- 1. Fractions 1/855/6011/121 2/10 11/ 12
2. What is a fraction? Loosely speaking, a fraction is a
quantity that cannot be represented by a whole number.Why do we
need fractions?Consider the following scenario.Can you finish the
whole cake?If not, how many cakes did youeat?1 is not the
answer,neither is 0.This suggest that we need anew kind of number.
3. Definition:A fraction is an ordered pair of whole numbers, the
1st oneis usually written on top of the other, such as or .
anumerator bdenominatorThe denominator tells us how many congruent
piecesthe whole is divided into, thus this number cannot be 0.The
numerator tells us how many such pieces arebeing considered. 4.
Examples:How much of a pizza do we have below? we first need to
know the size of the original pizza.The blue circle is our whole.-
if we divide the whole into 8congruent pieces,- the denominator
would be 8.We can see that we have 7 ofthese pieces.Therefore the
numerator is 7,and we have 7 of a pizza. 8 5. Equivalent fractionsa
fraction can have many different appearances, theseare called
equivalent fractions In the following picture we have of a
cakebecause the whole cake is divided into two congruentparts and
we have only one of those parts.But if we cut the cake into
smallercongruent pieces, we can see that1 2=2 4Or we can cut the
original cakeinto 6 congruent pieces, 6. Equivalent fractionsa
fraction can have many different appearances, theseare called
equivalent fractions Now we have 3 pieces out of 6 equal pieces,
butthe total amount we have is still the same.Therefore, 1 2 3 = =
2 4 6If you dont like this, we can cutthe original cake into 8
congruentpieces, 7. Equivalent fractionsa fraction can have many
different appearances, theyare called equivalent fractions then we
have 4 pieces out of 8 equal pieces, but thetotal amount we have is
still the same. Therefore,123 4= = =246 8We can generalize this to
11 n= whenever n is not 0 22 n 8. How do we know that two fractions
are the same? we cannot tell whether two fractions are the same
untilwe reduce them to their lowest terms.A fraction is in its
lowest terms (or is reduced) if wecannot find a whole number (other
than 1) that can divideinto both its numerator and
denominator.Examples: 6 is not reduced because 2 can divide into10
both 6 and 10. 35is not reduced because 5 divides into 40both 35
and 40. 9. How do we know that two fractions are the same?More
examples:110 is not reduced because 10 can divide into260 both 110
and 260.8is reduced. 15 11is reduced 23To find out whether two
fraction are equal, we need toreduce them to their lowest terms.
10. How do we know that two fractions are the same?Examples:Are
14and 30 equal? 21 4514reduce14 7 22121 7 330reduce30 5 6 reduce 6
3 24545 5 99 3 3Now we know that these two fractions are
actuallythe same! 11. How do we know that two fractions are the
same?Another example: 24 30Are andequal? 40 4224 reduce 24 2 12
reduce 12 4 34040 2 2020 4 5 30reduce 30 65 42 42 67This shows that
these two fractions are not the same! 12. Improper Fractions and
Mixed NumbersAn improper fraction is a fraction 5with the numerator
larger than orequal to the denominator.3Any whole number can be4
7transformed into an improper 4 , 1fraction.1 7A mixed number is a
whole3number and a fraction together 2 7An improper fraction can be
converted to a mixednumber and vice versa. 13. Improper Fractions
and Mixed NumbersConverting improper fractions intomixed numbers:5
2- divide the numerator by the denominator 1- the quotient is the
leading number, 3 3- the remainder as the new numerator. 7 311
1More examples:1 ,2 4 4 5 5Converting mixed numbers 3 2 7 3 17into
improper fractions. 2 7 77 14. How does the denominator control
afraction? If you share a pizza evenly among two people, you will
get 1 2 If you share a pizza evenly among three people, you will
get13 If you share a pizza evenly among four people, you will get14
15. How does the denominator control afraction?If you share a pizza
evenly among eightpeople, you will get only 18Its not hard to see
that the slice you getbecomes smaller and smaller.Conclusion:The
larger the denominator the smaller thepieces, and if the numerator
is kept fixed, the largerthe denominator the smaller the fraction,
a ai.e. whenever b c. b c 16. Examples: 2 22Which one is larger, or
? Ans : 7 55 8 88Which one is larger,or? Ans : 2325 2341 4141Which
one is larger, or ? Ans : 135267 135 17. How does the numerator
affect a fraction?Here is 1/16 , here is 3/16 , here is 5/16 , Do
you see a trend? Yes, when the numerator gets larger we have more
pieces. And if the denominator is kept fixed, the larger numerator
makes a bigger fraction. 18. Examples:7 57Which one is larger,or?
Ans : 1212 12 8 13 13Which one is larger,or? Ans : 2020 2045
6363Which one is larger, or ? Ans : 100100 100 19. Comparing
fractions with differentnumerators and different denominators.In
this case, it would be pretty difficult to tell just fromthe
numbers which fraction is bigger, for example 3 5 812This one has
less pieces This one has more piecesbut each piece is larger but
each piece is smallerthan those on the right. than those on the
left. 20. 3 5 812One way to answer this question is to change the
appearanceof the fractions so that the denominators are the same.In
that case, the pieces are all of the same size, hence thelarger
numerator makes a bigger fraction.The straight forward way to find
a common denominator is tomultiply the two denominators together:3
3 123655 8 40and8 8 1296 12 12 8 96Now it is easy to tell that 5/12
is actually a bit bigger than 3/8. 21. A more efficient way to
compare fractionsWhich one is larger,From the previous example, we
see that we dontreally have to know what the common denominator 75
or ? turns out to be, all we care are the numerators.118 Therefore
we shall only change the numerators bycross multiplying. 751187 8 =
56 11 5 = 55 75 Since 56 > 55, we see that118 This method is
called cross-multiplication, and make sure that you remember to
make the arrows go upward. 22. Addition of Fractions- addition
means combining objects in two or more sets- the objects must be of
the same type, i.e. we combine bundles with bundles and sticks with
sticks.- in fractions, we can only combine pieces of the same size.
In other words, the denominators must be the same. 23. Addition of
Fractions with equal denominatorsExample: 1 38 8+ = ?Click to see
animation 24. Addition of Fractions with equal denominatorsExample:
1 38 8 += 25. Addition of Fractions with equal denominatorsExample:
1 3 8 8 += (1 3) which can be simplified to 1 The answer is 8
2(1+3) is NOT the right answer because the denominator tells(8+8)us
how many pieces the whole is divided into, and in thisaddition
problem, we have not changed the number of pieces inthe whole.
Therefore the denominator should still be 8. 26. Addition of
Fractions with equal denominatorsMore examples2 1 35 5 5 6 7133 110
10 10 10 6 81415 15 15 27. Addition of Fractions withdifferent
denominatorsIn this case, we need to first convert them into
equivalentfraction with the same denominator.Example: 1 2 3 5An
easy choice for a common denominator is 35 = 151 1 5 5 2 2 3 63 3
515 5 5 3 15Therefore, 12 5611 3515 1515 28. Addition of Fractions
withdifferent denominatorsRemark: When the denominators are bigger,
we need tofind the least common denominator by factoring.If you do
not know prime factorization yet, you have tomultiply the two
denominators together. 29. More Exercises:3 1 3 2 1 6 1 6 17= = =4
8 4 2 8 8 88 83 2 3 7 2 5 21 10 21 10 31= = =5 7 5 7 7 5 35 35
35355 4 5 9 4 6 45 24= =6 9 6 9 9 6 54 5445 24 69 15=15454 54 30.
Adding Mixed Numbers 1 3 1 3Example: 3 2 3 2 5 5 5 5 1 3 3 2 5 5 1
3 55 4 5 5 4 5 5 31. Adding Mixed NumbersAnother Example: 4 3 4 3 2
1 2 1 7 8 7 8 4 3 2 1 7 8 4 8 3 7 356 5353 3 3 5656 32. Subtraction
of Fractions- subtraction means taking objects away.- the objects
must be of the same type, i.e. we can only take away apples from a
group of apples.- in fractions, we can only take away pieces ofthe
same size. In other words, the denominators must be the same. 33.
Subtraction of Fractions with equal denominatorsExample: 113 12 12
3 from 11 This means to take away12121112 (Click to see animation)
34. Subtraction of Fractions with equal denominatorsExample: 11 3
12 123 from 11 This means to take away 1212 11 12 35. Subtraction
of Fractions with equal denominatorsExample: 11 3 12 123 from 11
This means to take away 1212 36. Subtraction of Fractions with
equal denominatorsExample: 11 3 12 123 from 11 This means to take
away 1212 37. Subtraction of Fractions with equal
denominatorsExample: 11 3 12 123 from 11 This means to take away
1212 38. Subtraction of Fractions with equal denominatorsExample:
11 3 12 123 from 11 This means to take away 1212 39. Subtraction of
Fractions with equal denominatorsExample: 11 3 12 123 from 11 This
means to take away 1212 40. Subtraction of Fractions with equal
denominatorsExample: 11 3 12 123 from 11 This means to take away
1212 41. Subtraction of Fractions with equal denominatorsExample:
11 3 12 123 from 11 This means to take away 1212 42. Subtraction of
Fractions with equal denominatorsExample: 11 3 12 123 from 11 This
means to take away 1212 43. Subtraction of Fractions with equal
denominatorsExample: 11 3 12 123 from 11 This means to take away
1212 44. Subtraction of Fractions with equal denominatorsExample:
11 3 12 123 from 11 This means to take away 1212 45. Subtraction of
Fractions with equal denominatorsExample: 11 3 12 123 from 11 This
means to take away 1212 46. Subtraction of Fractions with equal
denominatorsExample: 11 3 12 123 from 11 This means to take away
1212 47. Subtraction of Fractions with equal denominatorsExample:
11 3 12 123 from 11 This means to take away 1212 48. Subtraction of
Fractions with equal denominatorsExample: 11 3 12 123 from 11 This
means to take away 1212 49. Subtraction of Fractions with equal
denominatorsExample: 11 3 12 123 from 11 This means to take away
1212 50. Subtraction of Fractions with equal denominatorsExample:
11 3 12 123 from 11 This means to take away 1212 51. Subtraction of
Fractions with equal denominatorsExample: 11 3 12 123 from 11 This
means to take away 1212 52. Subtraction of Fractions with equal
denominatorsExample: 11 3 12 123 from 11 This means to take away
1212 53. Subtraction of Fractions with equal denominatorsExample:
11 3 12 123 from 11 This means to take away 1212 54. Subtraction of
Fractions with equal denominatorsExample:11 312 123 from 11 This
means to take away 1212Now you can see that there are only 8 pieces
left,therefore 11 311 3 82 12 1212 123 55. Subtraction of
FractionsMore examples:15 7 15 7 8 116 16 16 16 2646 9 4 7 542854
28 26797 9 9 7 63636363 7 11 7 23 11 10 7 23 11 10161 11010 2310 23
23 10 10 2323051Did you get all the answers right? 230 56.
Subtraction of mixed numbersThis is more difficult than before, so
please take notes. 1 1Example: 3 1 4 2Since 1/4 is not enough to be
subtracted by 1/2, we better convertall mixed numbers into improper
fractions first.1 1 3 4 11 2 13 14 24 2133 42136 4473 144
