chapter 1: introduction to fractions skills for …
TRANSCRIPT
1
CHAPTER 1: INTRODUCTION TO FRACTIONS SKILLS FOR TRADES
Construction related measurements and calculations require knowledge of fractions. Fractions are a way of expressing a part of something. If you cut a piece of wood, you are using a fraction of the whole board. If you go home from work at lunch, you are only working a fraction of the day. There are two parts to every fraction, the numerator and the denominator.
The Numerator
(top number) refers to a specific part of the whole.
The Denominator
(bottom number) identifies the number of equal parts into which the whole has been divided.
Example: The fraction 83 tells you what part of the figure below is shaded. Three of the
eight equal parts are shaded. Therefore, the shaded portion represents 83 of an inch.
Darken or write fractions that represent the part of each inch that is indicated:
1) ________ 2) 8"5
3) 8"7
2
FRACTIONS IN WRITTEN FORM
Write the correct fraction for each of these. _______ 1) A yard contains 36 inches. 13 inches is what fraction of a yard? _______ 2) There are 12 inches in a foot. 7 inches is what fraction of the whole? _______ 3) 59 cents is what fraction of a dollar? _______ 4) There are 16 ounces in a pound. What fraction of a pound is 7 ounces? _______ 5) Alicia worked 11 months of the year. What fraction of the year did Alicia
work? _______6) Laura has to cut 150 2 x 4’s. This morning she cut 103. What fraction
of the job has she completed? _______7) There are 4 quarts in a gallon of paint. Tabitha used 1 quart to paint the
entryway. What fraction of the gallon did she use? _______8) 2,000 people applied to the pipefitters’ apprenticeship program. The
program accepted 147 people. What fraction of the applicants were accepted?
3
EQUIVALENT FORMS OF FRACTIONS-VOCABULARY
PROPER FRACTION- The numerator is less than the denominator.
Examples: 53
74
1615
A proper fraction is less than all the parts into which the whole is divided. The value of a proper fraction is always less than one. IMPROPER FRACTION- The numerator is equal to or larger than the denominator.
Examples: 58
23
99
An improper fraction may include all the parts into which a whole has been divided ( 8
8),
or it may include more than the total parts of the whole. The value of an improper fraction is either equal to or more than one. UNREDUCED FRACTION-The numerator is less than the denominator, but because it could be reduced, it isn’t a proper fraction.
Examples: 62
129
4832
MIXED NUMBER- A whole number is written next to a proper fraction
Examples: 833
1095
6512
4
REDUCING/SIMPLIFYING FRACTIONS Reducing or simplifying a fraction means writing it in its lowest term. Anyone who uses money is familiar with the concept of reducing fractions. A quarter is 25 out of 100
pennies 10025
or one of four quarters 41 . 10 dimes equals 1 dollar, therefore one dime
equals 101 of a dollar. 10 out of 100 pennies has the same value, 10 cents, and is
written not as 10010 but in its simplified form as
101 .
Below are examples of simplifying fractions.
Example 1: Simplify 10025
Step 1) First find a number that goes evenly into both the numerator and the denominator, in this case 25 is the only number that will work.
41
251002525 =
÷÷
Step 2) Next, see if another number will go evenly into the now reduced fraction. If no further reduction is possible, then the fraction is said to be at its lowest terms.
Example 2: Reduce 8154
Step 1) Find a number that goes evenly into the numerator and denominator. In this example, both numbers can be divided evenly by 9.
96
981954 =
÷÷
Step 2) Now that the fraction has been reduced once, check to see if it is possible to reduce it further. In this case, both numbers can be further divided by 3.
32
3936 =
÷÷
Step 3) Now check again to determine if the fraction can be reduced
further. If not, the fraction is at its lowest terms. 32 can not be reduced: it
is the answer.
Remember that no matter how much the numerator and denominator are reduced, the value of the fraction remains equal to the original just as 10 out of 100 pennies is equal to 1 out of 10 dimes.
5
Short cut: You can shorten the process if the numerator and denominator both end in zero, by crossing out the last zero in the numerator and the last zero in the denominator. This does not change the value, but does reduce the fraction.
Example 3: Simplify 10070
Step 1) Cross out a 0 at the end of each number.
Since you can only cross out one zero in the numerator, you can only cross out one zero in the denominator.
107
10070 =
Step 2) Check to see whether another number goes evenly into both the numerator and denominator of the fraction. Since no other number goes evenly into both 7 and 10 (besides 1), the fraction is reduced as far as it will go.
Example 4: Reduce 20080
Step 1) Cross out ending zeros.
208
20080 =
Step 2) Reduce further by dividing both the numerator and the denominator by 4.
52
44
208 =÷
Step 3) Check to determine if the fraction can be reduced further. It cannot.
Simplify each fraction to the lowest term.
1) 16
8 = ____ 2) 560
480= ____ 3) 400
8= ____ 4) 70
45= ____ 5) 32
4= ____
6)
8118
= ____
7) 7733
=____ 8) 3228
= ____ 9) 10035
=____ 10) 2510
= ____
11)
4928
= _____ 12) 6448
= ____ 13)12030
= _____ 14) 40040
= ____ 15) 21090
= _____
6
REVIEW
• Fractional parts show equal divisions of a whole
• Fractions with the same number on the top and bottom equal a whole
• Answers to fraction problems should always be in lowest terms.
ADDING AND SUBTRACTING FRACTIONS WITH THE SAME DENOMINATOR
With this much information you can easily add or subtract fractions with the same denominators (bottom numbers) by adding or subtracting the numerators (top numbers) and keeping the denominator the same. To work problems with mixed numbers (whole numbers and fractions), add or subtract the fractions first and then move to the whole numbers. Remember to reduce to their lowest terms all the fractions in your answers.
Example 1: Adding two fractions with the same denominator: 91
91 +
Step 1) Add the numerators: 1 + 1 = 2
Step 2) Write the sum from Step 1 over the denominator: 92
Example 2: Adding several fractions with a common denominator and reducing the
answer to lowest terms: 117
2214
22275
222
227
225 ===++ ++
Example 3: Adding mixed numbers to fractions with the same denominator:
31
315 +
Step 1) Add the fractions: 32
31
31 =+
Step 2) Place the whole number before the fraction: 325
(When adding two or more sets of mixed numbers, first add the fractions and then add the whole numbers separately). Example 4: Subtracting one fraction from another with a common denominator:
52
53 −
Step 1) Subtract the numerators: 123 =−
Step 2) Write the difference from Step 1 over the common denominator: 51
7
Example 5: 1012
1096 −
Step 1) Subtract the fractions and simplify/reduce: 54
108
101
109 ==−
Step 2) Subtract the whole numbers: 426 =−
Step 3) Put the whole number to the left of the fraction: 544
Work these problems. Simplify/reduce if possible:
1) =+83
82
_____ 2) =+171
173
_____ 3) =+42
41
_____
4) =++162
165
161 _____ 5) =−
6420
6442
_____ 6) =−162
167
_____
7) =+913
917 _____ 8) =−
317
3115
_____ 9) =−811
8110
_____
10) =++3612
364
368
_____ 11) =−4832
4846
_____ 12) =−653
6548
_____
13) =+195
199
_____ 14) =+3010
30102 _____ 15) =+
1328
1328 _____
16) =−2123
2193 _____ 17) =−
4056
40158 _____ 18) =−
102
1095 ______
19) 3261 + =
321015 _____ 20) =+
1879
18553 _____ 21) =−
913
9782 ______
22) =+16519
16562 _____ 23) =+
49154
49133 _____ 24) =+
99233
991011 _____
25) =−837
8513 _____ 26) =−
513
5421 _____ 27) =−
600303
6008041 _____
Extra Effort
28)
____49187
4942
49614 =++
29)
=−−5484
5453
544022 ______
30)
____8872
8834
888713 =−−
8
Some problems contain a whole number and a mixed number (or a fraction) where there is nothing to be added or subtracted from the fraction. In this case, deal with the fraction first by transferring it directly to your answer, then move to the whole numbers.
Example 1: 9 +72
=
Step 1) There is no fraction to be added to72
9 write 2/7 in the answer space
+ 2/7
Step 2) There is no whole number to be added to 9. 9 write 9 in the answer space answer: 9 2/7
+__2/7_
Example 2: 1+5112 =
Step 1) There is no fraction to be added to 1/5. 1 write 1/5 in the answer space
+12 1/5
Step 2) There is a whole number, 1, to be added to 12 1 write 13 in the answer space answer: 13 1/5
+12 1/5
Example 3: =− 69516
Step 1) There is no fraction to be subtracted from 5/9. 16 5/9 write 5/9 in the answer space
- 6____
Step 2) There is a whole number, 6, to be subtracted from 16 16 5/9 write 10 in the answer space answer: 10 5/9
- 6____
9
Work these problems:
1) =+327 _____ 2) =+ 11
43
_____ 3) =+169115 _____
4) =+ 3171117 _____
5) 14 + 4 +291510 = _____
6) =− 6857 _____
7) =− 2654 _____
8) =− 992199 _____
9) =− 33787782 _____
10) =+3263 _____
11) =− 815240 _____
12) =− 3836 _____
13) =+314 _____
14) =+838 _____
15) =+ 9174
_____
16) =− 9837777 ______
17) =+714955 _____
18) =− 9292100 _____
19) =− 7881431 _____
20) =−− 2216582 _____
You can only add or subtract fractions that have the same denominators. To this point all the fractions have had the same denominators, but that isn’t always the case. Turn the page to find out how to change problems with different denominators so that you can add or subtract them./
10
RAISING FRACTIONS
To add and subtract fractions with different denominators you will also need to learn how to raise fractions to higher terms. (This creates improper fractions. That’s okay – leave them alone for now) There are 2 ways to do this: one is by dividing the old denominator into the new.
Example 1: Raise 54
to 20ths. 20?
54 =
Step 1) In this problem 20 becomes the new denominator. To find the new numerator, divide the old denominator Into the new one: 20 ÷ 5 = 4
Step 2) Then multiply the numerator by that number, (4): 201644 =×
Step 3) Check your answer by dividing the raised fraction by 4.
54
420416 =
÷÷
The second way to raise fractions is by multiplying. Multiply the old denominator by whatever it takes to get to the new one, then multiply the original numerator by the same number. What you do to the bottom number, you must do to the top.
Example 2: Raise 97
to 27ths. 27?
97 =
Step 1) In this problem 27 becomes the new denominator. Multiply 9 X 3 to get 27
Step 2) Then multiply the numerator by that number, (3):
2721
3937 =
××
Step 3) Check your answer by dividing the raised fraction by 3.
97
327321 =
÷÷
Raise each fraction to higher terms.
1) 255
3 = 2) 3612
9 = 3) 328
7 = 4) 244
3 =
5) 279
7 = 6) 10010
5 = 7) 497
2 = 8) 486
3 =
9) 162
1 = 10) 183
2 = 11) 728
3 = 12) 153
2 =
13) 568
7 = 14) 484
1 = 15) 306
5 = 16) 505
4 =
11
CREATING MIXED/WHOLE NUMBERS FROM FRACTIONS
When adding fractions, the answer may be an improper fraction, such as 59
, a fraction that is greater than the whole. To finish solving a problem resulting in an improper fraction, you must convert it to a mixed number by dividing the denominator into the numerator and writing the remainder, if any, over the original denominator.
Example 1: Change 426
to a mixed/whole number.
Step 1) Divide the denominator into the numerator: 2
24
6264
Step 2) Write the 6 as a whole number. (There is a remainder of 2) Step 3) Write the remainder as a fraction over the original denominator: 4
26
Step 4) Reduce answer: 426 =
216
Example 2: Change 756
to a whole number
Step 1) Divide the denominator into the numerator 8
567
Step 2) Write 8 as a whole number (There is no remainder)
Step 3) No reducing necessary. The answer is 8. Practice changing improper fractions into mixed/whole numbers
1) 1122
= _____ 2) 935
= _____ 3) 5
100=_____
4) ____5
50 = 5) _____742 = 6) _____
241 =
7) 5
13= _____ 8)
90104
= _____ 9) 1212
= _____
10) _____3
77 = 11) _____8
345 = 12) _____27691 =
13) _____17447 = 14) _____
100000,1 = 15) _____
31563 =
12
HOW TO FIND THE COMMON DENOMINATOR A common denominator is that number which can be divided evenly by all the
denominators in a problem. The smallest number that can be divided evenly by all
denominators is called the lowest common denominator (LCD). First try using the largest denominator in the original problem as the LCD.
Example: =+41
81
Since 4 can go into 8 evenly, 8 can be used as the common denominator.
82
41
81
81 == and Rewritten, the problem becomes =+
82
81
Answer: 83
Solve these problems and reduce if necessary.
1) =+21
41
_____
2) =−31
65
_____
3) =+43
85
_____
4) =−41
127
_____
5) =++53
107
21
_____
6) =+253
54
____
7) =+5412
91
_____
8) =−114
7732
_____
9) =−83
3215
_____
10) =−72
217
_____
11) =++65
31
124
_____
12) =++2013
85
409
_____
13) =−73
5632
____
14) =++54
31
154
_____
15) =−21
3025
_____
16) =++8815
114
83
_____
17) =++65
31
124
_____
18) =−31
2411
_____
19) =−95
3627
_____
20) =++89
21
6421
_____
21) =++25
281
74
_____
22) =++53
209
107
_____ 23) =++3817
21
193
____ 24) =−−61
62
180110
_____
13
When none of the denominators will work as the (LCD), try:
• Going through the multiplication tables of the denominators to see if there
are common multiples
• Multiplying 2 small denominators together, then checking that number
against any other denominator (sometimes ½ of this will work—try it just to
see)
• Multiply all the denominators together ( Use this if all else fails: it always
works, but results in a VERY large number which means you will have a
lot of reducing/simplifying to do at the end)
Example 1: =++41
65
31
• Go through the multiplication table of the 3’s.
• 3 X 2 = 6 which cannot be divided by 4.
• 3 X 3 = 9 which cannot be divided by either 6 or 4.
• 3 X 4 = 12 which can
• Raise each fraction to 12ths.
be divided by 3, 6 and 4. Use 12 as the common
denominator.
,124
31 =
,1210
65 = ==
123
41
• Rewrite and add: 1217
123
1210
124 =++
• Answer (reduced): 1251
If there are whole numbers in the problem, deal with them separately
Example 2: 2 =+54
43
• Multiply the denominators to find the LCD: 4 x 5 = 20
• Raise each fraction to 20ths. 2016
54
2015
43 == and
• Rewrite and add: 2 20312
2016
2015 =+
• Answer: 32011
14
Solve these programs and simplify/reduce.
1) 3 83
1615 − = _____
2). 4 5107 1+ =_____
3) 5 165
43 + =_____
4) 7 61
32 + = _____
5) 7 210 114 − =____
6) 2 3
198 − = ____
7) 14 52
151 + =______
8) 8 32
2119 − = _____
9) 321
+77
= _____
10) 9 124
83 + =_____
11) 4
1662 + =_____
12) 3
25 1797 + =_____
13) 9 =− 41
53
______
14) =− 32
1615
_____ 15) 81
107 − =______
16) =− 72
43
____
17) =+ 32
217 _____
18) =− 9
4 552 ____
19) =+ 43
1048 _____
20) =− 83
659 _____
21) =+ 6
5434 ____
Extra effort
22) 32
+92
+1255 = ______
23) 52
+103
+1537 = _____
24) 735 +
1159 +
2112 = ______
15
BORROWING: A MULTI-STEP PROCESS
In some cases, to solve a subtraction problem you will have to borrow 1 from a whole number to create a needed fraction. This involves several critical steps:
1) LCD2)
(Identify/create the Lowest Common Denominator) Borrow
3) 1 from the larger whole number (never the smaller one)
Change• you may NOT change the value of the number, but you may state it differently
the borrowed 1 into the fraction you need it to be
($1 can be a single bill or 10 dimes or 4 quarters or 100 pennies) • fractions with the same top and bottom numbers equal 1:
11616;1
77;1
44 ===
4) Merge5)
the fraction from the borrowed 1 with that given in the problem (rewrite)
Subtract
Example 1: This is the easiest, involving only steps 1, 2, 3 & 5 (step 4 is not needed)
=−3268 Step 1) LCD
There is nothing above the
(you have only one: use 3rds as the LCD)
32
, yet you have to subtract it from
something.
Step 2) Borrow
1 from the 8, replacing the 8 with a 7 (+1) )1(78 +=
733 Step 3) Change
33 (You need more 3rds) Change the (+1) to
You now have a workable subtraction problem.
Step 4) is not needed
311
326
337 =− Step 5) Subtract
16
Example 2: The first fraction is smaller than the second fraction. Make it larger.
=−547
5113 Step 1) LCD
(you have only one: use 5ths as the LCD
51)1(1213 +→ Step 2) Borrow
1 from the 13, replacing it with 12 (+1)
)551( =+
Step 3) Change
the (+1) into 5/5
5612
55
5112 =+ Step 4) Merge and rewrite
the borrowed fraction with the problem
525
547
5612 =− Step 5)
Subtract
Practice borrowing by solving the following problems. Reduce if possible.
1) =−76
214 _____
2) =−546 _______
3) =−21
1031 ______
4) =−414 ______
5) =−107
529 ______
6) =−43
327 ______
7) =−157
3131 ______
8) =−326
3115 _____
9) =−3269 _______
10) =−5421 ______
11) =−
533
10313 ______ 12) =−
329
12719 ______
13) =−87142 _______
14) =−12712
4151 _____
15) =−215
11293 ______
16) =−11726
2152 ____
17) =−65263 _____
18) =−547
739 ______
19) =−97247 _____
20) =−
753
317 _____ 21) =−
432
8113 ______
22) =−1094
5318 _____ 23) =−
85
511 _____ 24) =−
75621 _____
17
25) =−433
12511 _____ 26) =−
1032 ______ 27) =−
972
5310 ____
WORD PROBLEMS: ADDITION AND SUBTRACTION Many of the trades use word problems on their qualifying tests. Look for key words that tell you whether you need to add or subtract. Words such as sum or total, or any problem that asks you to combine more than two numbers can be solved through addition. Difference, left, & remaining are some key words that tell you this is a subtraction problem. Figuring out an increase or decrease also may involve subtraction, as does the concept of something getting smaller (cut, shrunk, eaten or lost). In setting up subtraction problems the larger number always goes above the smaller one. To be correct, each answer must include numbers and words or symbols.
1) Vanessa cut off 833 inches from an 8” inch brick. How long was the remaining
piece?
2) Annette weighs 174 pounds. When she puts on her 4317 pound tool belt and her 5
lb. steel toed boots, how much does she weigh?
3) LaShawn cut a length of PVC into three pieces measuring, 16142 ”,
437 ” and
8113 ”.
How much PVC did she use?
18
4) Sue only used her truck for work. Monday morning the odometer on the truck showed 60,345
103 miles. Back at home Friday afternoon it showed 60,830
107 miles.
How far did she travel to work that week?
5) From an 80 pound bag of cement, Mary used 33 85 pounds to make concrete. How
much was left in the bag?
6) Kelly used up a box of nails to compete two projects. She used 24 41 lbs of nails for
the first project and 25 43 nails for the second one. How heavy was the box originally?
7) Before break, Crystal made up 3211 sets of nuts, bolts and washers. Before lunch,
she made up 9 more sets and after lunch, she finished another 5210 sets. How many
sets did she make?
8) If you lower the speed of a cement mixer from 12 rpms to 315 rpms, how much
slower does it spin?
9) Rosa’s 432 pound work boots weighed
8310 pounds after she worked in the mud
all day. How much was the increase?
19
PERIMETER
Perimeter is a measurement of the distance around an object. You will need to know how to measure perimeters to build a fence or to do other construction jobs that involve perimeters. To determine the perimeter of a shape, you must measure the sides, then add the sides together. Example: What is the perimeter of the following shape?
2 1/2'
16 1/12'
Add 2 1/2' + 2 1/2' (for each end of the figure) and 16 1/12' + 16 1/12' (for each side). The perimeter is then 37 1/6'. Use the symbol ' for feet and “ for inches. Note: Answers to perimeter problems must always include a unit of measurement. Determine the perimeter of the following shapes. 1) 2) P = _____ P = _____ 3) 4) 5) P = _____ P = _____ P = _____ 6) 7) 8) P = _____ P = _____ P = _____
413 '
2116 ’
216 '' 3
11 yd
317
ft
328 ft
8112 “ 6
16 yd 219
yd
311 yd
1612 “
8112 ''
318 yard
1016
yard
8110 miles
517
miles
20
PRACTICE PROBLEMS: ADDITION AND SUBTRACTION
1) =+61
31
_____
2) =+329
83
_____
3) =−43
87
_____
4) =−21
313 _____
5) =−95
951 _____
6) =+521
1077 _____
7) =−872
8110 _____
8) =++ 312511
21
_____
9) =−322
16713 _____
10) =−21
97
_____
11) =++ 11
22110
11216 ____ 12) =+
326
326 _____
13) =++1278
1278
1278 ___ 14) =−
643119
32131 __ 15) =++
217
539
324 ____
16) =++513
876
201313 ___ 17) =++
765
5117
352392 ___ 18) =++
5476
6133
302955 __
19) =−1054
10314 ____ 20) =−−
611
12116
9523 ____ 21) =−
1543
30715 _____
22) If a yard has 5 sides: 7’,2116 ’,
4113 ’,
2118 ’ and
316 ’, what is its perimeter?
Extra Effort
23) If the perimeter of a park is 41399 ’, and 3 sides are
21143 ’,
41142 ’, and
3155 ’,
what length is the 4th side?
24) The perimeter of a triangle is 96 yards. If one side is 3229 yards and another side
is 3130 yards, how long is the remaining side?
21
MULTIPLICATION OF FRACTIONS When multiplying fractions, you simply multiply the numerators together and the denominators together: no need for common denominators! Reduce/simplify if possible.
Example: =× 53
52
Step 1) Multiply the numerators. 2 x 3 = 6
Step 2) Multiply the denominators 5 x 5 = 25 Answer: 256
Multiply and reduce.
1) =×85
73
_____
2) =×75
32
_____
3) =×74
53
_____
4) =×95
31
_____
5) =×32
137
_____
6) =×21
65
_____
7) =×43
43
_____
8) =×87
103
_____
9) =×85
113
_____
10) =×41
21
_____
11) =×32
52
______
12) =×76
76
______
13) =×32
83
______
14) =×51
63
_______
15) =×32
21
______
Extra effort
16) =×25
74
______
17) =×1510
32
______
18) =×15
62
______
19) =×32
37
______
20) =×98
1113
______
21) =×49
3231
______
22
Shortcut
Cross canceling is a shortcut in multiplication of fractions. It is similar to simplifying or reducing. To cross cancel, divide any numerator and any denominator by a number that goes evenly into both. This process, though not required, makes the numbers smaller and, therefore, easier to multiply.
Example: 2112
87 ×
Step 1) Cancel 8 and 12 by 4. 8 ÷4 = 2 and 12 ÷ 4 = 3. Cross out the 8 and the 12 and replace them with 2 and 3.
213
27
21312
287 ×=→×→
Step 2) Cancel 7 and 21 by 7. Cross out the 7 and the 21 and replace them with 1 and 3.
3213
217
213
27
→×→=× Step 3) Multiply across by the new numbers.
Step 4) Simplify/ reduce.
21
63 =
Cross cancel and multiply.
1) =×2012
85
_____ 2) =×65
54
_____ 3) =×128
43
_____
4) =×97
73
_____ 5) =×103
125
_____ 6) =×31
1512
_____
7) =×109
65
_____ 8) =×2812
247
_____ 9) =×94
83
_____
10) =×103
115
_____ 11) =×154
125
_____ 12) =×1310
21
_____
13) =×207
354
_____ 14) =×93
153
_____ 15) =×169
278
_____
23
When there are three fractions in an equation, check to see if you can cross cancel to simplify the problem. Multiply the first two numerators together then multiply that answer by the third numerator. Do the same for the denominators. Simplify/reduce.
Example: =×× →→
482
53
714
22
53
71 ××
Multiply the first two numerators then multiply the answer times the last numerator.
331 =× 623 =× Multiply the first two denominators then multiply the answer times the last denominator.
3557 =× 70235 =×
Simplify/ reduce. 353
706 =
Multiply and simplify/reduce the following:
1) =××2
1
4
3
3
2 _____ 2) =××
3
1
8
5
9
4 _____ 3) =××
6
5
4
3
3
2 _____
4) =××8
7
6
5
5
3 _____ 5) =××
2
1
7
5
5
2 _____ 6) =××
7
6
6
5
4
3 _____
7) =××8
1
9
2
7
3 _____ 8) =××
4
1
3
1
2
1 _____ 9) =××
3
1
9
8
8
7 _____
10) =××5
2
10
3
12
5_____ 11) =××
9
7
8
5
8
3_____ 12) =××
5
3
8
7
14
9______
13) =××11
5
21
9
18
7_____ 14) =××
15
11
9
5
22
9 ____ 15) =××
8
3
12
1
7
4_____
16) =××12
5
4
3
20
1_____
17) =××
5
12
3
2
7
2_____ 18) =××
11
9
2
1
81
77______
24
CREATING IMPROPER FRACTIONS FROM WHOLE AND MIXED NUMBERS You cannot multiply fractions by whole or mixed numbers: you can only multiply a fraction by another fraction. To change a whole number into a fraction, use the given number as the numerator and put it over a 1 as its denominator. To change a mixed number into an improper fraction multiply the denominator by the whole number and add the numerator. Place the total over the original denominator. This changes the whole/mixed number into an improper fraction without changing its value. Example 1: Change 5 to an improper fraction. Place a 1 under the 5, creating
15 , which
is an improper fraction. Example 2: Change
433 to an improper fraction.
Step 1) Multiply the denominator by the whole number: 1243 =× Step 2) Add the result to the numerator: 12 + 3 = 15 Step 3) Place the total over the original denominator:
415
Example 3: Change 3
17 to an improper fraction. Step 1) 2137 =× Step 2) 21 + 1 = 22 Step 3)
322
Create improper fractions from the whole and mixed numbers below.
1) =414 _____
2) =3 _____
3) =327 _____
4) =748 _____
5) =542 _____
6) =1039 _____
7) =835 _____
8) 11 =_____
9) =4312 _____
10) =524 _____
11) =3220 _____
12) =5313 _____
13) =1132 _____
14) =854 _____
15) 14 =_____
16) =12532 ____
17) =
6123 _____ 18) =
7618 _____ 19) =
10943 ____ 20) =
9214 _____
25
Note: on this page all of the answers are left in improper fraction form!
MULTIPLYING WITH MIXED NUMBERS
Once you have converted whole or mixed numbers to improper fractions, multiply as shown in the following example:
Example 1: 20175 ×
Step 1) Write 5 as an improper fraction. 155 =
Step 2) Cross cancel 5 and 20 by 5. 42017
115
→×→
Step 3) Multiply the numerators and denominators.4
174
1711 =×
Step 4) Simplify/reduce 414
417 =
Example 2: =× 169
326
Step 1) Change 326 to an improper fraction. 3
20326 =
Step 2) Cross cancel 20 and 16 by 4. 49
35
4169
3520 ×=→×→
Cross cancel 9 and 3 by 3. 439
135 →→ ×
Step 3) Multiply the numerators and denominators. 415
4135=×=×
Step 4) Simplify/reduce. 4334
15 =
26
Multiply and reduce.
1) =× 31214
134 _____ 2) =× 7465
3 _____ 3) =× 10
75 _____
4) =× 7558
34 _____ 5) =× 41
872 _____ 6) =× 5
11312 _____
7) =× 9749 ______ 8) =× 21
1071024 _____ 9) =× 7
3312312 _____
10) =× 1211312 _____ 11) =× 149
5 921 _____ 12) =× 5466
59 _____
13) =× 32420
126 _____
14) =× 11514
310 _____
15) =× 116214
52 _____
16) =×× 43149
97 _____
17) =× 371516288 _____
18) =× 9
48721 _____
19) =×× 5339
8431 ____ 20) =×× 4
1215352
17 _____ 21) =×× 87
214
152
____
22) =×× 74327
219 ____ 23) =×× 13516
12756 _____ 24) =×× 6
59545
17 ____
27
WORD PROBLEMS: MULTIPLICATION
Use multiplication to solve word problems that ask you to: 1. Find a part of something 2. Use the information given about one thing to determine the weight/size/cost of
several things. Remember than fractional answers must be in simplified terms and include a word or symbol.
1) From home to work Lynn has to drive 75 miles. She usually stops for breakfast 32
of the way there. How far has she already driven when she stops for breakfast?
2) If one gallon of fresh water weighs 218 pounds, what is the weight of 2
11 gallons?
_____
3) For an entertainment center Kris plans to build, she wants four shelves that are
each 323 feet long. What is the total length of the four shelves Kris needs?
4) Mary gets paid $48/hour when she works overtime. If she worked 433 hours
overtime, how much would she be paid for the overtime work? ______________ Extra effort ( Hint: eliminate all but the essential numbers) 5) While on a 5 day vacation in Cancun, Aimee wanted to be outside 8 hours a day.
She spent 53
of her days lounging at the beach. How many days did Aimee spend lounging at the beach?
__________
6) If one fifty foot roll of 81
inch copper tubing costs $59.98, how much will 213 rolls
cost?
______
7) Nancy makes $14 an hour for regular time. When she works overtime she gets paid
time and a half. If she works 6 21
hours overtime in one week, how much extra
28
pay will she get for the overtime worked? _________________ DIVISION OF FRACTIONS
Now that you have mastered the processes required to multiply fractions, you are just one step away from division. To solve any division of fractions problem, remember:
1. You can only divide fractions by fractions, so change any whole/mixed numbers to improper fractions.
2. Invert (turn upside down) the fraction to the right of the division sign. 3. Change the division sign to a multiplication sign. 4. Multiply Example 1: 8
543 ÷
Step 1) is not needed
58
85 → Step 2) Invert the fraction on the right ( 8
5 ) to 58
58
43 × Step 3) change to x :
52
13 × Cancel 4 and 8 by 4.
511
56
52
13 ==x Step 4) Multiply and simplify/reduce
Example 2: 7
68 ÷
18 Step 1) Write 8 as a fraction.
67
76 → Step 2) Invert 7
6
Step 3) Change ÷ sign to x. 6
718
76
18 ×=÷ Cancel 8 and 6 by 2
. 3
19328
37
14 ==× Step 4) Multiply and simplify/reduce.
Example 3: 4
1312 ÷
41
37 ÷ Step 1) Change 3
12 to an improper fraction: 37
14
41 ÷ Step 2) Invert 4
1
14
37
14
37 ×=÷ Step 3) Change ÷ sign to x.
29
319
328
14
37 ==x Step 4) Multiply and simplify/reduce
Divide and simplify/reduce.
1) 21
32 ÷ = _____ 2)
161
161 ÷ = ____ 3)
1001
101 ÷ =____ 4) =÷
944 ____
5) 31615 ÷ =____ 6) 4
85 ÷ = _____ 7)
7618 ÷ = ____ 8)
611
1514 ÷ = ____
9) 324
97 ÷ = ____ 10)
87
127 ÷ = ____ 11) 5
2512 ÷ = ____ 12) 2
713 ÷ =____
13) 3112 ÷ = ____ 14)
2019
1092 ÷ = ____ 15)
229
1177 ÷ = ___ 16)
43
1271 ÷ = ___
17) 1271
924 ÷ = ____ 18)
1612
817 ÷ = ____ 19)
651
979 ÷ = ____ 20)
1211
322 ÷ =___
21) 323
833 ÷ _____ 22)
751
733 ÷ = _____ 23)
314
64402 ÷ = _____
Extra effort
24) 43
65
52 ÷÷ =___
25) 6454
722 ÷÷ ___
26) 43
87
641 ×÷ = ____
30
Word Problems: Division
To solve division of fractions word problems, first read the problem and determine-- “What is the entire thing to be evenly divided?” Write that number first, followed by the ÷ sign. The last number represents how many will be sharing or made from the first. The concept is: The whole divided/cut/packaged into how many
equal parts.
Remember to invert the number to the right of the ÷ sign. Always simplify your answer and include a word or symbol. 1) How many 7 2
1 inches pieces can be cut from 90 inches of rebar?
2) Kelly’s goal was to run 212 miles. She wanted to mark each 4
1mile with balloons.
How many would she need?
3) If Olivia needs 811 yards of denim to make 1 nail apron, how many can she make
from 433 yards of material?
4) How many 322 pound candles can be made from 3
210 pounds of candle wax?
5) How many 21
pound boxes of nails can be filled from 2116 pounds of nails?
6) Sheet metal 52 feet long is to be cut into 314 foot pieces. How many can be cut?
7) Leslie needs 412 gallons to paint a room. If all the rooms she’s painting are the
same size, how many can she paint with 9 gallons?
8) Marisol and Agnes are on a diet. They want to split 41
pound of chocolate evenly between them. How much would each get?
32
PRACTICE PROBLEMS: MULTIPLICATION AND DIVISION
1) 61
31 × = _____ 2)
32921× = _____ 3)
43
877 ÷ =_____
4) 315
213 ÷ = _____ 5)
98
2113 × = _____ 6)
521
1077 ÷ = _____
7) 273
973 ÷ = ______ 8)
4316
83 ×× = _____ 9)
313
957 ÷ =_____
10) 402524 ÷ = _____ 11)
15131
953 ÷ = _____ 12)
85
526
326 ×× = _____
13) 2116
3716 ÷ = _____ 14)
158
115
1211 ×× =_____ 15)
71
2193 ÷÷ = _____
16) 543× =_____ 17) 20
8011 × =_____ 18)
1094× = _____
Extra Effort
19) 517
623
30255 ×× =_____ 20)
755
71
838 ×÷ = _____ 21)
73
1511
3223 ×× = _____
22) 952
462310 × = _____ 23)
71
12511
167 ×× = _____ 24)
322
161113 ÷ = _____
33
Fractions packet answer key
Page 1
1) 81
2) 3) Page 2
1) 3613
2) 127
3) 10059
4) 167
5) 1211
6) 150103
7) 41
8) 2000147
Page 5
1) 21
2) 76
3) 501
4) 149
5) 81
6) 92
7) 73
Page 5 cont.
8) 87
9) 207
10) 52
11) 74
12) 43
13) 41
14) 101
15) 73
Page 7
1) 85
2) 174
3) 43
4) 21
5) 3211
6) 165
7) 10 92
8) 318
9) 91
10) 32
11) 247
Page 7 cont.
12) 139
13) 1914
14) 2 32
15) 16 134
16) 31
17) 2 41
18) 5 107
19) 16 21
20) 62 32
21) 79 32
22) 81 85
23) 7 74
24) 14 31
25) 6 41
26) 18 53
27) 38 121
28) 23 74
29) 15 21
30) 7 87
Page 9
1) 7 32
2) 11 43
3) 16 169
4) 20 1711
5) 28 2915
6) 1 85
7) 2 65
8) 21
9) 49 7877
10) 9 32
11) 32 152
12) 3 83
13) 4 31
14) 8 83
15) 9 174
16) 68 8377
17) 104 71
18) 8 92
19) 353 81
20) 59 65
Page 10
1) 2515
2) 3627
3) 3228
4) 2418
5) 2721
6) 10050
7) 4914
8) 4824
9) 168
10) 1812
11) 7227
12) 1510
13) 5649
14) 4812
15) 3025
16) 5040
34
Answer Key Continued
Answer Key Continued
Page 11 1) 2
2) 3 98
3) 20 4) 10 5) 6
6) 20 21
7) 2 53
8) 1 457
9) 1
10) 25 32
11) 43 81
12) 25 2716
13) 26 175
14) 10
15) 18 315
Page 12
1) 43
2) 21
3) 1 83
4) 31
5) 1 54
Page 12 cont.
6) 2523
7) 31
8) 774
9) 323
10) 211
11) 1 21
12) 1 21
13) 71
14) 1 52
15) 31
16) 1110
17) 1 21
18) 81
19) 367
20) 1 6461
21) 3 283
22) 1 43
23) 1 192
24) 91
Page 14
1) 3 169
2) 4 109
3) 6 161
4) 7 65
5) 7 143
6) 2 95
7) 14 157
8) 8 215
9) 4 21
10) 9 2417
11) 6 127
12) 25 92
13) 9 207
14) 4813
15) 4023
16) 2813
17) 8 61
18) 2 4511
19) 9 203
20) 9 2411
21) 5 127
Page 14 cont.
22) 6 3611
23) 7 109
24) 27 15459
Page 16
1) 3 149
2) 5 51
3) 54
4) 3 43
5) 8 107
6) 6 1211
7) 30 1513
8) 8 32
9) 2 31
10) 20 51
11) 9 107
12) 9 1211
13) 40 81
14) 38 32
15) 87 2215
16) 25 2219
17) 60 61
Page 16 & 17
18) 1 3522
19) 44 92
20) 3 2113
21) 10 83
22) 13 107
23) 4023
24) 14 72
25) 7 32
26) 1 107
27) 7 4537
Page 17 & 18
1) 4 85
inch 2)
196 43
pounds
3) 62 1615
inch
4) 485 52
miles
5) 46 83
lbs. 6) 50 lbs.
7) 31 151
sets
8) 6 32
rpms
9) 7 85
lbs.
35
Answer Key Continued
Page 19
1) 39 21
feet 2) 32 feet
3) 37 41
in. 4) 15 yds.
5) 21 32
yds.
6) 28 83
in.
7) 28 1513
yards
8) 34 2013
miles Page 20
1) 21
2) 3221
3) 81
4) 2 65
5) 1
6) 9 101
7) 7 41
8) 14 1211
9) 10 4837
10) 185
11) 37 225
12) 13 31
13) 25 43
14) 11 6435
15) 21 3023
16) 23 4029
17) 115 75
18) 165 1514
19) 9 54
20) 15 3617
21) 11 3029
22) 61 127
’
23) 58 61
’ 24) 36 yds. Page 21
1) 5615
2) 2110
3) 3512
4) 275
5) 3914
6) 125
7) 169
8) 8021
9) 8815
10) 81
Page 21 cont.
11) 154
12) 4936
13) 41
14) 101
15) 31
16) 1 73
17) 94
18) 1 32
19) 1 95
20) 1 995
21) 2 12823
Page 22
1) 83
2) 32
3) 21
4) 31
5) 81
6) 154
7) 43
8) 81
9) 61
Page 22 cont.
10) 223
11) 91
12) 135
13) 251
14) 151
15) 61
Page 23
1) 41
2) 545
3) 125
4) 167
5) 71
6) 2815
7) 841
8) 241
9) 277
10) 201
11) 19235
12) 8027
13) 665
14) 61
Page 23 cont.
15) 561
16) 641
17) 3516
18) 187
Page 24
1) 417
2) 13
3) 323
4) 760
5) 514
6) 1093
7) 843
8) 111
9) 451
10) 522
11) 362
12) 568
13) 1125
14) 837
15) 114
16) 12389
36
Page 24 cont.
17) 6139
18) 7132
19) 10439
20) 9128
Page 26
1) 11 21
2) 3 3533
3) 3 21
4) 25
5) 3223
6) 2 54
7) 43
8) 31 52
9) 42 10) 47
11) 4 91
12) 66 1513
13) 30 54
14) 15 117
15) 6
16) 2 41
17) 20 3728
Page 25/26 cont.
18) 10 76
19) 5 53
20) 87 43
21) 451
22) 25
23) 20 71
24) 27 31
Page 27 1) 50 miles
2) 12 43
lbs.
3) 14 32
feet 4) $180 5) 3 days 6) $209.93 7) $136.50 Page 29
1) 1 31
2) 1 3) 10 4) 9
5) 165
6) 325
Page 29 cont. 7) 21
8) 54
9) 61
10) 32
11) 12512
12) 1 74
13) 1 21
14) 3 191
15) 18 32
16) 2 91
17) 2 32
18) 3 115
19) 5 31
20) 2 1110
21) 36 22) 2
23) 10463
24) 2516
25) 89655
26) 2243
Page 30 1) 12 pieces 2) 10 markers
3) 3 31
aprons 4) 4 candles 5) 33 boxes 6) 12 pieces 7) 4 rooms
8) 81
pound Page 31
1) 181
2) 5 3229
3) 10 21
4) 3221
5) 12
6) 5 21
7) 34
8) 4 21
9) 2 154
10) 1253
11) 1 2119
12) 26 32
Page 31 cont.
13) 3721
14) 92
15) 49
16) 2 52
17) 2 43
18) 3 53
19) 140 20) 335
21) 8077
22) 26 65
23) 192137
24) 5 12817