Fractions A fraction is any part of a group, number or whole. Fractions are always written as Numerator Denominator A unitary fraction is one where the numerator is always 1 e.g 1 1 1 1 1...etc... 2 3 4 5 6 1
Equivalent Fractions Any fraction can be written in different ways as long as you multiply or divide the numerator and denominator by the same thing at the same time then the fractions are equivalent The most common way to calculate equivalent fractions is to simplify When simplifying fractions you need to always ask yourself 'what number can I divide into both the numerator and denominator without a remainder?' e.g simplify the following fraction to its simplest form: 16 = 4 20 5 4 4 Both 16 and 20 are in the 4 times tables so divide them both by 4 at the same time to simplify 2
Sometimes you can simplify a fraction in one step or you can simplify in more than one step e.g Simplify 28 to its simplest form 42 2 7 28 = 14 = 2 42 21 3 2 7 3
To begin with you need to have a grasp of what basic fractions look like. One way to think of a fraction is to think of them as 'out of' e.g 1 can mean '1 out of 4' 4 or 5 can mean '5 out of 8' 8 Using this we can find a fraction of a group of objects e.g circle 1 of the stars below this means '1 out of 3' 3 so we circle every 1 in 3 stars. 4
You then move on from showing fractions of numbers of objects to splitting a shape into a fraction You need to remember that when you have a fraction you divide by the denominator e.g show what 1 of the following shape looks like 5 To find 1 you split the 5 shape into 5 equal sized parts You then shade 1 of them in. 5
The next step is to shade in a fraction of a shape where the numerator is not 1 the simplest way of doing this is by splitting the shape according to its denominator and then shading in according to the numerator e.g shade in 5 of the following shape 6 First as the denominator of the fraction is 6 you split the shape into 6 equal sized parts Next as the numerator is 5 you shade in 5 of the parts 6
Fractions of amounts When finding a fraction of any amount we use the following method: divide the amount by the denominator (bottom number) FIRST then multiply your answer by the numerator (top number). e.g1 calculate the following: 4 of 84 7 Step 1: DIVIDE the amount by the denominator 84 7 = 12 Step 2: MULTIPLY your answer by the numerator 12 x 4 = 48 so the answer is 48 7
e.g2 calculate the following which involves a decimal answer: 3 of 62KG 5 Divide: 62 5 =? we need to use short division for this! 1 2 4 1 2 5 6 2 0 remember: 62.0 is the same as 62. You can carry remainders over until you get the answer. Just dont forget to put the decimal point into your answer! Multiply: 12.4 x 3 = 37.2 (use geolosia for this) so the answer = 37.2KG 8
Sometimes when you have to shade in a fraction of a shape it is not always easy to split it into pieces visually so we need to use a mathematical method. This uses the same method as fractions of amounts e.g Shade in 5 of the rectangle below 6 This rectangle is made up of 48 equal sized squares So we need to calculate 5 of 48 6 5 of 48 6 Divide first: 48 6 = 8 then Multiply: 8 x 5 = 40 So, we need to shade in 40 squares!! 9
Ordering fractions remember: numerator denominator e.g1 order the following fractions smallest to largest 7 2 8 9 10 10 10 10 2 7 8 9 10 10 10 10 Simply order the fractions according to the numerators The above example was fairly easy as all the denominators were the same (we call this common denominators) When we order fractions that have different denominators we need to make them all the same to be able to order our fractions. We need to find a Common denominator A common denominator is a number into which all the denominators will divide into without remainder. 10
e.g2 order the following fractions smallest to largest 3 10 16 3 4 12 20 5 We need to find a common denominator that all 4 of the denominators will divide into!! To do this we look at the largest denominator and write out its times tables 20, 40, 60, 80, 100... 4, 12, 20 and 5 can all be divided into 60 We now need to change all of the above fractions to ones with a common denominator. remember: what you do to the denominator you must also do to the numerator. x15 3 10 16 3 the fractions become 4 x5 x3 x12 12 20 5 45 x15 x5 50 48 x3 36 60 60 60 60 x12 So now that all the denominators are equal, it is easier to order the original fractions Answer = 3 3 16 10 5 4 20 12 11
Adding and subtracting Fractions At its most basic, adding and subtracting fractions involves adding or subtracting fractions that have the same denominators e.g Calculate: 8 5 10 10 = 8 5 10 = 3 10 You simply add or subtract the numerators leave the denominators alone!! After this you need to be able to add or subtract fractions that have different denominators. 12
Adding and subtracting fractions with different denominators When adding or subtracting fractions you need to make sure your fractions have common denominators (look at ordering fractions for a reminder) e.g1 calculate the following: 2 + 3 We first need to find a common denominator. 7 8 We do this by multiplying the two denominators together. 7 x 8 = 56 so our 'new' denominator is 56 Now we need to change each fraction to its equivalent fraction with our new denominator x8 2 = 16 7 56 x8 x7 3 = 21 8 56 x7 Remember: what you do to the denominator of each fraction, you must also do to the numerator! This gives us 16 + 21 56 56 so the answer = 37 56 remember: you only add or subtract the numerators (top numbers), the denominator in the answer is the same as the common denominator When subtracting fractions you follow the same method and at the end you subtract the numerators instead. 13
Multiplying Fractions When multiplying fractions we use the following method: e.g calculate 5 x 3 6 4 We simply multiply the numerators together, then you multiply the denominators together so 5 x 3 = 5 x 3 = 15 6 4 6 x 4 24 We can simplify this 15 = 5 24 8 14
Dividing fractions When dividing fractions we use the following method: e.g calculate 6 5 8 9 We invert (flip round) the fraction we are dividing by and change the calculation into a multiplication this gives us 6 x 9 = 54 8 5 40 We can simplify this to 27 20 15