined, such as 1 34 To make it easy to add and subtract them, just convert to Improper Fractions first: Quick Definition: An Improper fraction has a top number larger than or equal to the bottom number, 7 /4 (seven-fourths or seven-quarters) such as 74 or 43 (It is "top-heavy")

34 Adding Mixed Fractions I find this is the best way to add mixed fractions: convert them to Improper Fractions then add them (using Addition of Fractions) then convert back to Mixed Fractions: Example: What is ? Convert to Improper Fractions: Common denominator of 4: 2 34 = = stays as becomes 144 (by multiplying top and bottom by 2) Now Add: Convert back to Mixed Fractions: = = 6 14 When you get more experience you can do it faster like this: Example: What is Convert them to improper fractions: 3 58 = = 74

35 Make same denominator: 74 becomes 148 (by multiplying top and bottom by 2) And add: = 438 = 5 38 Subtracting Mixed Fractions Just follow the same method, but subtract instead of add: Example: What is ? Convert to Improper Fractions: Common denominator of 12: = = 536 Now Subtract: 634 becomes becomes Convert back to Mixed Fractions: = =

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