Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
List of important publications in mathematics wikipedia , lookup
John Wallis wikipedia , lookup
Large numbers wikipedia , lookup
Location arithmetic wikipedia , lookup
Collatz conjecture wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Vincent's theorem wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Positional notation wikipedia , lookup
Fractions MTH 3-07 a, b, c Fraction Terms 3 4 numerator denominator The denominator describes how many equal parts the shape has been cut into. The numerator describes how many of these parts are shaded. Equal Fractions The pairs of fractions shown in the diagram below are equal. 1 3 2 6 1 2 3 6 3 6 4 8 To create equal fractions numerically we use the following method: 1 4 12 1 4 12 x3 Work out what the denominator has been multiplied by (i.e. how many 4’s go into 12?) x3 1 3 4 12 Now multiply the numerator by the same number. x3 Similarly we have : 5 6 24 which gives x4 5 6 24 x4 and so 5 20 6 24 x4 And: x8 7 9 72 which gives 7 9 72 and so x8 7 56 9 72 x8 Exercise Copy and complete these equal fractions: a) 2 3 12 b) 4 5 30 c) 6 7 28 d) 3 8 40 e) 5 11 66 f) 7 12 36 g) 6 7 56 Simplifying Fractions This works in a similar way to equal fractions, except this time we are dividing the numerator and denominator of the fractions by the same number to change the fraction into its simplest form. e.g. Reduce 15 to its simplest form. 18 First we find the largest number which divides into 15 and 18. This is the highest common factor of the two numbers. In this case, 15 and 18 both divide by 3. ÷3 15 5 18 6 So we have: ÷3 Similarly we have: 27 45 ÷9 27 3 45 5 and the highest common factor of 27 and 45 is 9 so ÷9 Sometimes the highest common factor is not obvious, so we may have to repeat the process if we have not found the largest number, for example: Simplify ÷2 48 . 72 ÷2 ÷6 48 24 12 2 72 36 18 3 ÷2 ÷2 ÷6 This could be simplified in any of these ways: ÷6 ÷4 or 48 8 2 or 72 12 3 ÷6 ÷4 ÷8 ÷3 48 6 2 72 9 3 ÷8 ÷3 ÷ 24 or 48 2 72 3 ÷ 24 Obviously, it is preferable to find the highest common factor in the first place if we can since this takes less time. Exercise Reduce each of these fractions to their simplest form: a) 15 20 b) 12 16 c) 18 24 d) 14 21 e) 30 36 f) 18 45 g) 42 48 Comparing Fractions Knowing how to change the denominator of a fraction is important so that fractions can be ordered, compared, added and subtracted. 5 1 1 For instance, to work out whether or is bigger, we can convert into twelfths as follows: 12 4 4 x3 1 3 4 12 x3 Now we can compare the two fractions: 1 3 5 5 and so we can see that is larger. 4 12 12 12 e.g.1 Which is bigger, 7 5 or ? 9 6 Here we have to change both fractions so that their denominators are the same – here the first number that 6 and 9 both divide into is 18 so we change them into eighteenths as follows: 7 9 18 and x3 x2 So we have: x2 7 14 9 18 x3 and x2 And now we can see that 5 6 18 5 15 6 18 x3 5 is slightly bigger. 6 e.g. 2 Put this list of fractions in order of size, starting with the smallest. 2 5 7 3 , , , 3 6 12 4 In order to compare the size of the fractions we must change each one so that all of the denominators are the same. We need to find the lowest common denominator of our fractions i.e. the lowest common multiple of 3, 6, 12 and 4 = 12 2 8 3 12 5 10 6 12 x4 x2 3 9 4 12 7 12 x3 So now we can order the fractions: 7 2 3 5 , , , 12 3 4 6 Exercise 1. By using equal fractions, decide which of these fractions is bigger: a) 5 17 or 6 18 2. Put these fractions in order of size, starting with the smallest: a) b) 3 7 13 , , 5 10 20 b) 2 5 or 9 27 c) 7 8 or 10 15 5 19 11 , , 6 24 12 c) d) 3 7 or 4 9 4 11 5 9 , , , 5 15 6 10 Fraction of a Quantity A reminder: To find a fraction of an amount we divide by the denominator and then multiply the answer by the numerator as follows: 4 of 35 7 = 35 ÷ 7 x 4 =5x4 = 20 Note that this calculation could also be written as 35 x 4 . 7 For example, which is bigger? 5 of 84 12 5 of 84 12 = 84 ÷ 12 x 5 =6 x 5 = 30 or 72 x and 72 x 4 9 4 9 = 72 ÷ 9 x 4 =8x4 = 32 So 72 x 4 is bigger. 9 Examples 1. a) Evaluate: 2 of 36 3 f) 60 x 3 4 b) 4 of 72 9 g) 48 x c) 5 6 7 of 56 8 h) 81 x 7 9 d) i) 3 of 265 5 3 x 104 8 e) j) 4 of 343 7 4 x 612 9 Adding and Subtracting Fractions If the fractions have the same denominator then simply add/subtract the numerators of the fractions as follows: 2 3 5 7 7 7 Note that the denominators do not change. Likewise: 8 5 3 11 11 11 However, in order to either add or subtract fractions, all fractions must have the same denominator. In this case, the method of equal fractions must be used to convert fractions where necessary. For example: 2 3 5 10 Here we have to change 2 4 2 into tenths. We have 5 5 10 x2 2 3 5 10 4 3 10 10 7 10 Sometimes we have to convert both fractions, such as: 2 1 3 8 Look at the denominators – the first number that 3 and 8 both divide into is 24. This is known as the lowest common denominator and is the lowest common multiple of the two denominators. 2 16 1 3 and 3 24 8 24 x8 2 1 3 8 16 3 24 24 19 24 so x3 The same process applies to subtraction, so: 5 2 9 5 The lowest common denominator of 5 and 9 is 45. 5 25 2 18 and 9 45 5 45 x5 5 2 9 5 25 18 45 45 7 45 so x9 Exercise 1. Copy and complete the calculations below: a) 2 1 5 5 b) h) 1 1 2 3 i) 8 7 9 9 c) 2 1 5 15 d) 7 1 9 3 e) 3 2 10 5 f) 2 1 5 3 j) 2 1 3 4 k) 4 1 7 8 l) 2 5 3 7 m) 17 4 18 9 9 3 11 4 g) 4 2 7 21 n) 3 2 8 5 Mixed Numbers Mixed numbers are numbers which contain a mixture of whole parts and fractions e.g. 4 3 5 Some fractions have a numerator that is larger than the denominator – these are called improper or top heavy fractions. It is useful to be able to convert between mixed numbers and improper fractions. Note: One whole can be written as : 1 = 2 3 4 5 ..... 2 3 4 5 Mixed Numbers to Improper: 3 Convert 4 to an improper fraction. 5 5 For every whole part we have so here we could write : 5 3 5 5 5 5 3 23 4 = + + + + = 5 5 5 5 5 5 5 23 5 It is quicker to realise that in 4 whole parts there will be 4 x 5 = 20 fifths, giving a total of when we add on the extra 3 . 5 So, for example, 2 6 x 7 2 44 6 = 7 7 7 and 9 3 9 8 3 75 . 8 8 8 Examples: 1. Convert these mixed numbers to improper fractions: a) 3 2 5 b) 2 1 7 c) 8 3 7 d) 6 5 6 e) 10 4 9 f) 5 7 8 g) 12 5 11 Improper to Mixed Number: To convert back to a mixed number we need to know how many wholes can be made with the given fractions: i.e. 7 3 3 1 1 2 3 3 3 3 3 or 7÷3=2r1 so 7 1 2 3 3 Here we can see that there are 2 wholes since 3 can divide into 7 twice. There would be a remainder of 1 third. 19 4 4 4 4 3 3 4 4 4 4 4 4 4 4 or 19 ÷ 4 = 4 r 3 so 19 3 4 4 4 So to convert an improper fraction into a mixed number we divide the numerator by the denominator. The remainder of this is the fractional part of the mixed number. Examples: 1. a) Convert each improper fraction back into a mixed fraction: 11 2 b) 8 3 c) 13 4 d) 21 5 e) 36 7 f) 50 8 g) 45 6 Adding and Subtracting with Mixed Numbers – Extension Work If we want to add or subtract mixed numbers we always deal with the whole number part first, and then add/subtract the fractions as before, remembering that they must have the same denominators first. For example: 2 1 6 3 5 2 4 5 9 10 10 9 9 10 7 2 5 2 8 3 21 16 3 24 24 5 3 24 Note: 6 + 3 = 9 Note: 5 – 2 = 3 With both addition and subtraction there can occasionally be an extra complication as follows: 1 3 4 2 2 5 6 5 6 10 10 11 6 10 1 6 1 10 1 7 10 1 1 8 5 3 2 2 3 3 6 6 6 2 3 2 6 6 6 5 2 6 Convert the improper fraction here We can’t take 3 from 2 here. We need to break down one of the wholes into sixths. Examples: 1. Copy and complete these fraction calculations: a) 2 1 2 1 7 7 b) 2 2 9 3 5 5 c) 7 8 10 4 9 9 d) 5 1 3 2 2 11 e) 1 5 7 4 4 8 f) 2 2 9 1 9 3 g) 4 3 8 4 7 5 h) 2 2 5 -3 5 3