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ACADEMIC STUDIES
MATH
Support Materials and Exercises for
FRACTIONS Book 2
The Addition and Subtraction of Fractions
SPRING 1999
13%
13
1 17%2 3
736%
26
2 35%3 3
4
25%
15'
35
2
ADDING FRACTIONS WITH COMMON DENOMINATORS
Like whole numbers (1, 77, 2000), we can add, subtract, multiplyand divide fractions.
To add fractions that have the same denominators just add the numerators and keep the same denominator that you started with.
.....The answer is called the SUM.
Example:
two fifths + one fifth = three fifths
Have you noticed?... Like whole numbers, we are finding out howmuch of something we have altogether. When we add fractions,we are asking ourselves...how much of the WHOLE do we havealtogether? In this case, we want to know how many fifths wehave altogether.
24
14
34
24%
14'
34
25%
25'
45
14%
14'
24'
12
58%
18'
68'
34
3
Example 2: How many fourths of this circle have beenshaded?
of this circle has been shaded one way,
and has been shaded another way.
Altogether, of the circle has been shaded.
2 + 1 = 3 (numerator), and keep the original
denominator (4).
two quarters + one quarter = three quarters
Example 3:
2 + 2 = 4 (numerator), and keep the originaldenominator (5).
two fifths + two fifths = four fifths
REMEMBER: YOUR ANSWER MAY NOT BE IN THESIMPLEST FORM, SO YOU WILL HAVE TO REDUCE IT!!
34%
24'
54
54'1 1
4
1 14
23
23
43
43'1 1
3
4
CHANGING IMPROPER FRACTIONS TO MIXED NUMBERSWHEN ADDING FRACTIONS WITH COMMON DENOMINATORS.
Sometimes when fractions are added, the sum is an improperfraction. This means of course that you end up with more thanone whole “thing” as an answer. This “improper” fraction must bechanged to a mixed number. This is its reduced form.
Take a look...
Example 1
To arrive at the answer , just divide 4 into 5.
“Four goes into five once, (this will be the whole number),with one left over (this will be the numerator of the newfraction) and you keep the same denominator that youstarted with.”
What a mouthful!
Example 2
cup of flour + cup of flour = cups of flour total.
107%
47'
147'2
34%
14'
44'1
5
“Four goes into five once with one left over, and keep the same denominator, four.
Sometimes, an improper fraction does not have aremainder. That is, when we divide, we end up with awhole number only.
Watch!
Example 3
Divide. “Seven goes into fourteen twice with no remainder.In other words, there are exactly two groups of seven in
fourteen.”
Example 4
Divide. “Four goes into four once with no remainder.
25%
25
38%
58
54%
34
1012
%1
12
47%
37
35%
15
49%
19
24%
14
28%
38
68%
18
36%
26
25%
35
23%
13
612
%3
12
17%
47
48%
38
86%
46
57%
17
25%
15
410
%2
10
411
%3
111014
%3
14
3040
%5
401525
%4
25
313
%9
13100200
%50
200
6
Exercise 1
Add these fractions and reduce your answers if needed.
1) 14)
2) 15)
3) 16)
4) 17)
5) 18)
6) 19)
7) 20)
8) 21)
9) 22)
10) 23)
11) 24)
12) 25)
13) 26)
64
125
2012
607
226
144
256
325
85
134
125
167
4030
106
1812
72
10040
5513
152
4510
73
3312
2013
247
95
4410
14%?' 4
4?% 3
7'
67
28%
78'? ?% 5
9'1
410
%?' 710
23%
13'? 3
8%?' 3
8?% 1
6'
12
reduced
28%?' 3
4reduced ?% 4
10'
35
reduced ?% 2030
'1
612
%?' 23
reduced ?% 49'
89
1020
%1320
'? ?% 37'1 1
7reduced
7
Exercise 2
Change these improper fractions to mixed numbers. Reduce tolowest terms.
26) 2) 3) 4) 5) 6)
7) 8) 9) 10) 11) 12)
13) 14) 15) 16) 17) 18)
19) 20) 21) 22) 23) 24)
25) 26)
Exercise 3
Complete the equation by filling in the missing fraction. Reduceto lowest terms.
1) or 1 2) 3) 4)
5) 6) 7) 8)
9) 10) 11)
12) 13) 14) 15)
14%
24
614
%4
14
25%
15
13%
13
610
%4
1015%
15%
15
37%
27
2027
%1
27
29%
59
38%
38
510
%3
1035%
15
512
%3
1246%
16
812
%2
1238%
58
8
Exercise 4
Connect the pairs of equivalent fractions.
27
37
57
27
38
9
ADDING FRACTIONS WITH DIFFERENT DENOMINATORS
Fractions with different denominators cannot be added until bothdenominators have been changed to the same number.
The fractions and can be added, because the denominators
are the same. By adding the numerators and keeping the same
denominator, we arrive at the answer .
The fractions and cannot be added yet, because the
denominators are different.
To add fractions with different denominators, follow these
simple steps:
a) Find a common denominator for all fractions by creating
equivalent fractions.
b) Add the new numerators together.
c) Put this sum over the new denominator.
d) Reduce your answer if possible.
14%
35
14'
?20
14'
520
35'
?20
35'
1220
1720
1720
10
Example 1:
step1: Find a common denominator for 4 and 5 by creating
equivalent fractions.
The common denominator is 20.
step 2: Add the new numerators together. 5 + 12 = 17
step 3: Put the new numerator over the original denominator.
step 4: Reduce if possible. The fraction cannot be
reduced.
38%
23
38'
?24
38'
924
23'
?24
23'
1624
2524
2524
11
24
38%
23'1
124
11
Example 2:
step 1: Find an common denominator for 8 and 3 by creating equivalent fractions.
The common denominator is 24. This is because 24 can be divided by 8 evenly and 3 evenly!
step 2: Add the new numerators together. 9 + 16 = 25
step 3: Put the new numerator over the new denominator.
step 4: Reduce if possible. can be reduced to .
14%
512
%12
14'
?12
14'
312
512
'?
12512
'5
12
12'
?12
12'
612
12
Example 3: You can add more than two fractions at a time.Just follow the same steps!
step 1: Find a common denominator for 4, 12 and 2 bycreating equivalent fractions.
The common denominator is 12. Hint: choose the largest denominator, 12 and go through the times tables for 12 dividing each product by the other twodenominators.
12 x 1 = 12 Can 12 be divided by 4 evenly and 2evenly? yes.12 ÷ 4 = 312 ÷ 2 = 6
Because 12 can be divided evenly by all denominators, you don’t have to go any further! You can use 12 as the new denominator!
step 2: Add the new denominators together. 3 + 5 + 6 = 14
1412
1412
12
121
16
34%
45
26%
78
47%
23
13
step 3: Put the new denominator over the new denominator.
step 4: Reduce if possible. can be reduced to , whichequals .
You will be happy to know that there is a shortcut to findingcommon denominators!
All you have to do is multiply the original denominatorstogether! This product is the new denominator! Be carefulthough, it may not be the smallest common denominator.
Take a look!
Example 1: 4 x 5 = 20
Example 2: 6 x 8 = 48
Example 3: 7 x 3 = 21
Get the idea? This works with more than two denominators too!
23%
56%
34
23%
35
13%
67
56%
67
318
%59
24%
58
410
%12%
23
12%
79
24%
13%
112
34%
67
510
%24%
68
89%
910
36%
39%
13
13%
16
12%
35%
910
24%
13
1020
%710
%35
112
%34
79%
78
56%
68
46%
37
14
Example 4: 3 x 6 x 4 = 72
Did you notice? 72 is not the smallest common denominator, 12is. But, you can still reduce your answer at the end.
Exercise 5
Practice finding common denominators for the following fractions.
1) 11)
2) 12)
3) 13)
4) 14)
5) 15)
6) 16)
7) 17)
8) 18)
9) 19)
10) 20)
23%
47'
110
%35%
12
?21
%1221
?10
%?
10%
?10
56%
68
23%
45
2024
%?
2410?%
8?
12%
910
25%
410
%1215
?10
%9
10?
30%
?30
%?
30
45%
56
1012
%524
%56
2430
%?
3020?%
5?%
20?
35%
49
47%
12
27?%
20?
?14
%?
14
15
Exercise 6
Fill in the missing numbers.
1) 6)
2) 7)
3) 8)
4) 9)
5) 10)
116
%116
'2
16
16
Exercise 7
Answer the following questions using the diagram below.
1 6 3 5 2
8 7 4
Example: section 1 + section 8 =
1) Write a fraction for each section of the diagram that tellshow much space of the whole is occupied.
2) Use fractions to show how much of the whole eachnumbered section represents. . .
a. 1 and 2b. 4 and 2c. 3 and 7 d. 3 and 5 and 7e. 5 and 4 and 2f. 3 and 5 and 6 and7
3) Which section(s) takes up the most amount of space?
4) Which section(s) takes up the least amount of space?
25%
12
78%
34
23%
46
36%
59
1012
%68
17%
210
27%
514
2030
%1020
56%
34
710
%12
38%
57
78%
40
79%
35
56%
712
16%
210
89%
12
25%
48
37%
2021
925
%45
1012
%34
Exercise 8
Express each of the following sums in the simplest form.
5) 11)
6) 12)
7) 13)
8) 14)
9) 15)
10) 16)
11) 17)
12) 18)
13) 19)
14) 20)
312
1023
3012
3523
18
ADDING MIXED NUMBERS
The following fractions are “mixed numbers”:
A mixed number, also called : “mixed fraction”, is a number that ismade up of more than one kind of number - that’s why it’s called“mixed”. It always has a whole number part as well as a fractionpart.
fraction part
whole number part
You may be asking... when will I ever have to add mixed
numbers?? I’m glad you asked!
You may want to add up the number of hours you worked in the
last two weeks to see if you are owed overtime. Let’s say you
worked hours one week, and hours another week. You
would of course need to add these amounts together to arrive ata total.
427%1
37
27%
37'
57
557
557
548%
28
48%
28'
68
568
568'5
34
138%2
38%6
18
38%
38%
18'
78
878
19
To add mixed numbers that have common denominators,follow these simple steps:
Example:
1. Add the whole numbers. 4+1= 52. Add the fractions.
3. Write the whole number beside the fraction.4. Reduce the answer if possible.
The fraction cannot be reduced.
Example 2: * you can also add a mixed number and a proper fraction. The whole number in the second fraction is 0.
a. Add the whole numbers. 5+0=5b. Add the fractions.
c. Write the whole number beside the fraction.
d. Reduce the fraction if possible.
Example 3:
a. 1+2+6=8
b.
c. This answer cannot be reduced.
124%5
14
1279%2
19
437%
27
7410
%42
10
3510
%31
109
920
%420
%62
20
8915
%43
156
525
%121225
19%4
29%6
49
3818
%115
18
317%10
57
2919
%67
19
24
12%3
512
%112
128
311
%6
11
6916
%43
1610
50100
%2030
100
148%5
38
1100200
%950
200
101
10%3
310
%72
1017
314
%1314
%45
14
20
Exercise 9: Add these mixed numbers. Always reduce wherepossible.
1) 11)
2) 12)
3) 13)
4) 14)
5) 15)
6) 16)
7) 17)
8) 18)
9) 19)
10) 20)
235%2
25
35%
25'
55'1
1079%3
29
79%
29'
99'1
61
10%3
610
%13
10
110
%610
%3
10'
1010
'1
21
Sometimes when you add the fraction parts you get a wholenumber as an answer. If this happens, just add that wholenumber to the other one. Take a look!
Example:
a. 2+2=4
b. *Remember-any number divided by itself is 1.
c. 4+1=5 The answer is 5.
Example 2:
a. 10+3=13
b.
c. 13+1=14
Example 3:
a. 6+3+1=10
b.
c. 10+1=11
325%6
35
101120
%5320
%66
207
16%1
26%3
36
8715
%68
159
910
%31
105
512
%10312
%74
12
21220
%168
206
39%1
39%2
39
1128%12
68
203045
%101045
%25
451
414
%71014
33
10%4
410
%53
10
77
1012
35
467
829
1037
458
12
106
78
127
2012
125
157
103
72
114
259
22
Exercise 10: Add the groups of mixed numbers below. Eachsum is a whole number.
1) 2) 3)
4) 5) 6)
7) 8) 9)
10) 11) 12)
Exercise 11: Change these mixed numbers to improperfractions.
1) 2) 3) 4)
5) 6) 7) 8)
Exercise 12: Change these improper fractions to mixednumbers.
1) 2) 3) 4)
5) 6) 7) 8)
119%3
39%6
49
2312
%71112
33
12%
512
%102
12
14
10%3
310
%23
1015
120
%7720
%82
20916
%43
16
103
15%
615
%11
15%9
215
45
10%6
210
%113
10
71025
%4425
%105
25%
125
738%
38
239%2
69
610
%44
102
58%4
38%0 3
37%4
17
503080
%202580
41118
%1
187
310
%7310
%73
10
137%?'6
67
?%7915
'181415
10418
%?'161218
?%9725
'201625
8610
%?'119
10?%13
1120
'211620
?%337'10
77
12%?'2 4
512
%?'8
12
23
Exercise 13: Mixed practice. Reduce answers if possible.
1) 2) 3)
4) 5) 6)
7) 8)
9) 10) 11)
12) 13) 14)
15) 16) 17)
Exercise 14: Fill in the missing fraction.
1) 2) 3)
4) 5) 6)
7) 8) 9)
325%2
45
25%
45'
65
65'1
15
5%115'6
15
138%4
88
38%
88'
118
118'1
38
5%138'6
38
1610
%38
10
4%14
10'5
410
'5256
10%
810
'1410
52514
10'1
410
24
You might also get an improper fraction when you add the fraction partof mixed numbers. Just change it to a mixed number, then add the twonumbers together.
1) Add the whole numbers. 3+2=52) Add the fractions.
3) Change the improper fraction to a mixed number.
4) Add the whole number and mixed number.
Example 2:
1) 1+4=5
2)
3)
4)
Example 3:
1) 3+1=4 4.2)
The answer is .3)
126%3
56
43
10%7
910
1049%5
89
47
12%1
812
227%6
17%
67
1120
%88
20%
420
7711
%39
112
615
%11015
9412
%81112
3%212'5
12
3%2'5,5%12'5
12
7%45
106
712
%10 12%84
124
59%6
224%10 9
816
%5 121015
%10 1%11
100
25
Exercise 15: Add the following mixed numbers. Don’t forget toreduce your answers!
1) 2) 3)
4) 5) 6)
7) 8) 9)
Finally, mixed numbers can be added to whole numbers and vice-versa. All you have to do is add the whole numbers together and keep thefraction. this makes sense because you are adding whole amounts plusanother part of a whole. example:
Exercise 16: Practice adding these kinds of numbers...andreducing!
1) 2) 3) 4)
5) 6) 7) 8)
215%1
610
15'
?10
15'
210
610
'?
10610
'6
10
210
%610
'8
10
38
103
810
'345
345
26
Adding Mixed Numbers with Unlike Denominators
Sometimes the mixed numbers have differentdenominators. So the first step is to changethe fractions to equivalent fractions with a common denominator. Then go on as usual.
example:
1) Change fractions to equivalent fractions that have acommon denominator.
b. Add the whole numbers. 2+1=3f. Add the fractions.
7) Combine the whole number and fraction.8) Reduce.
The answer is .
more examples...
324%4
512
24'
?12
24'
612
512
'?
12512
'5
12
612
'512
'1112
71112
3610
%8
10
610
'?
20610
%1220
810
'?
20810
'1620
1220
%1620
'2820
32820
'3%18
20
4820
'425
425
27
example 2:
1)
2) 3+4=73)
4) This is the answer.
example 3:
1.
2. 3+0=3
3.
4.
5.
The answer is .
238%4
124
1036%9
13
935%4
26
779%2
127
1314
%2
285
510
%86
20
6430
%2
604
28%3
56
1048%5
59
335%9
89
227%11
36
610
%32
100
1112
%21248
314%10
58
378%4
310
112%6
410
757%
34
234%9
732
6410
%77
309
69%1
16
234%
810
225%10
58
456%9
48
313%5
212
612
%434
28
Exercise 17: Add these mixed numbers.
Reduce your answers!
1) 2) 3)
4) 5) 6)
7) 8) 9)
10) 11) 12)
13) 14) 15)
16) 17) 18)
19) 20) 21)
22) 23) 24)
25)
45
25−
45
25− 2
5
57
27−
57
27− 3
7
1315
815
515
15− = =
29
SUBTRACTION
Subtracting fractions is very similar to adding fractions. Theequation is always set up the same. The only difference is theoperation we use. You subtract instead of add.
Subtract fractions with like denominatorsLet’s look at the following equation:
=
The denominators are the same. We call these “likedenominators.” In subtracting like denominators, we simplysubtract the second numerator from the first. 4-2=2. The newnumerator is 2. The denominator stays the same. So, =
Let’s try another one. = the denominators are likedenominators, so all that we have to do is subtract the secondnumerator from the first. 5-2=3. The new numerator is 3. Thedenominator stays the same. So,
= .
Don’t forget that when working with fractions, you must ALWAYS reduce your answer to its lowest terms. ( )
811
511− 2
171
17− 23
13−
49
39− 14
23923− 4
525−
715
515− 9
137
13− 2332
532−
710
310− 21
249
24− 1528
1328−
715
215− 12
21521− 19
403
40−
49
14− 11
12960− 9
1515−
1636
936− 7
365560
960− 46
602330
915
315− 6
1525
23
14− 15
2127− 5
623−
615
13− 14
2525− 7
1813−
710
23− 9
1024
100− 913
826−
30
Exercise 18: Now you try a few:1) = 6) = 11) =
2) = 7) = 12) =
3) = 8) = 13) =
4) = 9) = 14) =
5) = 10) = 15) =
Subtracting when the denominators are different
Subtracting fractions that have different denominators takesadditional steps than fractions with like denominators. You mustfirst find a common denominator; next, you create equivalentfractions using this common denominator; then, you subtract.
(*note: the common denominator is in brackets)
Ex: = (36) = (30) =(15)
= = =
= =
Exercise 19: Now, you try.
1) = 4) = 7) =
2) = 5) = 8) =
3) = 6) = 9) =
4 258
14−
4 258
28−
58
28
38− =
4 258
28− 3
8
9 81214
27−
9 81214
414− 12
141214
27
414= =,
814
1214
414− 8
14
814
47
9 81214
27− 4
7
31
*Let’s review the steps up to this point*
Step 1: find the common denominatorStep 2: create equivalent fractionsStep 3: subtract the numeratorsStep 4: reduce to lowest terms
Subtracting Mixed Numbers
When subtracting mixed numbers, the fractions are alwayssubtracted before the whole numbers.
ex: = Step 1: find the common denominator (8)
= Step 2: create equivalent fractions
Step 3: subtract the fraction4-2= 2 Step 4: subtract the whole number
So, = 2 Step 5: reduce to lowest terms if necessary
Let’s look at another one: = find the common denominator = 14
= create equivalent fractions { }
subtract the fraction { = }1 subtract the whole number {9-8=1} =1 reduce if necessary
=1
So, =1
15 423
39− 15 46
939− 3
913
18 2318
19− 92 8120
3012−
12 11925
15− 10 23
423−
43 1559
218− 42 182
312−
7 31415
35− 101 574
202
50−
5 214
34−
5 214
34−
14
141 1+ =
1 14
54=
32
Here’s one more without the steps outlined. See if you can followit:
= =11 =11How did you make out? Could you follow the steps? If youcould, good for you! If not, go back and review the steps outlinedabove. Exercise 20: Try theseD. = 5) =
E. = 6) =
F. = 7) =
G. = 8) =
Borrowing
With fractions, just like with regular subtraction, the time willcome when you need to borrow from a whole number in order tocomplete the subtraction process.
Let’s look at the following equation: . The first numeratoris smaller than the second. What we need to do is borrow onefrom our whole number (5) in order to complete the problem.
Let’s take a look at the steps for borrowing:Step one: borrow one from the whole number 5 L 4
Step two: add the 1 you borrowed to the fraction
Step three: change the mixed number from step two into animproper fraction (remember to make animproper fraction, you multiply the whole numberby the denominator and then add the numerator)
11
4
54
5 21
4
3
4−
4 254
34− 2
2
4 21
2
6 238
1124−
6 2924
1124−
5 23324
1124−
5 23324
1124− 3 22
24
3 2224 3 11
12
33
NOTE: when borrowing one (1) from a whole number, theeasiest way to make an improper fraction is tosimply add the denominator to the numerator. The sum becomes the new numerator. This cutsout the multiplication step. Think of it this way:any number times one equals itself (9x1=9,10x1=10, 429x1=429, etc.). So, = 4x1+1 or =
4+1=
Step four: subtract as usual
= = =
Example 2-find a common denominator (24)
-create equivalent fractions
-borrow 1 from your whole number and create animproper fraction (by multiplying the denominatorby the whole number that you borrowed (1) andadding the numerator or by simply adding thedenominator to the numerator)
= -subtract (fraction first, then whole number)
= -reduce if necessary
14 10316
516− 50 104
79
10−
3 213
23− 100 992
2534−
41 391851
1951− 15 3
50225−
34 514
23− 73 431
27
10−
78 1825
56− 1 1
525−
5 320
1920− 24 57
2545−
2 112
23− 4 3
789−
17 1516
57− 1
875
2950−
14 335
78− 84 438
1912−
3 1715
23− 5 3
57
10−
34
Exercise 21: Now, you try some
a. = 11) =
b. = 12) =
c. = 13) =
d. = 14) =
e. = 15) =
f. = 16) =
g. = 17) =
h. = 18) =
i. = 19) =
j. = 20) =
33
2929
4394391 1 1= = =, ,
12
5 5 122
= 2 25
439
439
439439
25
10
50
6
185
321321
4
4
15
2
3 2114
66
6
5 23
3
513
1414
58
3 416
6
12525
1010
1813
2 4516
16
6 723 7 67
5
35
Fractions that equal one
At this time, we need to look at fractions that equal 1 (one). Anyfraction that has the same numerator and denominator equals 1(one): . Any number over itself equals one. Thismakes sense if you think of it this way: a fraction stands for thenumerator divided by the numerator. That is why when you aremaking a mixed number out of an improper fraction, you divide
the numerator by the denominator ( R2 = ). When you
take the fraction what you are saying is 439 divided by 439 which equals 1;therefore, =1.
*This is true for any number over itself. *
Exercise 22: Change the following fractions to mixed numbersor whole numbers.
a. 11)
b. 12)
c. 13)
d. 14)
e. 15)
f. 16)
g. 17)
h. 18)
i. 19)
j. 20)
23
25 152
3−
24 1533
23−
24 1533
23− 1
3
13
14 7 415−
13 71515
415−
1115
36
Subtract fractions from whole numbers
Let’s say that you are doing a craft project and you need 15 cm of ribbon. You have25 cm of ribbon to work with. How much ribbon would you have left over after yourcraft project? This problem requires you to subtract a fraction from a whole number.
In order to do this, you need to borrow one (1) from your wholenumber. This leaves 24. To make a fraction out of the one (1)that you borrowed, you simply place any number over itself in afraction (remember any number over itself equals one).
Because the first step in subtracting fractions is to locate a common denominator, the easiest thing to do is to use the denominator of the fraction that is already in your equation; so, for the equation , you would choose three (3) as your denominator. Your equation would become . Now you would follow
through with your subtracting: =9 ;
therefore, you would have 9 cm of ribbon left over.
Example 2-borrow from the whole number (14-1=13) and find acommon denominator (15)
6 -subtract (and reduce to lowest terms if necessary)
5 34− 1
9
417 57 3 4
7−
43 5 35− 4
5
38 4350− 18 2 16
17−38 70 35 8
15−23
425
417
521
12
219
38
2526
89
3100
37
**Keep these steps in mind when you subtract fractions:**
Step 1: find the common denominator/borrow from a wholenumber if necessary
Step 2: create an equivalent fractionStep 3: borrow from the whole number if necessaryStep 4: subtract the fractionsStep 5: subtract the whole numbers
Exercise 23: Now you try!
a. = 11) 92-47 =
b. 15-3 = 12) =
c. = 13) 10-5 =
d. = 14) =
e. 22-4 = 15) =
f. 32-14 = 16) 26- =
g. 45-21 = 17) 68-42 =
h. 8- = 18) 92-3 =
i. 36-5 = 19) 5-4 =
j. 100-57 = 20) 71-1 =
25%
25'
45
86%
46'
126'2 2
4%
14'
34
54%
34'
84'2 2
5%
15'
35
68%
18'
78
47%
37'
77'1 4
11%
311
'7
1125%
35'
55'1
49%
19'
59
3040
%540
'3540
'78
612
%312
'9
12'
34
28%
38'
58
313
%913
'1213
48%
38'
78
36%
26'
56
38%
58'
88'1 5
7%
17'
67
23%
13'
33'1 10
12%
112
'1112
410
%210
'6
10'
35
17%
47'
57
35%
15'
45
1014
%314
'1314
1525
%425
'1925
100200
%50200
'150400
'38
38
Answer Key
Exercise 1 page 5:
11) 9) 17)
12) 10) 18)
13) 11) 19)
14) 12) 20)
15) 13) 21)
16) 14) 22)
17) 15) 23)
18) 16) 24)
25)
26)
64'1 2
4'1 1
2134'3 2
4'3 1
25513
'4 313
125'2 2
5125'2 2
5152'7 1
2
2012
'1 812
'1 23
167'2 2
74510
'4 510
'4 12
607'8 4
74030
'1 1030
'1 13
73'2 1
3
226'3 4
6'3 2
3106'1 4
6'1 2
33312
'2 912
'2 34
144'3 2
4'3 1
21812
'1 612
'1 12
2013
'1 713
256'4 1
672'3 1
2247'3 3
7
325'6 2
510040
'2 2040
'2 12
95'1 4
5
85'1 3
54410
'4 410
'4 25
14%
34'
44'1 2
3%
13'
33'1 10
30%
2030
'1
37%
37'
67
38%0' 3
86
12%
212
'23
28%
78'
98'1 1
826%
16'
12
49%
49'
89
49%
59'1 2
8%
48'
34
1020
%1320
'2320
'1 320
410
%310
'7
102
10%
410
'35
57%
37'1 1
7
39
Exercise 2 page 6:
a. 10) 18)
b. 11) 19)
c. 12) 20)
d. 13) 21)
e. 14) 22)
f. 15) 23)
g. 16) 24)
h. 17) 25)
i. 26)
Exercise 3 page 6:
1) 6) 11)
2. 7) 12)
3. 8) 13)
4. 9) 14)
5. 10) 15)
14%
24'
38%
38
29%
59'
2027
%127
25%
15'
15%
15%
15
510
%310
'35%
15
610
%410
'38%
58
512
%312
'13%
13
37%
27'
614
%414
812
%212
'46%
16
23%
47
45%
56
23%
45
47%
12
1421
%1221
2430
%2530
1015
%8
15814
%7
14
56%
68
35%
49
25%
410
%1215
2024
%1824
2745
%2045
1230
%1230
%2430
12%
910
110
%35%
12
1012
%524
%56
510
%9
101
10%
610
%5
102024
%524
%2024
40
Exercise 4 page 7:
Exercise 5 page 13:
1. 15 6) 90 11) 21 16) 182. 42 7) 6 12) 18 17) 103. 8 8) 12 13) 30 18) 204. 18 9) 12 14) 12 19) 185. 28 10) 24 15) 40 20) 42
Exercise 6 page 14:
1. 4) 7) 10)
2. 5) 8)
3. 6) 9)
116
18
216
116
14
116
14
116
316
716
616
816
516
816
25'
410
1012
'2024
56'
1012
12'
510
68'
1824
34'
912
410
%510
'9
102024
%1824
'3824
'1 1424
'1 712
1012
'912
'1912
'1 712
23'
46
27'
414
38'
2156
46'
46
514
'5
1457'
4056
46%
46'
86'1 2
6'1 1
34
14%
514
'9
142156
%4056
'6156
'1 556
41
Exercise 7 page 15:
a. section 1 = section 5 =
section 2 = section 6 =
section 3 = section 7 =
section 4 = section 8 =
b. sections 1 and 2 = sections 3,5 and 7 =
sections 4 and 2 = sections 5,4 and 2 =
sections 3 and 7 = sections 3,5,6 and 7 =
c. Sections 3 and 4 take up the most amount of space.
d. Sections 1,6,7,and 8 take up the least amount of space.
Exercise 8 page 16:a. 3) 5)
b. 4) 6)
79'
3545
36'
1836
56'
1012
35'
2745
59'
2036
712
'7
12
3545
%2745
'6245
'1 1745
1836
%2036
'3836
'1 236
'1 118
1012
%712
'1712
'1 512
16'
530
17'
1070
89'
1618
210
'6
30210
'1470
12'
918
530
%630
'1130
2470
'1235
1618
%918
'2518
'1 718
25'
1640
2030
'4060
37'
921
48'
2040
1020
'3060
2021
'2021
1640
%2040
'3640
'910
4060
%3060
'7060
'1 1060
'1 56
921
%2021
'2920
'1 920
925
'9
25710
'7
101012
'1012
45'
2025
12'
510
34'
912
925
%2025
'2925
'1 425
710
%510
'1210
'1 210
'1 15
1012
%912
'1912
'1 712
78'
78
78'
78
34'
68
41'
328
78%
68'
138'1 5
878%
328'
418'5 1
8
42
g. 12) 17)
h. 13) 18)
i. 14) 19)
j. 15) 20)
k. 16)
1 24%5 1
4'6 3
46 5
25%12 12
25'18 17
25
4 37%
27'4 5
73 8
18%11 5
18'14 13
18
3 510
%3 110
'6 610
'6 35
2 919
%6 719
'8 1619
8 915
%4 315
'12 1215
'12 45
8 311
%611
'8 911
19%4 2
9%6 4
9'10 7
910 50
100%20 30
100'30 80
100'30 4
5
3 17%10 5
7'13 6
71 100
200%9 50
200'10 150
200'10 3
4
2 412
%3 512
%11 212
'16 1112
17 314
%1 314
'4 414
'22 1114
6 916
%4 316
'10 1216
'10 34
1 48%5 3
8'6 7
8
10 110
%3 310
'7 210
'20 610
'20 35
12 79%2 1
9'14 8
9
7 410
%4 210
'12 610
'12 35
9 920
%4
20%6 2
20'15 15
20'15 3
4
43
Exercise 9 page 19:
a. 14)
b. 15)
c. 16)
d. 17)
e. 18)
f. 19)
g. 20)
h.
i.
j.
k.
l.
m.
3 25%6 3
5'9 5
5'9%1'10 2 2
20%16 8
20'18 20
20'18%1'19
10 1120
%5 320
%6 620
'21 2020
'21%1'22 6 39%1 3
9%2 3
9'9 9
9'9%1'10
7 16%1 2
6%3 3
6'11 6
6'11%1'12 11 2
8%12 6
8'23 8
8'23%1'24
8 715
%6 815
'15 1515
'15%1'16 20 3045
%10 1045
%2 545
'32 4545
'32%1'33
9 910
%3 110
'12% 1010
'12%1'13 1 414
%7 1014
'8 1414
'8%1'9
5 512
%10 312
%7 412
'15 1212
'15%1'16 3 310
%4 410
%5 310
'12 1010
'12%1'13
7 710
'7710
10 37'
737
12 35'
635
4 58'
378
4 67'
347
1 210
'1210
8 29'
749
6 78'
558
127'1 5
7103'3 1
3
2012
'1 812
'1 23
72'3 1
2
125'2 2
5114'2 3
4
157'2 1
7259'2 7
9
44
Exercise 10 page 21:a. 7)
b. 8)
c. 9)
d. 10)
e. 11)
f. 12)
Exercise 11 page 21:a. 5)
b. 6)
c. 7)
d. 8)
Exercise 12 page 21:a. 5)
b. 6)
c. 7)
d. 8)
1 19%3 3
9%6 4
9'10 8
95 3
7
2 312
%7 1112
'9 1412
'9%1 212
'10% 12
11 515
3 312
%5
12%10 2
12'13 10
12'13 5
66 8
18
1 410
%3 310
%2 310
'6 1010
'6%1'7 11 925
15 120
%7 720
%8 220
'30 1020
'30 12
3 310
916
%4 316
'4 1216
'4 34
8 520
10 315
%615
%1 115
%9 215
'20 1215
'20 45
7 47
4 510
%6 210
%11 310
'21 1010
'22 1 12
7 1025
%4 425
%10 525
%1
25'21 20
254 3
12
7 38%
38'7 6
8'7 3
4
2 39%2 6
9'4 9
9'4%1'5
610
%4 410
'4 1010
'4%1'5
2 58%4 3
8%0'6 8
8'6%1'7
3 37%4 1
7'7 4
7
50 3080
%20 2580
'70 5580
'70 1116
4 1118
%118
'4 1218
'4 23
45
Exercise 13 page 22: Exercise 14 page 33:
i. 1)
ii. 2)
iii. 3)
iv. 4)
v. 5)
vi. 6)
vii. 7)
viii. 8)
ix. 9)
x.
xi.
xii.
xiii.
xiv.
xv.
xvi.
7 310
%7 310
%7 310
'21 910
1 26%3 5
6'4 7
6'4%1 1
6'5 1
6
4 310
%7 910
'11 1210
'11%1 210
%12 210
'12 15
10 49%5 8
9'15 12
9'15%1 3
9'1 1
3
4 712
%1 812
'5 1512
'5%1 312
'6 14
2 27%6 1
7%
67'8 9
7'8%1 2
7
1120
%8 820
%4
20'8 23
20'8%1 3
20'9 3
20
7 711
%3 911
'10 1611
'10%1 511
'11 511
2 615
%1 1015
'3 1615
'3%1 115
'4 115
9 412
%8 1112
'17 1512
'17%1 312
'18 312
'18 14
7%4 510
'11 510
'11 12
2 24%10'12 2
4'12 1
2
6 712
%10'16 712
9 816
%5'14 816
'14 12
12%8 412
'20 412
'20 13
12 1015
%10'22 1015
'22 23
4 59%6'10 5
91%1 1
100'2 1
100
46
xvii.
Exercise 15 page 35:
a.
b.
c.
d.
e.
f.
g.
h.
i.
Exercise 16 page 24:
1) 5)
2) 6)
3) 7)
4) 8)
2 38'
924
924
%1
24'
1024
%6'6 1024
'6 512
4 124
'1
24
10 36'
36
36%
26'
56%19'19 5
6
913'
26
935'
1830
1830
%1030
'2230
'11230
%13'141230
'1425
426'
1030
779'
2127
2127
%127
'2227
%9'9227
2127
'1
27
1314
'6
28628
%2
28'
828
%1'18
28'1
27
228
'2
28
5510
'1020
1020
%6
20'
1620
%13'131620
'1345
8620
'6
20
47
Exercise 17 page 27:
1. +
2.
+
3. +
4. +
5. +
6. +
6 430
'8
60860
%2
60'
1060
%6'6 1060
'6 16
260
'2
60
4 28'
624
624
%2024
'2624
'1 24'1 1
2%7'8 1
2
3 56'
2024
10 48'
3672
3672
%4072
'1 472
%15'16 472
'16 118
5 59'
4072
3 35'
2745
2745
%4045
'6745
'1 2245
%12'13 2245
9 89'
4045
2 27'
1242
1242
%2142
'3342
%13'13 3342
11 36'
2142
610
'60
10060100
%2
100'
62100
%3'3 62100
'3 3150
3 2100
'2
100
48
7. +
8. +
9) +
10) +
11) +
12) +
1112
'4448
4448
%1248
'5648
'1 448
%2'3 848
'3 16
2 1248
'1248
3 14'
28
28%
58'
78%13'13 7
8
10 58'
58
3 78'
3540
3540
%1240
'4740
'1 740
%7'8 740
4 310
'1240
1 12'
510
510
%410
'9
10%7'7 9
10
6 410
'4
10
7 57'
2028
2028
%2128
'4128
'1 1328
%7'8 1328
34'
2128
2 34'
2432
2432
%732
'3132
%11'11 3132
9 732
'7
32
49
13) +
14) +
15) +
16) +
17) +
18) +
6 410
'1230
1230
%730
'1930
%13'13 1930
7 730
'7
30
9 69'
3654
3654
%954
'4554
%10'10 4554
1 16'
954
2 34'
1520
1520
%1620
'3120
'1 1120
%2'3 1120
810
'1620
2 25'
1640
1640
%2540
'4140
'1 140
%12'13 140
10 58'
2540
4 56'
4048
4048
%2448
'6448
'1 1648
%13'14 1648
'14 13
9 48'
2448
3 13'
412
412
%2
12'
612
'12%8'8 1
2
5 212
'2
12
612
'6
126
12%
912
'1512
'1 312
%4'5 312
'5 14
4 34'
912
50
19) +
20) +
21) +
22) +
23) +
24) +
25) +
811
511− 3
112124
924− 12
2412
49
39− 1
91221
521− 7
2113
715
515− 2
1523
13− 1
3
710
310− 4
1025
45
25− 2
5
715
215− 5
1513
2332
532− 18
329
16
217
117− 1
171528
1328− 2
281
14
1423
923− 5
231940
340− 16
4025
913
713− 2
13
23
14− 8
123
12− 512
910
24100− 90
10024
100− 66
1003350
615
13− 6
155
15− 115
56
23− 15
181218− 3
1816
710
23− 21
302030− 1
307
1813− 7
186
18− 118
1521
27− 15
21621− 9
2137
913
826− 18
26826− 10
265
13
1425
25− 14
251025− 4
25
18 2318
19− 18
3
182
2
18− 16
1
18
12 119
25
1
5− 129
2511
5
25− 1
4
25
43 1559
218− 43
10
1815
2
18− 28
3
1828
1
6
7 31415
35− 7
14
153
9
15− 4
5
154
1
5
92 812030
12− 92
20
3081
15
30− 11
5
3011
1
6
51
Exercise 18 page 29: z. = 9) = =
aa. = 10) = =
bb. = 11) =
cc. = = 12) =
dd. = = 13) = =
6) = 14) = =
7) = 15) = =
8) =
Exercise 19 page 29:1) = = 6) = = =
2) = = 7) = = =
3) = = 8) = =
4) = = = 9) = = =
5) = =
Exercise 20 page 31:
a. = =
b. = =
c. = = =
d. = = =
5) = = =
10 234
23− 10
9
122
8
12− 8
1
12
42 1823
12− 42
4
618
3
6− 24
1
6
101 57420
250− 101
20
10057
4
100− 44
16
10044
4
25
14 103
16
5
16− 1319
1610
5
16− 3
14
163
7
8=
3 213
23− 2
4
32
2
3−
2
3
41 391851
1951− 40
69
5139
19
51− 1
50
51
34 514
23− 34
3
125
8
12− 33
15
125
8
12− 28
7
12
78 182
5
5
6− 7812
3018
25
30− 77
42
3018
25
30− 59
7
30
5 320
1920− 4
23
20
19
20− 4
4
204
1
5=
2 112
23− 2
3
61
4
6− 1
9
61
4
6−
17 1516
57− 17
7
4215
30
42− 16
49
4215
30
42− 1
19
42
14 33
5
7
8− 1424
403
35
40− 13
64
403
35
40− 10
29
40
3 1715
23− 3
14
301
20
30− 2
44
301
20
30− 1
24
301
4
5=
50 1047
910− 50
40
7010
63
70− 49
110
7010
63
70− 39
47
70
100 99225
34− 100
8
10099
75
100− 99
108
10099
75
100−
33
100
153
50
2
25− 153
50
4
50− 14
53
50
4
50− 14
49
50
52
6) = =
7) = =
8) = = =
Exercise 21 page 33:
a. = =
b. = =
c. = =
d. = = =
e. = = =
f. = =
g. = = =
h. = = =
i. = = =
j. = = =
11) = = =
12) = = =
13) = = =
73 4312
710− 73
5
1043
7
10− 72
15
1043
7
10− 29
8
1029
4
5=
1 15
25−
6
5
2
5−
4
5
24 5725
45− 24
7
255
20
25− 23
32
255
20
25− 18
12
25
43
7
8
9− 427
63
56
63− 3
90
63
56
63− 3
34
63
1875
2950− 1
16
150
87
150−
166
150
87
150−
79
150
84 43819
12− 84
16
3843
19
38− 83
54
3843
19
38− 40
35
38
5 35
710− 5
6
10
7
10− 4
16
10
7
10− 4
9
10
2510 2
5
102
1
2= 50
6 82
68
1
3=
185 3
3
5321321
44
152 7
1
2
3 2114 4
7
214
1
3= 6
6
6
5 11
52
3
3
513
1414
5 83 7
2
34 16
6 64
66
2
3=
125
25
10
10
18 132 24
1
245 16
16
6 723 7 67
5 202
5
53
14) = = =
15) = =
16) = = =
17) = = =
18) = = =
19) = = =
20) = = =
Exercise 22 page 34:
a. = 11) =
b. = 12) =1
c. =1 13) =
d. = 14) =1
e. = 15) =3
f. =17 16) =1
g. = 17) =
h. =5 18) =1
i. = 19) =46
j. =30 20) =
5 34− 4
4
4
3
4− 1
1
4
417 14
17
173
4
17− 11
13
17
43 5 35− 42
5
55
3
5− 37
2
5
3843
50− 3750
50
43
50− 37
7
50
38 21
8
84
3
8− 17
5
8
23 31
3
314
2
3− 17
1
3
417 44
17
1721
4
17− 23
13
17
1
2 72
2
1
2− 7
1
238 35
8
85
3
8− 30
5
8
89 99
9
957
8
9− 42
1
9
19 91
9
947
1
9− 44
8
9
57 34
7− 567
73
4
7− 53
3
7
45 9
5
55
4
5− 4
1
5
18 2 1617− 17
17
172
16
17− 15
1
17
70 35 815− 69
15
1535
8
15− 34
7
15
4
25 2525
25
4
25− 25
21
25
54
Exercise 23 page 36:
a. = =
b. 15-3 = =
c. = =
d. = =
e. 22-4 = =
f. 32-14 = =
g. 45-21 = =
h. 8- = =
i. 36-5 = =
j. 100-57 = =
11) 92-47 = =
12) = =
13) 10-5 = =
14) = =
15) = =
16) 26- = =
521 67
21
2142
5
21− 25
16
21
219 91
19
193
2
19− 88
17
19
2526 4
26
264
25
26−
1
26
3
100 70100
1001
3
100− 69
97
100
55
17) 68-42 = =
18) 92-3 = =
19) 5-4 = =
20) 71-1 = =
56
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