Fractions

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Fractions
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• 1/2 or ½ or 12

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• 1/2 or ½ or 12
• What do these familiar symbols
• really mean?

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

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• 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)?

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• 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).

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• 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?

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• 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 .

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• 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.

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• 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. (22) / (4 2) 1/2 which is where we
  started.

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• or, similarly,
• 7/14 means How many 14s in 7?

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• 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.

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• 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

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• 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?

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• 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!

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• 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?

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• We can make complicated fractions easier

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

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• 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).

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• e.g. 2358720 / 4717440

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

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• 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

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• 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

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• 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

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• 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

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• 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

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• (24/6) / (48/6) 4/8 1/2

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• (24/6) / (48/6) 4/8 1/2
• (4/4) / (8/4) 1/2

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• 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.

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• 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

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• 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?

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• 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 lets have a go.

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

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• 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

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• Now its tedious and we have to try successive
  prime divisors to both numbers

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• Now its tedious and we have to try successive
  prime divisors to both numbers
• /2 ? X (odd numbers) 10829/29393

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• Now its 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)

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• Now its 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 dont end in 0 or 5)

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• Now its 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 dont end in 0 or 5)
• /7 OK! ? (10829/7) /(29393/7) 1547/4199

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• Now its 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 dont end in 0 or 5)
• /7 OK! ? (10829/7) /(29393/7) 1547/4199
• /11 ? X

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• Now its 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 dont end in 0 or 5)
• /7 OK! ? (10829/7) /(29393/7) 1547/4199
• /11 ? X
• /13 OK! ? (1547/13)/(4199/13) 119/323

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• Now its 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 dont 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

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• Now its 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 dont 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 dont have any factors.

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• So, our complicated fraction means How many
  nineteens in seven? or How can we divide seven
  into nineteen parts?

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• 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.

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• 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.

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• 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.

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• 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

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• 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

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• 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

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• 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

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• 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

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• similarly 3/8

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• similarly 3/8
• 0.375 if we do long division or use a calculator

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

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• 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

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• 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

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• 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.

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• but

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• but, some fractions are awkward

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• but, some fractions are awkward
• 5/7 0.7142857142857.

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• 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.

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• 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 cant do much better.