Topic 3: Fractions. Topic 1 Integers. Topic 2 Decimals. Topic 3 Fractions. Topic 4 Ratios. Topic 5 Percentages. Topic 6 Algebra


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1 Topic : Fractions Topic Integers Topic Decimals Topic Fractions Topic Ratios Topic Percentages Duration / weeks Content Outline PART (/ week) Introduction Converting Fractions to Decimals Converting Decimals to Fractions Equivalent Fractions Simplifying Fractions Mixed, Improper and Proper Fractions Lowest Common Denominator PART ( week) Addition/Subtraction of Fractions with Common Denominators Addition/Subtraction of Fractions with Different Denominators Multiplication of Fractions Reciprocals Division of Fractions Cancelling Fractions and Chance Topic 6 Algebra Topic 7 Equations and Formulae Topic 8 Measurement
2 Topic : Fractions Introduction Fractions are used to describe anything that has been divided into equal sized pieces. Some examples are a block of chocolate, an apple or a litre of lemonade. Fractions state how many pieces something is divided into or shared between and how many are chosen. How does it work? A fraction represents a part of a whole. top number or NUMERATOR bottom number or DENOMINATOR The Numerator refers to the number of parts you have. The Denominator is the number of equal size pieces whole is divided into. Example : Consider the following: The shaded portion of the pie represents 6 or /6 or 6 of the total pie; there are pieces chosen out of the 6 pieces the shape is divided into. The shaded portion of the box represents (or ). There are pieces chosen out of the pieces the shape is divided into. This piece is NOT broken up into equally shaped pieces. This cannot be described as the fraction.
3 Converting Fractions to Decimals A fraction can be considered to indicate a division: numerator numerator denominator denominator To change a fraction to a decimal, divide the numerator by the denominator. Example : Convert to a decimal. Some common fractions you may know and not need to work out. Step : Step : Consider the fraction as a division. Perform the division: Example : Convert to a decimal. Step : Step : Consider the fraction as a division. Perform the division: This is written in short form as 0.6 This is called a recurring decimal
4 units tenths hundredths thousandths ten thousandths 6 Example : Convert 9 to a decimal. Step : Consider the fraction as a division and perform the division Step : Perform the division: If two or more digits repeat then we place a line above the group of digits that repeat. Converting Decimals to Fractions Look at the following place value table: decimal. fraction
5 7 To change a decimal to a fraction express the fraction in terms of its place value. The number of places after the decimal point tells us how many zeros to use in the number under the fraction line, i.e. whether it should be 0, 00, 000 Example : Step : Convert 0. to a fraction. Since there are two digits after the decimal point, the fraction will have a denominator of Step : Where possible, simplify the fraction. Simplifying fractions is explained in more detail to follow is called a decimal fraction and 7/0 is called a common fraction Example : Convert 0.6 to a fraction. Step : Since there are three digits after the decimal point, the fraction will have a denominator of Step : Simplify the fraction
6 8 Equivalent Fractions Equivalent Fractions are equal in value, even though they may look different. You can imagine it like this: A block of wood is cut up into pieces (half) of the wood ( out of the pieces). of the wood ( out of the pieces). 8 of the piece of wood ( out of the 8 pieces). You can see that the fractional amount is the same size no matter how many pieces it is cut into. You can see that can be cut and shown as or 8 Why are they the same? 8 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 multiply or divide numerator by you must also do to the denominator! 8 These are all equivalent, they represent the same amount.
7 9? Example : Express as 6 Step : Look at the fractions you want to change. It will always be changed by multiplying or dividing. To work it out, ask yourself: What do I need to do to change the denominator from a 6 to a? To make something smaller, you need to break it down (divide). 6 Step : Whatever you do to the top, you must do to the bottom. In this example we divide the bottom by (to make it a ), so we will also have to divide the top by to find the equivalent fraction. becomes 6 Example : Express as? Step : To work it out, ask yourself: What do I need to do to change the denominator from a to a? To make something larger, you need to multiply it. x Step : Whatever you do to the top, you must do to the bottom. In this example we multiply the bottom by (to make it a ), so we will also have to multiply the top by to find the equivalent fraction. x x 9 becomes Important to remember: The top and bottom of the fraction must always be a whole number. Hence, the number you pick to divide by must always divide evenly (ie no remainders) for both the top and bottom numbers. You only multiply or divide, never add or subtract, to get an equivalent fraction.
8 60 Simplifying Fractions Simplifying (or reducing) fractions means making the fraction as simple as possible. Why say threesixths ( 6 ) when you can say a half ( )? Sometimes we encounter fractions with extremely large denominators. It is useful to reduce such fractions to a more manageable size. We can use the principle of equivalent fractions to do this. To simplify a fraction, divide the numerator and denominator by the highest number that can divide into both numbers exactly. Example : Simplify 08 Step : Step : When simplifying, the fraction will always be changed by dividing since we want to remove common factors. To work it out, ask yourself: What number goes into both the numerator and the denominator? Whatever you do to the top, you must do to the bottom. In this example the numerator and denominator have been divided by, then and then to get the final answer. You may have been able to do this in less steps Step : Check: Can this be broken down any further? What number goes into both the numerator and the denominator? Only goes into and 9 and this will not change the result. The fraction 08 has been reduced to the equivalent fraction 9.
9 6 Mixed, Improper and Proper Fractions Proper fractions have a numerator smaller than the denominator. These are probably the fractions with which we are most familiar. For example:, 8 and 0 Improper fractions have a numerator larger than the denominator (they are top heavy ). 7 For example:, and Mixed numbers contain both a whole number part and a fraction part. For example: and Example : If we have and ½ buckets of water, we are using mixed numbers. Each whole is made up of halves, so wholes make 6 halves. And then there is another half. So and ½ buckets is the same amount as 7 halves We need to be able to make the conversion between mixed and improper fractions without drawing pictures. 7 To convert a mixed fraction to an improper fraction: Multiply the whole number part by the fraction's denominator. Add that to the numerator Write the result on top of the denominator.
10 6 Example : Express as an improper fraction. Step : The answer will be in fifths (since the denominator is ). First we need to calculate how many fifths are in wholes. x 0 Step : Step : We add the fifths. 0 + ( + ) fifths fifths To convert an improper fraction to a mixed fraction: Divide the numerator by the denominator. Write down the whole number answer Then write down any remainder above the denominator. Example : Express 9 as a mixed number. Step : The question asks you to change 9 quarters into a whole number part and a fraction part. The answer will be in quarters (since the denominator is ). Step : First we need to calculate how many times goes into 9. Hint 9?? You may need to do long division. 9 remainder remainder Step : This means that goes into 9 times with a remainder of, so 9
11 6 Lowest Common Denominator Finding a lowest common denominator allows you to compare the same type of fractions, ones that have the same size pieces. It is useful to be able to find the lowest common denominator of fractions. Once again equivalent fractions will be used. Example : Express 8 and as equivalent fractions with the lowest common denominator. Step : Using equivalent fractions we can see that and Look at the denominators of the fractions. Is there one that is the same? 8 (eighths) and 6 (sixteenths) appear in both, but 8 is smaller (lower). Step : 8 is the lowest common denominator of 8 and. So the answer is 8 and 8 SHORT CUT: Step : You don t need to calculate all the numerators when asked to find the LCD, just look at the denominators. will have equivalent fractions with denominators of 8 8, 6,, (8 times table) will have equivalent fractions with denominators of, 8,, 6, 0 ( times table) The lowest common denominator will be 8. Step : Express both 8 and with a denominator of 8. does not need to be changed 8 X becomes X 8 Step : So the answer is 8 and 8
12 6 Example : Which is the larger fraction: or? Step : Find the lowest common denominator (LCD) of the two fractions. can have equivalent fractions with denominators of, 8,, 6, 0 can have equivalent fractions with denominators of, 6, 9,, The LCD of quarters and thirds is. Step : Express both fractions with the common denominator of : 9, 8 Step : It is now obvious which is the larger of the two fractions: 9 8 means that mathematical symbol meaning greater than When you wish to add, subtract, or determine the larger of two or more fractions, you will use the principle of equivalent fractions with lowest common denominators.
13 6 Addition and Subtraction of Fractions with Common Denominators There are three types of questions involving adding or subtracting fractions: easy, not so easy and messy. The difference is with the denominators. When fractions have the same denominators it means the individual pieces are of equal value. When you add them, you are counting how many quarters of an orange you have, squares of chocolate in each child s hand or sandwich halves. How do we turn this into the rules we need for maths? Example : Imagine an orange cut into quarters. There are two pieces left. Say the question out loud: one quarter plus one quarter gives Example : A chocolate bar is cut into pieces. You want pieces and your friend wants pieces. Step : Count up the pieces you want: + 7 Step : The piece sizes do not change, so the denominator stays as twelfths. 7 Say the question out loud: Five twelfths plus two twelfths gives 7 twelfths. To add or subtract fractions with common denominators Add the numerators and leave the denominator the same. Give your answer as a simplified, mixed number (if needed). Improper fractions are converted to mixed numbers. If the answer to a question was 9, it is more commonly called.
14 66 Addition and Subtraction of Fractions with Different Denominators There is a bit of work to do before the answer can be calculated. To add or subtract fractions with different denominators find a common denominator by using equivalent fractions then follow the rule of similar denominators above. Let s try out an example that we can imagine easily (a good way to work out any problem). Example : Step : Calculate Problem!! The question involves halves and quarters. Solution: Find a common denominator. The denominators for would be,, 6, 8 The denominators for would be, 8,, 6 The common denominator of a half () and a quarter () is. Step : Step : Using equivalent fractions: X becomes X Change both the fractions to quarters Now that the denominators are the same, you can add the numerators, leaving the denominator the same.
15 67 Example : Step : Calculate Find a common denominator. The denominators for fifths () would be, 0,, 0, The denominators for quarters () would be, 8,, 6, 0, The common denominator for and would be 0. Step : X X Using equivalent fractions 8 X 0 X 0 Step : Change the denominator to Now that the denominators are the same, you can subtract the numerators. Example : Step : Put integers and fractions together. ( ) ( ) WHEN ADDING: If integers are involved, add the integers, then add the fractions. Step : Add integers and fractions separately. ( ) ( ) 6 6 6
16 68 Example : 7 6 WHEN SUBTRACTING: If integers are involved, it is better to change the mixed number to an improper fraction before carrying out Multiplication of Fractions Compared to adding and subtracting, multiplying fractions is a breeze! To multiply fractions: Change mixed numbers into improper fractions Multiply the numerators together Multiply the denominators together Give your answer as a simplified, mixed number of means ' Example : What is half of a half? Answer: If you think about halving half an apple, you get the answer of one quarter If you were to write the above as a mathematical statement, you would have:
17 69 Example : Answer: What is one third of a quarter? Look at the diagram below, where a quarter has been divided into three parts One quarter has been divided into equal parts.... and the answer is a twelfth, If you were to write the above as a mathematical statement, you would have: Example : Calculate Step : Multiply the numerators and denominators together. Step : Simplify the fraction What number goes into both 6 and 0? Answer: does In this case, divide the numerator and denominator by NOTE: In the example above you can remove common factors before multiplying. If you do this you don t need to simplify the fraction and the multiplication will be easier. 0
18 70 Example : Calculate Step : Change to improper fractions. Step : Multiply the numerators and denominators together. Step : Simplify the fraction. Step : Give answer as mixed number What number goes into both 9 and? Answer: none, it cannot be Example : Calculate Whole numbers are best written in fraction form Important to remember: You do not need common denominators to multiply fractions
19 7 Reciprocals The reciprocal of a fraction can be found by inverting or flipping the fraction. It is used in division of fractions. Proper fractions: the reciprocal of is Mixed numbers: the reciprocal of an improper fraction 9 so the reciprocal is 9 is found by changing the mixed number to Whole numbers: the reciprocal of is found by writing as so the reciprocal of is Division of Fractions To divide fractions, we use the fact that division is the inverse process to multiplication. To divide fractions: Change mixed numbers into improper fractions Invert (flip) the number after the division sign Change the to x Multiply as before Consider the following examples: Example : How many halves in 7? Think about it, there are halves in every whole. So there must be halves in 7. The answer is bigger than the question! Step : Write the question as a mathematical statement 7 Step : Multiply by the reciprocal 7
20 7 Example : If I have $0 and divide it into equal shares, how much is each share? Answer: 0 If I take a share of $0, how much do I have? Answer: of 0 0 Example : Calculate 6 7 Step : Turn this into a multiplication problem. change the division into a multiplication sign. flip the fraction that follows the division sign Step : Simplify and give answer as a mixed number Example : Calculate 8 Step : Change each number to an improper fraction. Step : Turn into a multiplication problem. Step : Simplify. What number goes into both 7 and 9? Answer: 9 does
21 7 Cancelling When simplifying fractions, we divided both numerator and denominator by common factors. This technique can be used to advantage when multiplying fractions. This process is sometimes called cancelling, and should be continued until no further common factors remain between numerators and denominators. Important to remember: Cancelling cannot be used when additions or subtractions appear in numerator or denominator. Examples: a) b) c)
22 7 Order of Operations and Fractions The rules we used in Topic for order of operations apply to fraction calculations. Example : Step : 9 8 using BEDMAS calculate the brackets first need to get a common denominator when adding fractions Step : 69 7 multiplication is next Fractions and Chance Card games, horse races, rolling dice, insurance and weather forecasts are some of the everyday occurrences which rely on describing in numbers the chance of an event happening. Fractions are ideal for this purpose. Example : Here is a hand of playing cards. If you ask a friend to select one card at random (without looking) what would be the chance of selecting a King? Answer: Your answer should be one out of, which is easily written as the fraction
23 7 Example : Here is another hand of cards. If you selected one at random, what would be the chance of selecting a? Also, what would be the chance of selecting an Ace? You probably had little trouble deciding that the chance of selecting a was out of or and the chance of selecting an Ace was out of or. Chance of an event Number of times the event occurs Number of possible outcomes A list of all possible outcomes is called a sample space. In the last example, the sample space is A, A, A,, so the number of possible outcomes is. Since of these outcomes were Aces, we write: Chance of Ace Example : A coin is thrown 0 times and the following is recorded where H is heads and T is tails. H T T H H T H T T T In this experiment, what is the chance of having a tail appear? Chance of an event Number of times the event occurs Number of possible outcomes The number of times the event occurred was 6, and the sample space was 0, so Chance of throwing a Tail 0 6
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