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Fractions

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Fractions

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    2. Fractions

    3. 1/2 or ½ or 1÷2

    11. 1/2 or ½ or 1÷2 What do these familiar symbols really mean?

    12. How can we divide something (one, on the top line) into two parts (the bottom line)?

    13. How can we divide something (one, on the top line) into two parts (the bottom line)? ­ or How many 2s are there in one (thing)?

    14. We get the same answer (one half) whenever the bottom of this sort of symbol (the denominator) is twice as big as the top half (numerator).

    15. We get the same answer (one half) whenever the bottom of this sort of symbol (the denominator) is twice as big as the top half (numerator). e.g. 2/4 How many 4s in 2?

    16. We get the same answer (one half) whenever the bottom of this sort of symbol (the denominator) is twice as big as the top half (numerator). e.g. 2/4 How many 4s in 2? Same as before, 1/2 .

    17. We get the same answer (one half) whenever the bottom of this sort of symbol (the denominator) is twice as big as the top half (numerator). e.g. 2/4 How many 4s in 2? Same as before, 1/2 . Or, we could have got this by dividing top and bottom of 2/4 by 2.

    18. We get the same answer (one half) whenever the bottom of this sort of symbol (the denominator) is twice as big as the top half (numerator). e.g. 2/4 How many 4s in 2? Same as before, 1/2 . Or, we could have got this by dividing top and bottom of 2/4 by 2. i.e. (2÷2) / (4 ÷ 2) = 1/2 which is where we started.

    19. or, similarly, 7/14 means ‘How many 14s in 7?’

    20. or, similarly, 7/14 means ‘How many 14s in 7?’ and, in the same way as before, ,we can do our dividing process, this time by 7.

    21. or, similarly, 7/14 means ‘How many 14s in 7?’ and, in the same way as before, ,we can do our dividing process, this time by 7. (7 ÷ 7) / (14 ÷ 7) = 1/2

    22. Is it correct (legal?, fair?) to treat the top and bottom of a fraction in the same way, and have the value of the fraction remain unchanged?

    23. Is it correct (legal?, fair?) to treat the top and bottom of a fraction in the same way, and have the value of the fraction remain unchanged? YES!

    24. Is it correct (legal?, fair?) to treat the top and bottom of a fraction in the same way, and have the value of the fraction remain unchanged? YES! So how can we make use of this?

    25. We can make complicated fractions easier

    26. We can make complicated fractions easier if both top and bottom (numerator and denominator) can be divided evenly by the same number(s)

    27. We can make complicated fractions easier if both top and bottom (numerator and denominator) can be divided evenly by the same number(s) (we call this having common factors).

    28. e.g. 2358720 / 4717440

    29. e.g. 2358720 / 4717440 This has top and bottom which can be successively divided by 13, 7, 10, 9, 12, 6 & 4 to give

    30. e.g. 2358720 / 4717440 This has top and bottom which can be successively divided by 13, 7, 10, 9, 12, 6 & 4 to give (2358720/13) / (4717440/13) = 181440/362880 = 1/2

    31. e.g. 2358720 / 4717440 This has top and bottom which can be successively divided by 13, 7, 10, 9, 12, 6 & 4 to give (2358720/13) / (4717440/13) = 181440/362880 = 1/2 (181440/7) / (362880/7) = 25920/51840 = 1/2

    32. e.g. 2358720 / 4717440 This has top and bottom which can be successively divided by 13, 7, 10, 9, 12, 6 & 4 to give (2358720/13) / (4717440/13) = 181440/362880 = 1/2 (181440/7) / (362880/7) = 25920/51840 = 1/2 (25920/10) / (51840/10) = 2592/5184 = 1/2

    33. e.g. 2358720 / 4717440 This has top and bottom which can be successively divided by 13, 7, 10, 9, 12, 6 & 4 to give (2358720/13) / (4717440/13) = 181440/362880 = 1/2 (181440/7) / (362880/7) = 25920/51840 = 1/2 (25920/10) / (51840/10) = 2592/5184 = 1/2 (2592/9) / (5184/9) = 288/576 = 1/2

    34. e.g. 2358720 / 4717440 This has top and bottom which can be successively divided by 13, 7, 10, 9, 12, 6 & 4 to give (2358720/13) / (4717440/13) = 181440/362880 = 1/2 (181440/7) / (362880/7) = 25920/51840 = 1/2 (25920/10) / (51840/10) = 2592/5184 = 1/2 (2592/9) / (5184/9) = 288/576 = 1/2 (288/12) / (576/12) = 24/48 = 1/2

    35. (24/6) / (48/6) = 4/8 = 1/2

    36. (24/6) / (48/6) = 4/8 = 1/2 (4/4) / (8/4) = 1/2

    37. So far we have only used 1/2 as our example but we can use this same method to simplify any fraction, however difficult it looks.

    38. So far we have only used 1/2 as our example but we can use this same method to simplify any fraction, however difficult it looks. eg 54145/146965 = 7/19

    39. So far we have only used 1/2 as our example but we can use this same method to simplify any fraction, however difficult it looks. eg 54145/146965 = 7/19 How do we know what number to divide by?

    40. So far we have only used 1/2 as our example but we can use this same method to simplify any fraction, however difficult it looks. eg 54145/146965 = 7/19 How do we know what number to divide by? This is a bit complicated, but let’s have a go.

    41. Both top and bottom, Numerator and Denominator end in 5, and so we can divide both by 5 and get a whole number answer.

    42. Both top and bottom, Numerator and Denominator end in 5, and so we can divide both by 5 and get a whole number answer. (54145/5) / (146965/5) = 10829/29393

    43. Now it’s tedious and we have to try successive prime divisors to both numbers

    44. Now it’s tedious and we have to try successive prime divisors to both numbers /2 ? X (odd numbers) 10829/29393

    45. Now it’s tedious and we have to try successive prime divisors to both numbers /2 ? X (odd numbers) 10829/29393 /3 ? X (digit sum not a multiple of 3)

    46. Now it’s tedious and we have to try successive prime divisors to both numbers /2 ? X (odd numbers) 10829/29393 /3 ? X (digit sum not a multiple of 3) /5 ? X (they don’t end in 0 or 5)

    47. Now it’s tedious and we have to try successive prime divisors to both numbers /2 ? X (odd numbers) 10829/29393 /3 ? X (digit sum not a multiple of 3) /5 ? X (they don’t end in 0 or 5) /7 OK! ? (10829/7) /(29393/7) = 1547/4199

    48. Now it’s tedious and we have to try successive prime divisors to both numbers /2 ? X (odd numbers) /3 ? X (digit sum not a multiple of 3) /5 ? X (they don’t end in 0 or 5) /7 OK! ? (10829/7) /(29393/7) = 1547/4199 /11 ? X

    49. Now it’s tedious and we have to try successive prime divisors to both numbers /2 ? X (odd numbers) /3 ? X (digit sum not a multiple of 3) /5 ? X (they don’t end in 0 or 5) /7 OK! ? (10829/7) /(29393/7) = 1547/4199 /11 ? X /13 OK! ? (1547/13)/(4199/13) = 119/323

    50. Now it’s tedious and we have to try successive prime divisors to both numbers /2 ? X (odd numbers) /3 ? X (digit sum not a multiple of 3) /5 ? X (they don’t end in 0 or 5) /7 OK! ? (10829/7) /(29393/7) = 1547/4199 /11 ? X /13 OK! ? (1547/13)/(4199/13) = 119/323 /17 OK! ? (119/17)/(323/17) = 7/19

    51. Now it’s tedious and we have to try successive prime divisors to both numbers /2 ? X (odd numbers) /3 ? X (digit sum not a multiple of 3) /5 ? X (they don’t end in 0 or 5) /7 OK! ? (10829/7) /(29393/7) = 1547/4199 /11 ? X /13 OK! ? (1547/13)/(4199/13) = 119/323 /17 OK! ? (119/17)/(323/17) = 7/19 Why have we stopped? Because both 7 and 19 are prime numbers and so they don’t have any factors.

    52. So, our complicated fraction means ‘How many nineteens in seven?’ or ‘How can we divide seven into nineteen parts?’

    53. So, our complicated fraction means ‘How many nineteens in seven?’ or ‘How can we divide seven into nineteen parts?’ It is useful to have some idea about how big a fraction is in simple terms.

    54. So, our complicated fraction means ‘How many nineteens in seven?’ or ‘How can we divide seven into nineteen parts?’ It is useful to have some idea about how big a fraction is in simple terms. 7/19 is a bit awkward, but it will be a bigger than one third since both (6/18) and (7/21) equal a third.

    55. Using decimals is just a more convenient way of expressing fractions, where we choose a particular denominator – always a simple power of ten. i.e. 10, 100, 1000, 10000 etc.

    56. Using decimals is just a more convenient way of expressing fractions, where we choose a particular denominator – always a simple power of ten. i.e. 10, 100, 1000, 10000 etc. 1/2 = 0.5000…

    57. Using decimals is just a more convenient way of expressing fractions, where we choose a particular denominator – always a simple power of ten. i.e. 10, 100, 1000, 10000 etc. 1/2 = 0.5000… = 5/10

    58. Using decimals is just a more convenient way of expressing fractions, where we choose a particular denominator – always a simple power of ten. i.e. 10, 100, 1000, 10000 etc. 1/2 = 0.5000… = 5/10 = 50/100

    59. Using decimals is just a more convenient way of expressing fractions, where we choose a particular denominator – always a simple power of ten. i.e. 10, 100, 1000, 10000 etc. 1/2 = 0.5000… = 5/10 = 50/100 = 500/1000

    60. Using decimals is just a more convenient way of expressing fractions, where we choose a particular denominator – always a simple power of ten. i.e. 10, 100, 1000, 10000 etc. 1/2 = 0.5000… = 5/10 = 50/100 = 500/1000 = 5000/10000 etc

    61. similarly 3/8

    62. similarly 3/8 = 0.375 if we do long division or use a calculator

    63. similarly 3/8 = 0.375 = 375/1000 (as many zeros as there are numbers after the decimal point.)

    64. similarly 3/8 = 0.375 = 375/1000 (as many zeros as there are numbers after the decimal point.) 375/1000: Divide by 5 ? (375/5)/(1000/5) = 75/200

    65. similarly 3/8 = 0.375 = 375/1000 (as many zeros as there are numbers after the decimal point.) 375/1000: Divide by 5 ? (375/5)/(1000/5) = 75/200 Divide by 5 again ? (75/5)/(200/5) = 15/40

    66. similarly 3/8 = 0.375 = 375/1000 (as many zeros as there are numbers after the decimal point.) 375/1000: Divide by 5 ? (375/5)/(1000/5) = 75/200 Divide by 5 again ? (75/5)/(200/5) = 15/40 Divide by 5 again ? (15/5)/(40/5) = 3/8.

    67. but

    68. but, some fractions are awkward

    69. but, some fractions are awkward 5/7 = 0.7142857142857…….

    70. but, some fractions are awkward 5/7 = 0.7142857142857……. is not possible to treat in this way unless we allow for an infinite number of decimal places.

    71. but, some fractions are awkward 5/7 = 0.7142857142857……. is not possible to treat in this way unless we allow for an infinite number of decimal places. We could use an approximation of 714/1000 = 357/500 ~ 355/500 = 71/100 = 0.71, but we can’t do much better.

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