Basic Math Review
1
Whole Numbers
PLACE VALUE
Ones
Tens
Hundreds
,
Thousands
Ten thousands
Hundred thousands
,
millions
Ten millions
Hundred millions
,
Billions
Whole numbers are written with the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The number 66 has two digits.
The number 24,070 has five digits. The value of a digit depends on its position in the number. Every
position has a place value. The chart below gives the names of the first ten whole number places.
The 6 at the left in 66 (66) is in the tens place. It has a value of 6 tens or 60. The 6 at the right (66) is in
the units or ones place. It has a value of 6 ones or 6. The digit is still 6, but the value is different
because of its position.
The 2 in 24,070 is in the ten thousands place. It has a value of 2 ten thousands or 20,000. The 4 is in the
thousands place. It has a value of 4 thousands or 4,000. The 7 is in the tens place. It has a value of 7
tens or 70. Notice that the value of the hundreds place and the value of the units place are both 0 in
24,070.
EXAMPLE: Find the value of 8 in 2,840.
2,840
100
x8
800
Step 1 Step 2
Step 1: Tell the name of the place 8 is in. 8 is in the hundreds place.
Step 2: Multiply 8 by the place name.
Large numbers are usually separated in groups of three figures from the right to the left like this:
7,654,321
Numbers are read left to right by naming each group appropriately.
In the number 4761, you would start from the right hand side of the number and mark off every three
digits with a comma as follows:
4,761 The number is read “four thousand, seven hundred sixtyone.”
In the number 3057010, you would start at the right hand side and mark off every three digits with a
comma as follows:
3,057,010
The number is read “three million,
fifty-seven thousand, ten.”
2
In the number 1357926183, the commas would appear as follows: 1,357,926,183 The number is read
“one billion, three hundred fifty-seven million, nine hundred twenty-six thousand, one hundred eightythree.
Note: The word “and” is not used when reading or writing whole numbers. It is used to indicate the
placement of a decimal point. For example, 5,008 is read “five thousand eight,” not “five thousand and
eight.”
Note: Compound numbers from 21 to 99 are written with a hyphen, such as twenty-one and ninetynine.
1.
Write 1954 in words:
________thousand, ________hundred ___________________
2.
1917 is written as
________thousand, ________hundred ___________________
3.
1812 is written as
________thousand, ________hundred ___________________
4.
25,416 is written as
________thousand, ________hundred ___________________
5.
14,703 is written as
________thousand, ________hundred ___________________
6.
10,908 is written as
________thousand, ________hundred ___________________
7.
12,100 is written as
________thousand, ________hundred
8.
12,008 is written as
________thousand, _______________
9.
10,092 is written as
________thousand, _______________
10.
123,456 is written as _________________thousand, ________hundred ____________
11.
756,100 is written as _________________thousand, ________hundred
12.
1,658,325 is written as
______million, ____________________thousand,
_____________hundred, _____________________
Write in figures:
13.
one million, two hundred forty-three thousand, five
hundred fifteen __________________________________
3
14.
two hundred ten thousand, six hundred twelve
___________________________________________________
15.
seven hundred forty-five _________________________
16.
fifty thousand, sixty-eight ______________________
17.
forty million, thirty-six ________________________
18.
five million, seven thousand, two hundred thirty-eight
____________________________________________________
19.
twelve billion, fifteen million ___________________
20.
three hundred thirteen million, seven hundred ten thousand
__________________________________________
Answers:
1.
one thousand, nine hundred, fifty-four
2.
one thousand, nine hundred, seventeen
3.
one thousand, eight hundred, twelve
4.
twenty-five thousand, four hundred sixteen
5.
fourteen thousand, seven hundred three
6.
ten thousand, nine hundred eight
7.
twelve thousand, one hundred
8.
twelve thousand, eight
9.
ten thousand, ninety-two
10.
one hundred twenty-three thousand, four hundred fifty-six
11.
seven hundred fifty-six thousand, one hundred
12.
one million, six hundred fifty-eight thousand, three hundred twenty-five
13.
1,243,515
14.
210,612
15.
745
16.
50,068
17.
40,000,036
18.
5,007,238
19.
12,015,000,000
20.
313,710,000
4
ADDITION
Addition with whole numbers is the basis for the other operations. Addition must be learned
well before progress can be made in developing math skills. The digits are the symbols 0, 1, 2, 3, 4, 5,
6, 7, 8, and 9. The numbers that are added together are called the addends, and the answer to an
addition problem is called the sum or total.
12 <----addend
+5 <----addend
17 <----sum or total
You must start with mastery of the following addition facts, because they are the foundation for
further review. If necessary, go over them a second or third time. Speed is important, but accuracy
must come first.
1)
0
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
2)
1
2
5
2
2
2
9
2
7
2
8
2
3
2
6
2
4
2
0
2
3)
1
3
3
3
5
3
7
3
9
3
2
3
4
3
6
3
8
3
0
3
4)
2
4
4
4
6
4
8
4
1
4
3
4
5
4
9
4
0
4
7
4
5)
9
5
7
5
5
5
3
5
1
5
8
5
6
5
0
5
4
5
2
5
6)
9
6
5
6
1
6
0
6
3
6
7
6
2
6
4
6
6
6
8
6
7)
8
7
4
7
6
7
2
7
0
7
7
7
9
7
5
7
1
7
3
7
8)
2
8
6
8
4
8
8
8
3
8
1
8
5
8
0
8
9
8
7
8
5
9)
5
9
3
9
4
9
2
9
9
9
1
9
0
9
7
9
8
9
6
9
Answers:
1.
1
2
3
4
5
6
7
8
9
10
2.
3
7
4
11
9
10
5
8
6
2
3.
4
6
8
10
12
5
7
9
11
3
4.
6
8
10
12
5
7
9
13
4
11
5.
14
12
10
8
6
13
11
5
9
7
6.
15
11
7
6
9
13
8
10
12
14
7.
15
11
13
9
7
14
16
12
8
10
8.
10
14
12
16
11
9
13
8
17
15
9.
14
12
13
11
18
10
9
16
17
15
Adding Numbers With More Than One Digit
To add numbers with more than one digit, write the numbers one under another with the ones’
column in a straight line; begin by adding on the right. First, add the numbers in the ones’ place (the
right-hand number). Next, add the numbers in the tens’ place (the number second from the right).
Then, add the numbers in the hundreds’ place (third from the right). Proceed in this way for as many
places as there are in the number. The first problem in each row has been done as an example.
1)
32
25
57
41
38
23
45
21
32
63
36
34
50
2)
633
264
897
371
518
337
242
451
324
382
211
322
444
3)
327
342
669
432
263
216
432
534
425
274
211
322
552
6
4)
1256
2132
3388
8491
1504
6371
2322
1233
4731
5679
3110
1256
8613
5)
4536
3343
7879
3215
3584
4167
3522
3156
5732
7135
2564
1254
6325
6)
205
342
547
306
573
514
302
307
450
620
307
340
506
Answers
1.
57
79
68
53
99
84
2.
897
889
579
775
593
766
3.
669
695
648
959
485
874
4.
3388
9995
8693
5964
8789
9869
5.
7879
6799
7689
8888
9699
7579
6.
547
879
816
757
927
846
CARRYING IN ADDITION
In the problems on the preceding page, the sum of each column as less than 10. Now we will
review problems where the sum of a column is more than 9 and we must carry to the next column.
Study the example in the box and work the problems that follow in the same manner.
538
374
912
Add the ones’ column: 8 + 4 = 12
Write the 2 under the 4; carry the 1 to the
tens’ column: 1 + 3 + 7 = 11
Write the 1 under the tens’ column; carry the
1 to the hundreds’ column: 1 + 5 + 3 = 9
Write the 9 under the 3.
7
1)
525
356
881
338
243
516
247
429
354
354
237
536
245
2)
337
427
212
359
327
555
235
719
335
528
432
472
3)
234
139
234
358
217
374
249
515
235
518
225
226
4)
636
246
223
348
325
458
319
272
475
315
216
274
5)
356
428
579
316
328
643
312
178
354
217
159
321
6)
325
206
626
206
509
249
409
321
802
158
307
529
7)
475
336
329
584
347
486
735
398
616
595
534
497
Answers
1.
881
581
763
783
591
781
2.
764
571
882
954
863
904
3.
373
592
591
764
753
451
4.
882
571
783
591
790
490
5.
784
895
971
490
571
480
6.
531
832
758
730
960
836
7.
811
913
833
1133
1211
1031
8
Add:
1)
1
5
9
6
7
8
7
3
8
9
9
9
7
8
7
5
6
8
8
9
9
7
7
4
2)
57
14
35
42
29
34
39
43
75
57
34
18
3)
43
8
5
38
8
27
4
59
37
4
38
7
4)
764
457
391
509
745
985
794
529
645
727
264
337
5)
639
284
854
765
385
437
574
245
729
154
224
355
6)
1576
834
1497
943
1769
825
1398
745
7)
2415
1385
4014
3505
3052
4705
2345
1654
8)
5307
6099
7209
4095
4096
8405
3690
7059
9
Answers:
1.
21
25
35
27
31
27
2.
71
77
63
82
132
52
3.
51
43
35
63
41
45
4.
1221
900
1730
1323
1372
601
5.
923
1619
822
819
883
579
6.
2410
2440
2594
2143
7.
3800
7519
7757
3999
8.
11,406
11,304
12,501
10,749
SUBTRACTION
Subtraction is the opposite of addition. It means to take from or take away. So, 7 - 5 means to
take 5 from 7. In subtraction, the upper number is called the minuend; the lower number is the
subtrahend. The answer to a subtraction problem is called the difference.
96  minuend
-31  subtrahend
65  difference
Below is a chart containing the basic subtraction facts that you should know.
Subtraction Facts
To use the chart to find an answer, such as 9 - 6, find the 9 on the left hand side and go along that row until
you reach the column with 6 at the top of the chart. That number (3) will be the answer to 9 - 6.
10
The following exercise will help you master basic subtraction facts. As with the addition facts,
speed is important, but accuracy must come first.
Subtract:
1)
2
0
5
1
3
3
9
2
9
0
6
3
2
1
3
2
1
0
10
3
2)
10
2
7
0
4
3
3
1
5
3
8
2
4
0
8
1
2
2
9
1
3)
8
0
4
2
9
3
3
0
11
3
1
1
8
3
4
1
7
3
6
2
4)
6
1
5
2
6
0
7
1
10
1
0
0
11
2
7
2
12
3
5
0
5)
4
4
6
5
10
6
8
4
13
6
11
5
10
4
7
5
7
6
10
5
6)
15
6
14
6
6
4
5
5
12
6
13
4
13
5
9
6
11
4
14
5
7)
12
5
5
4
11
6
9
5
8
6
7
4
9
4
8
5
6
6
12
4
8)
9
7
11
9
8
8
11
8
7
7
12
9
16
9
14
7
8
7
14
9
11
Answers:
1.
2
4
0
7
9
3
1
1
1
7
2.
8
7
1
2
2
6
4
7
0
8
3.
8
2
6
3
8
0
5
3
4
4
4.
5
3
6
6
9
0
9
5
9
5
5.
0
1
4
4
7
6
6
2
1
5
6.
9
8
2
0
6
9
8
3
7
9
7.
7
1
5
4
2
3
5
3
0
8
8.
2
2
0
3
0
3
7
7
1
5
Sometimes one or more of the place numbers in the subtrahend are larger than those in the minuend.
This makes borrowing necessary. Study the illustrations carefully to be sure you understand how to
proceed with the practice problems that follow.
52
- 17
Borrow 1 ten from the 5 tens, leaving 4 tens
4 12
Add the 1 ten you borrowed to the 2 ones;
you get 12 ones.
52
- 17
35
12 ones - 7 ones = 5 ones
4 tens - 1 ten = 3 tens
Subtract:
1)
74
25
Answers:
1.
49
80
36
44
26
75
49
47
38
86
39
85
47
57
38
19
12
603
- 178
5 9 13
-
603
178
425
Since the 8 is larger than the 3 in the ones column
we must borrow.
But there is a zero in the tens place,
so we borrow 1 from the 6 hundreds leaving 59 tens.
We then have 13 in the ones column.
13 ones - 8 ones = 5 ones
9 tens - 7 tens = 2 tens
5 hundreds - 1 hundred = 4 hundreds
Subtraction with borrowing:
1)
647
258
752
269
843
544
2)
616
487
3)
1,564
849
1,293
454
1,863
945
1,472
958
4)
3,714
2,319
7,413
3,249
5,604
2,395
4,317
2,809
5)
1,232
584
1,678
945
1,498
953
1,589
855
6)
1,677
734
1,776
942
1,265
452
6,415
2,329
754
275
1776
798
625
349
1396
479
513
234
635
247
1655
727
13
7)
73,050
27,455
46,940
24,946
19,053
8,954
23,006
4,999
8)
36,174
16,925
86,502
26,590
74,763
64,767
306,050
145,059
Answers:
1.
389
483
299
276
279
2.
129
479
978
917
928
3.
715
839
918
514
4.
1,395
4,164
3,209
1,508
5.
648
733
545
734
6.
943
834
813
4,086
7.
45,595
21,994
10,099
18,007
8.
19,249
59,912
9,996
160,991
388
MULTIPLICATION
Suppose you walked 4 miles every day for one week. How far did you walk altogether? One way to
find the answer to this question is to add: 4 + 4 + 4 + 4 + 4 + 4 + 4 = 28. However, an easier way to do
this kind of ‘repeated addition’ is to multiply: 4  7 = 28.
In a multiplication problem, the two numbers being multiplied are called factors. The answer to the
problem is called the product.
factor  factor = product


4  7 = 28

Below is the multiplication table, showing multiplication facts from 1 through 9. You should memorize
this table before doing the exercises on the following pages.
Keep in mind that multiplication is a short-cut for addition. The questions immediately following the
multiplication table should help you understand the relationship between addition and multiplication.
This understanding will help in memorizing the multiplication facts.
14
0
1
2
3
4
5
6
7
8
0
0
0
0
0
0
0
0
0
0
1
0
1
2
3
4
5
6
7
8
2
0
2
4
6
8
10
12
14
16
3
0
3
6
9
12
15
18
21
24
4
0
4
8
12
16
20
24
28
32
5
0
5
10
15
20
25
30
35
40
6
0
6
12
18
24
30
36
42
48
7
0
7
14
21
28
35
42
49
56
8
0
8
16
24
32
40
48
56
64
9
0
9
18
27
36
45
54
63
72
9
0
9
18
27
36
45
54
63
72
81
Look at the rows in the table above:
1)
Each number in the 9 row is how much more than the number before it? ______
2)
Each number in the 8 row is how much more than the number before it? ______
3)
Each number in the 5 row is how much more than the number before it? ______
4)
Each number in the 3 row is how much more than the number before it? ______
Now look at the columns:
5)
Each number in the 8 column is how much more than the number above it? _____
6)
Each number in the 4 column is how much more than the number above it? _____
7)
8)
Each number in the 6 column is how much more than the number above it? _____
Each number in the 8 column is how much less than the number below it? ______
9)
Each number in the 9 column is how much less than the number below it? ______
Answers:
1.)
9
2.)
8
3.)
5
4.)
3
5.)
6.)
4
7.)
6
8.)
8
8
9.)
9
Once you have mastered the multiplication table, practice your multiplication facts by doing the
following exercise. Work for accuracy first, then speed. When you have finished, check your answers
with the table.
15
Multiply:
1)
5
2
7
2
9
2
5
3
5
4
6
3
6
4
9
4
7
2
3
5
7
3
2)
6
4
4
8
5
9
5
8
9
5
3
4
2
8
6
3
9
4
9
2
5
6
3)
5
4
7
5
3
7
4
6
9
5
3
5
5
8
7
3
7
2
4
8
5
6
4)
6
6
7
7
8
8
3
9
4
7
9
5
6
7
9
9
9
7
9
5
9
3
5)
8
2
8
4
8
6
8
8
9
6
8
9
6
5
6
7
7
8
4
9
7
5
6)
6
9
5
7
8
4
8
8
7
3
4
5
8
6
7
9
9
6
0
0
2
0
Answers:
1.
10
14
18
15
20
18
24
36
14
15
21
2.
24
32
45
40
45
12
16
18
36
18
30
3.
20
35
21
24
45
15
40
21
14
32
30
4.
36
49
64
27
28
45
42
81
63
45
27
5.
16
32
48
64
54
72
30
42
56
36
35
6.
54
35
32
64
21
20
48
63
54
0
0
MULTIPLYING WITH LARGER NUMBERS
Once you have mastered the basic multiplication facts, you can proceed to multiplying with
larger numbers. Study the following examples and explanations to review how it is done.
16
34
x 2
68
Multiply each digit of 34 by 2.
2 X 4 ones = 8 ones
2 X 3 tens = 6 tens

number carried
316
x 4
1264
4 x 6 ones = 24. Put the 4 in the ones
place and carry the 2 to the tens column.
4 x 1 ten = 4 tens, plus the 2 tens that
were carried is 6 tens.
4 x 3 = 12 hundreds.
Multiplying with one and two digit numbers:
1)
12
4
22
3
34
2
23
3
11
7
24
2
2)
21
7
31
6
41
8
53
3
64
2
83
3
12
12
Answers:
1.
48
12
13
23
23
33
33
42
21
61
11
66
68
69
77
48
2.
147
186
328
159
128
249
3.
144
156
529
1089
882
671
3)
17
21
x 32
42
63
672
2 ones x 21 = 42 ones, so the last digit
of 42 is placed in the ones column.
3 tens x 21 = 63 tens, so the last digit
of 63 is placed in the tens column.
Now, add those products, making sure the
42 and the 63 are lined up properly: the
42 ends under the 2 of the 32; the 63 ends
under the 3 of the 32.
1 2
1
327
x 402
654
000
1308
131454
numbers carried when multiplying by 4
number carried when multiplying by 2
2 ones x 327 = 654 ones
0 tens x 327 = 000 tens
4 hundreds x 327 = 1308 hundreds
adding those products
Multiplying with carrying:
1)
48
7
59
7
78
9
95
9
86
8
67
8
2)
649
9
769
8
825
6
892
8
483
9
536
7
3)
785
7
849
8
675
9
935
5
684
7
324
8
4)
53
25
46
47
90
58
19
39
38
28
37
57
18
5)
35
60
18
70
56
20
49
30
28
40
37
50
6)
845
69
497
18
936
37
764
46
354
55
319
23
7)
2675
128
9735
849
3629
436
5649
328
8)
1578
912
2815
715
6273
1085
51,763
605
Answers:
1.
336
413
702
855
688
536
2.
5841
6152
4950
7136
4347
3752
3.
5495
6792
6075
4675
4788
2592
4.
1325
2162
5220
741
1064
2109
5.
2100
1260
1120
1470
1120
1850
6.
58,305
8946
34,632
35,144
19,470
7337
7.
342,400
8,265,015
1,582,244
1,852,872
8.
1,439,136
2,012,725
6,806,205
31,316,615
19
DIVISION
If you have mastered the multiplication skills, division should come fairly easily for you because
division is the opposite of multiplication. For example, since
3  6 = 18, 18  6 = 3 and 18  3 = 6.
The parts of a division problem each have a name:
6
3 1 8
divisor
quotient
dividend
Remember that any number multiplied by zero equals zero. When zero is divided by any
number, the answer is zero. So, 7  0 = 0 and 0  7 = 0. Be careful, though, of problems like 7  0!
Since there is no number such that 0  (?) = 7, there is also no numerical answer to 7  0. Remember,
division by zero is not possible!
The following exercises are designed to help you develop your skill in division. The
multiplication in the first column is put there to remind you that multiplication and division are
opposites. So, if the question asks
56  7 = ?, think 7  ? = 56.
1.
4  8 = ______
32  4 = ______
32  8 = ______
2.
2  8 = ______
16  2 = ______
16  8 = ______
3.
5  7 = ______
35  5 = ______
35  7 = ______
4.
3  9 = ______
27  3 = ______
27  9 = ______
5.
5  8 = ______
40  5 = ______
40  8 = ______
6.
3  6 = ______
18  3 = ______
18  6 = ______
7.
4  9 = ______
36  4 = ______
36  9 = ______
8.
5  9 = ______
45  5 = ______
45  9 = ______
9.
4  7 = ______
28  4 = ______
28  7 = ______
10.
5  4 = ______
20  5 = ______
20  4 = ______
11.
50=0
0  5 = ______
5  0 = ______
20
12.
4  0 = ______
0  4 = ______
13.
5 30
3 24
4 20
5 25
4 24
5 15
14.
5 40
4 28
2 14
8 40
3 21
2 16
15.
3 27
2 18
4 32
5 45
4 36
39
16.
9 27
8 24
9 18
7 21
9 36
7 14
17.
5 45
7 42
7 63
7 49
7 28
6 42
18.
8 48
8 64
8 40
6 30
3 12
8 56
19.
9 54
8 72
9 81
9 45
9 63
9 72
Answers:
1.
32
8
4
2.
16
8
2
3.
35
7
5
4.
27
9
3
5.
40
8
5
6.
18
6
3
7.
36
9
4
8.
45
9
5
4  0 = ______
21
9.
28
7
4
10.
20
4
5
11.
0
0
undefined
12.
0
0
undefined
13.
6
8
5
5
6
3
14.
8
7
7
5
7
8
15.
9
9
8
9
9
3
16.
3
3
2
3
4
2
17.
9
6
9
7
4
7
18.
6
8
5
5
4
7
19.
6
9
9
5
7
8
Here are a few problems that are more challenging. The first one is done as an example.
Think: 7 x ? = 56
Think: 7 x ? = 7
81
7
567
20.
3 156
2 124
3 123
2 142
21.
3 189
4 168
4 324
5 155
22.
4 248
5 255
8 408
6 306
23.
7 147
9 180
6 186
5 355
24.
6 546
9 279
9 189
8 488
Answers:
20.
52
62
41
71
21.
63
42
81
31
22.
62
51
51
51
23.
21
20
31
71
24.
91
31
21
61
22
A.
B.
CARRYING IN DIVISION
Sometimes, part of a division problem doesn’t come out even, which makes carrying necessary.
Study the illustration on the next page to see how it is done.
Divide 9,657 by 13.
Step 1
How many 13's in 96?
( 7 )
Step 2
Multiply.
( 7 x 13 = 91 )
Step 3
Subtract.
( 96 - 91 = 5 )
Step 4
Bring down the 5 which is the next digit.
Now, repeat the steps.
How many 13's in 55?
( 4 )
Multiply.
( 4 x 13 = 52 )
Subtract.
( 55 - 52 = 3 )
Bring down the 7 which is the next digit.
C.
Repeat the steps again.
9 1
5 5
7 4
13 9,5 6 7
9 1
5 5
5 2
3 7
7 4 2
How many 13's in 37?
D.
7
13 9,6 5 7
( 2 )
Multiply.
( 2 x 13 = 26 )
Subtract.
( 37 - 26 = 11 )
R 11
13 9,6 5 7
9 1
5 5
5 2
3 7
2 6
1 1
Check your work.
Multiply the divisor by the quotient.
Add the remainder.
7 4 2
x 1 3
9 6 4 6
+
1 1
2 2 2 6
7 4 2
9 6 5 7
9 6 4 6
The result should equal the dividend.
23
Practice with Carrying
The following problems will give you practice doing division problems that involve carrying.
Some of these problems have remainders and some don’t. Be sure to check each problem.
1)
4 96
3 72
7 441
6 342
2)
5 443
7 591
9 2313
8 5224
3)
3 7372
6 1049
4 19089
4 15638
4)
23 391
41 943
47 2491
58 2088
5)
69 3249
75 4595
232 15080
314 20410
6)
23 23046
34 34102
503 302856
604 424652
7)
56 5836
59 16180
78 27080
401 17638
8)
66 3142
87 2763
75 51255
64 64155
24
Answers:
1.
24
24
63
57
2.
88 R3
84 R3
257
653
3.
2457 R1
174 R5
4772 R1
3909 R2
4.
17
23
53
36
5.
47 R6
61 R20
65
65
6.
1002
1003
602 R50
703 R40
7.
104 R12
274 R14
347 R14
43 R395
8.
47 R40
31 R66
683 R30
1002 R27
AVERAGES
The average of a list of numbers is found by adding the numbers and dividing their sum by how
many numbers there are in the list.
Example:
To find the average of 96, 88, 83, and 93, first add the numbers; then divide the sum
by 4, which is how many numbers are in the list:
96 + 88 + 83 + 93 = 360
360  4 = 90
So, 90 is the average of the list of numbers.
Here are a few problems to give you a little practice in finding averages.
1.
Find the average of each set of numbers:
a.) 3, 5, 7, 8, 4, 9
b.) 56, 48, 37, 28, 41
c.) 142, 139, 156, 187
2.
If you receive scores of 79, 85, 96, 93, and 67 on your tests during the semester, what is your
average score?
25
3.
Six stores had the following prices on paper towels: $1.39, $1.28, $1.45, $1.53, $1.45, and $1.30.
What was
the average price of paper towels?
Answers:
1.
a)
b)
c)
2.
84
3.
$1.40
6
42
156
SOLVING WORD PROBLEMS
Before trying to solve a word problem, read the problem carefully and be sure you understand what the
problem is asking. You may need to read the problem several times. Look for indicator words -words that indicate what operations will be necessary to solve the problem. Some of the most common
indicator words are listed below.
Addition
plus
more
more than
added to
increased by
sum
total
sum of
increase of
gain of
Multiplication
product
double
triple
times
of
twice
twice as much
Subtraction
less
subtract
subtracted from
difference
less than
fewer
decreased by
loss of
minus
take away
Division
divided by
divided into
quotient
goes into
divide
divided equally
per
26
Equals
is
the same as
equals
equal to
yields
results in
are
After you have decided what calculations need to be done, estimate a reasonable answer. Then
solve the problem. Finally, if the answer you get is reasonable, check your work to be sure your
calculations are correct. If your answer is not reasonable, reread the problem and begin again.
Here are some problems for you to try. They use all of the whole number operations you have
been practicing in this review packet. Don’t forget to label your answers.
1.)
Last year, the town of Montvale spent $16,593,650 for police and fire protection. The town
spent $1,108,212 for welfare and health. How much more did the town spend for police and fire
protection than for welfare and health?
2.)
John, Tom, Bill, and Fred started a record store. At the end of one year, they had a profit of
$27,936. They decided to share the profit equally. How much did each person get?
3.)
Ezra Hicks died in 1978 after his 94th birthday. In what year was he born?
4.)
Cheryl made a gross salary of $12,500 last year. Her employer deducted $2,518 from her salary.
What was Cheryl’s net income last year?
5.)
Sara bought 12 towels at a sale. She wanted to divide them equally among herself and her three
sisters. How many towels did each of the sisters get?
6.)
Bert Roberts is an auto mechanic. Last year, he made $16,456. Betty Roberts is a clerk. Last
year, she made $11,294. Their son Henry works part-time at a grocery store. He made $3,367
last year. Find the combined income of the Roberts family last year.
7.)
Sandy paid $63 for three pairs of jeans. What was the price of one pair of jeans?
8.)
It costs about $1,850 a year to educate one student at Franklin High School. There are 230
students in the school. Find the total cost for educating all the students at Franklin High School
for one year.
27
9.)
In 1950, there were 14,273 people living in Elmford. In 1980, there were 8,498 more people
living in Elmford than in 1950. How many people lived in Elmford in 1980?
10.)
Mr. Schmidt ordered nine boxes of dress shirts to sell in his store. There were 12 shirts in each
box. He sent two boxes of shirts back because they were the wrong colors. How many shirts did
Mr. Schmidt keep?
11.)
In 1975, there were 1,036,000 divorces in the U.S. In 1977, there were 1,091,000 divorces.
How many more divorces were there in 1977 than in 1975?
12.)
On weekends Karen works as a waitress. On Friday she made $63. On Saturday, she made $79.
On Sunday she made $68. Find the average amount she made per day.
13.)
Fred pays $12 for a gallon of paint. How much does he pay for 13 gallons of paint?
14.)
Mount Hood is 11,239 feet high. Mount Whitney is 14,494 feet high. How much higher is
Mount Whitney than Mount Hood?
15.)
Al wants to split a board 102 inches long into six equal pieces. How long will each piece be?
16.)
Oregon became a state in 1859. For how many years had Oregon been a state in 1980?
17.)
The yearly budget for the town of Staunton is $150,000. By the end of June, the town had spent
$89,480. How much money was left in the budget for the rest of the year?
18.)
Mr. and Mrs. Wiley bought new furniture marked $650. They agreed to pay for the furniture by
making 15 equal monthly payments of $52 each. How much more than $650 did the furniture
cost them?
19.)
In one month, the Simpsons spent $462 for utilities and mortgage payments, $436 for income
taxes and Social Security, $194 for car payments and gasoline, $323 for food, and $245 for
everything else. How much did they spend altogether that month?
20.)
George makes $295 for a five-day work week. How much does he make each day?
21.)
The Fulton County Stadium can hold 52,532 people. One Saturday afternoon, the stadium was
full. The average price for a ticket was $4.00. Find the total price paid for tickets at the stadium
that day.
28
22.)
In 1970 there were 2,809,813 people living in Los Angeles, 715,674 in San Francisco, and
697,027 in San Diego. What was the total number of people living in the three cities?
23.)
Mr. Seltzer ordered 3 cases of tomato soup, 4 cases of chicken soup, and 2 cases of bean soup for
his store. Each case held 12 cans. How many cans of soup did he order altogether?
24.)
Of the 432 students who graduated from Central High School, 184 went to college and 93 went
into the army. How many of the graduates did not go into the army?
25.)
On Thursday night, 1,129 people went to the All-City basketball tournament. On Friday night,
1,347 people went to the tournament. On Saturday, 1,406 people went. What was the average
number of people per night at the tournament?
Answers:
1.
$15,485,438
16.
121 years
2.
$6984
17.
$60,520
3.
1884
18.
$130
4.
$9982
19.
$1660
5.
3 towels
20.
$59
6.
$31,117
21.
$210,128
7.
$21
22.
4,222,514 people
8.
$425,500
23.
108 cans
9.
22,771 people
24.
339 graduates
10.
84 shirts
25.
1294 people
11.
55,000 divorces
12.
$70
13.
$156
14.
3,255 feet
15.
17 inches
29
Fractions
TERMS AND DEFINITIONS
A. Fraction—is a statement, in numbers rather than in words, that an object, any kind of object, has
been divided into equal parts. It represents a part of a whole object.
B. Denominator—tells the reader into how many equal parts the object has been divided.
C. Numerator—tells the reader how many parts of the divided object the fraction is representing.
PARTS OF A FRACTION
A. Denominator—is the bottom part of the fraction. Remember it tells you into how many equal
parts an object has been divided. Hint: Denominator is “down,” it begins with a “d” just as
“down” does.
B. Fraction Line—is the line separating denominator from numerator. It says “divide” just as ÷
says “divide.”
C. Numerator—is the top part of the fraction. Remember, it tells you how many parts of the
divided object you are concerned with.
Example:
1
2
1
→Numerator—we are concerned with one part
-→Fraction Line—an object has been divided
2
→Denominator—an object has been divided into 2 equal parts
3
-5
3
5
→Numerator—we are concerned with 3 parts
→Fraction Line—an object has been divided
→Denominator—an object has been divided into 5 equal parts
Assignment sheet #1-Parts of fractions
1. In the following fractions identify the numerator and the denominator.
3
→___________________
4
→___________________
5
8
→___________________
→___________________
11
16
→___________________
→___________________
2. Write the fractions to meet the following requirements.
a. _______ A fraction has a numerator of 3 and a denominator of 10.
b. _______A fraction has a denominator of 12 and a numerator of 7.
30
c. _______A fraction has a numerator of 13 and a denominator of 18.
3. A carpenter cuts a piece of lumber into three equal boards and uses them for shelves for a bookcase.
a. _______Write the fraction that represents the part of the original piece of lumber that the
carpenter used for one shelf.
b.
c.
d.
e.
f.
_______Which number of the fraction in (a) is the numerator?
_______Which number is the denominator?
_______What part of the original piece of lumber did the carpenter use for two shelves?
_______What part of the original piece of lumber did the carpenter use for the entire bookcase?
_______When numerator and denominator are identical, the fraction becomes the whole number
1, right?
Answers:
1. a. 3 numerator
4 denominator
b. 5 numerator
8 denominator
c. 11 numerator
16 denominator
2. a.
b.
c.
3. a.
3
10
7
12
13
18
1
3
b. 1
c. 3
d.
e.
2
3
3
3
f. right
31
Assignment sheet #2—Parts of fractions-Word problems
Directions: Write a fraction which fits the requirements of each of the following statements.
1.
A plumber had four equal lengths of pipe. He used three of them.
A. _______What fraction of the total number of pipes did the plumber use?
B. _______What part of the pipes was left?
2. The student’s tuition was divided into five equal monthly installments.
A. _______What part of the tuition did the student pay the first month of school?
B. _______After three months of school had passed, what part of the tuition had the student
paid?
C. _______What fraction was left to pay?
3. There are 12 inches in a foot.
A. _______ Seven inches is what part of a foot?
B. _______ What fraction represents the rest of the foot?
4. One $20 bill has the same value as twenty one-dollar bills.
A. _______ $13 is what fraction of the $20 bill?
B. _______What part of the $20 bill is left after you have spent $13?
C. _______ You spend the rest of the $20. Write the fraction that represents all of the $20
which you have spent.
D. _______Do you have any of your $20 bill left? When numerator and denominator are
identical, the fraction is equal to 1, which is one $20 bill you have spent!
Answers:
3
1. a. 4
1
b.
4
1
2. a. 5
3
b. 5
2
c. 5
7
3. a. 12
5
b. 12
13
4. a. 20
7
b. 20
20
c. 20 = 1
d. No
32
Test #1—FRACTION PARTS AND MEANINGS
1. Match these terms with the correct definition.
_______a. Fraction
_______b. Denominator
_______c. Numerator
_______d. Fraction line
2. In the fraction
9
1. Tells into how many equal parts an object has been divided
2. This says a division has taken place
3. This tells us an object has been divided into equal parts
4. Tells how many parts of a divided object we are concerned with
, identify:
16
a. _______the numerator
b. _______the denominator
3. Write the fractions that represent the following statements.
a. _______I ate three of my five chocolates. What part did I eat?
b. _______The instructor gave A’s to 17 of his 35 students. What part of the class had A’s?
c. _______Only one of the nurse’s three patients got well. What fraction of the sick people
died?
d. _______The mechanic discovered that five of the spark plugs in the eight-cylinder engine
needed to be replaced. What fraction of the plugs needed replacing?
e. _______An interior decorator completed 9 of the 10 draperies required for the room’s
windows. What fraction of the draperies were ready to hang?
Answers:
1. a. 3
3. a.
b. 1
c. 4
b.
d. 2
2. a. 9
c.
b. 16
d.
e.
3
5
17
35
2
3
5
8
9
10
33
KINDS OF FRACTIONS
Types
Proper—a fraction with numerator smaller than denominator
Examples:
4
2
11
9
3
12
Improper—a fraction with denominator smaller than numerator
Examples:
6
7
14
4
3
9
Mixed number—a whole number with a fraction
1
5
12
Examples:
1
46
8 10
A FRACTION which is the equivalent of the whole number “1” has the same numerator and
denominator.
Examples:
3
15
1
3
15
1
EQUIVALENT FRACTIONS are two or more fractions which have different numbers and which are
read differently, but which represent the same value. Equivalent fractions are a useful and necessary
tool to solve problems.
Examples:
1
2
2
4
8
4
8
16
= = =
=
16
, etc.
32
Equivalent fractions are easy to understand with the use of a ruler.
1 inch (greatly magnified!)
1/16
1/8
2/16
1/4
2/8
4/16
3/8
6/16
1/2
2/4
4/8
8/16
5/8
10/16
3/4
6/8
12/16
7/8
14/16
16/16
8/8
4/4
2/2
34
ASSIGNMENT SHEET #3—PROPER AND IMPROPER FRACTIONS, MIXED NUMBERS, AND
EQUIVALENT FRACTIONS
1.
Name the following fractions by labeling “P” for proper, “I” for improper, and “M” for mixed
number.
_______a.
3
4
_______b. 1
_______c.
_______d.
5
9
7
10
_______g.
8
5
9
4
_______h.
6
8
5
_______i. 9
7
_______e. 2
2.
_______f.
1
_______j.
3
1
2
11
12
Using the “ruler” above, name one or more fractions that are equivalent to the given fractions.
1
a.
8
6
b.
8
2
c.
4
=
=
=
d.
16
12
16
=
8
=
4
2
e. 1 = =
4
2
2
=
16
Answers:
1. a. P
f.
10
16
=
8
=
16
8
f. I
b. M
g. I
c. P
h. P
d. I
i. M
e. M
j. P
2. a. 2
=
4
d. 6,3
b. 3
e. 4, 8, 16
c. 1, 8
f. 5
Test #2—KINDS OF FRACTIONS
1. Write three different fractions which are equivalent to the number “1.”
____________________
_____________________
_____________________
35
2.
Identify each of the following at “P” for proper, “I” for improper, “M” for mixed number.
_______a.
_______b.
9
8
1
8
13
_______f.
9
_______c. 3
_______d.
______e.
3
4
7
16
7
5
_______g.
12
11
_______h. 9
1
2
3. Write two different equivalent fractions for each of the following fractions.
1
=
2
12
=
16
=
=
2
=
8
=
2
=
2
=
Answers:
1. answers will vary
2. a. I
b. P
c. M
d. P
e. P
f. I
g. I
h. M
3. answers will vary
RAISING AND REDUCING FRACTIONS
Change a mixed number to an improper fraction. (This operation is necessary to solve problems.)
Multiply the denominator of the fraction by the whole number and add the numerator to the result. This
result is the new numerator; the denominator remains the same.
36
1
9
2
2
4 =
Examples:
(4x2)+1
2
2
11
3
3
3 =
=
9
(3x3)+2
2
3
=
11
3
Change an improper fraction to a mixed number. This operation is necessary to obtain an answer in the
correct, or simplest form. Divide the denominator into the numerator, making a fraction of any
remainder, using the divisor as denominator.
10
Examples:
3
=3
1
14
3
9
=1
5
9
Reduce a fraction to lowest terms. The fraction is in its lowest terms, or in the simplest form, if
numerator and denominator cannot be divided evenly by any number except “1.”
3
Examples:
9
4
8
2
5
=
=
=
1
3
1
2
2
5
Divide both numerator and denominator by 3.
Divide both numerator and denominator by 4.
Numerator and denominator cannot be divided.
ASSIGNMENT SHEET #4—RAISING AND REDUCING FRACTIONS
1.
Raise each of the following mixed numbers to improper fractions.
a. 3
b. 6
c. 1
d. 7
e. 4
1
2
2
3
3
4
2
25
1
3
f. 8
g. 2
h. 5
i. 3
j. 4
3
8
4
9
4
5
2
7
1
8
37
2. Convert each of the following improper fractions to mixed numbers. Reduce the resulting
fraction if necessary.
a.
b.
c.
d.
e.
12
f.
9
4
g.
3
6
h.
4
7
i.
5
18
j.
7
Answers:
7
1. a.
20
c.
d.
e.
2.
7
4
11
3
15
6
14
7
h.
4
177
25
13
3
1
a. 1
i.
j.
67
8
22
9
29
5
23
7
33
8
f. 4
3
b. 1
1
g. 2
3
1
h. 3
2
d. 1
e. 2
9
g.
3
c. 1
8
f.
2
b.
32
2
i. 2
5
4
1
4
2
3
1
2
j. 2
7
38
TEST #3—RAISING AND REDUCING FRACTIONS
1.
Raise each of the following mixed numbers to an improper fraction.
a. 1
b. 7
c. 3
d. 5
2.
a.
b.
c.
d.
e.
2
e. 2
3
5
f. 9
7
1
g. 1
4
1
h. 4
6
3
4
2
9
3
8
3
5
Convert the following improper fractions to mixed numbers, reducing the resulting fraction if
necessary.
8
f.
4
9
g.
5
14
h.
3
12
i.
8
15
j.
9
11
7
7
4
6
6
8
3
22
5
Answers:
1.
a.
b.
c.
5
3
54
7
13
4
d.
e.
f.
31
6
11
4
g.
h.
11
8
23
5
83
9
39
2
2
a. 2
e. 1
3
4
b. 1
4
f. 1
5
7
2
c. 4
2
i. 2
3
2
j. 4
5
3
g. 1
3
4
1
d. 1
h. 1
2
MULTIPLYING FRACTIONS
Multiplying fractions is the simplest of all fraction arithmetic operations.
A. Multiply the numerators
B. Multiply the denominators
C. Reduce the result if necessary.
Examples:
2 4 2𝑥4
8
a. x =
=
3 5 3𝑥5
15
1 3 1𝑥3
b. x =
8 4 8𝑥4
6 1 6𝑥1
c. x =
=
=
3
32
6
=
2
7 3 7𝑥3
21 7
Cancel is a handy device used to reduce fractions before multiplying them so as to obtain smaller
numbers to multiply.
A. Examine both fractions to see if any pairs, diagonally (kitty-corner), or vertically (up and down), -NEVER horizontally (across)—can be divided evenly by the same number.
B. Multiply the results.
Examples: Cancel diagonally.
3
3 6
𝑥
4 and 6 can both be divided by 2
4 7
2
3
2
3
𝑥 =
9
7 14
Answer
40
1
5
7
𝑥
8 10
5 and 10 can both be divided by 5
2
1
8
7
𝑥 =
7
2 16
Answer
1
5
16
𝑥
4
5 and 5 can both be divided by 5
5
1
1
1
16
𝑥
4
1
4
4
1
1 1
𝑥 =
1 4
4 and 16 can both be divided by 4
Answer
Example: Cancel diagonally.
3
6
9
𝑥
3
4
6 and 4 can both be divided by 2
2
1
3
9
𝑥
3
2
9 and 3 can both be divided by 3
3
1
3
3
1
1
1
𝑥
1
2
1 1
𝑥 =
2 2
3 and 3 can both be divided by 3
Answer
Multiply mixed numbers and whole numbers.
A.
B.
C.
D.
E.
F.
Convert the mixed number to an improper fraction.
Give the whole number the denominator “1.”
Cancel where possible.
Multiply the resulting fractions.
Convert the result to a mixed number.
Reduce the result if necessary.
41
Examples:
1
2 𝑥3
4
9 3
27
4 1
4
x =
=6
3
4
3
4 𝑥2
5
23 2
46
5 1
5
x =
=9
1
5
2
3 𝑥3
9
1
29 3
29
9 1
3
x =
3
=9
2
3
Multiply mixed numbers.
A.
B.
C.
D.
E.
Convert all mixed numbers to improper fractions.
Cancel where possible.
Multiply the fractions.
Convert results to mixed numbers.
Reduce if necessary.
Examples:
1 3
2 𝑥1
5 4
11 7
77
5 4
20
x =
=3
17
20
2 1
3 𝑥4
3 8
11
11 33
3
1
x
8
=
121
8
= 15
1
8
1
2
5 𝑥2𝑥1
4
9
7
1
21 2
11
4 1
9
x 𝑥
2
3
=
77
6
= 12
5
6
42
ASSIGNMENT SHEET#5—MULTIPLYING FRACTIONS
Directions: Multiply the following fractions and mixed numbers. Give the results in the lowest terms.
1.
5
6
9
𝑥
2. 4 x
20
5
4.
7
1
1
8
9
10
𝑥
4
27
6.
7
24
𝑥
4
8
9
Answers:
3
1.
8
1
2.
1
3.
72
4.
5.
6.
4
15.
7
8
1
6
1
7.
2
8.
10
9.
1
2
10
3
9
1
3
𝑥
7
8
4
2
3
𝑥8
𝑥
7
32
3
4
3
16
𝑥3
1
7
13. 2
2
1
2
1
14. 1
15.
3
10. 7
4
9
7
16. 2
4
15
1
14. 6𝑥
7
5
8. 12𝑥
13.
4
7. 5 𝑥
9
12.
21
16
8
11. 24𝑥
8
3
5. 𝑥
8
10. 3 𝑥2
16
3. 10 𝑥7
3
9. 𝑥
7
20
2
3
5
16
21
32
16. 6
7
8
11. 18
12.
14
27
43
TEST #4—MULTIPLYING FRACTIONS
Directions: Multiply the following fractions and mixed numbers. Be sure the answer is in the lowest
terms.
1
1
3 1
1. 2 𝑥1
6. 𝑥
8
5
4 2
4
2.
5
𝑥
2
7
3
8
3. 𝑥
1
2
3
5
3
6
4
7
5. 1 𝑥1
20
8
21
1
4
5
7
8. 5 𝑥1
1
4. 4 𝑥3
5
7. 𝑥
3
9. 9𝑥
10.
7
18
4
8
𝑥16
Answers:
11
1. 2
6.
20
2.
3.
4
7.
15
7
12
11
4.
14
5.
3
15
1
3
8
25
42
8.
8
9.
3
6
35
1
2
10. 8
4
DIVIDING FRACTIONS
Dividing fractions
A. Invert (turn over) the divisor (the number or fraction following the division sign).
B. Change the sign to multiply.
C. Cancel where possible
D. Multiply the resulting fractions and give the result in the lowest terms.
Example 1:
1 1
÷
2 4
2
1 4 2
𝑥 = =2
2 1 1
1
44
Example 2:
1 2
÷
3 5
1 5 5
× =
3 2 6
Dividing mixed numbers and whole numbers
A. Convert the mixed numbers to improper fractions.
B. Make each whole number into a fraction by giving it a denominator of “1.”
C. Invert the divisor and change the division sign to multiply.
D. Cancel where possible.
E. Multiply the fractions and give the result in the lowest terms.
Example 1:
1
1
2
4
1
3
4
3 2 ÷ 1 or
7
Or
2
÷
=
7
2
4
1
7
=2 𝑥
1
4
7
=8
4
1
7 1 7
𝑥 =
2 4 8
Example 2:
1
4 5 ÷ 3 or
21
Or
5
1
5
3
1
4
=
21
5
3
1
=
21
3
1
7
2
● 3 = 5 = 15
3
÷
1
7
21 1
𝑥
5 3
1
7 1 7
2
𝑥 = =1
5 1 5
5
Example 3:
1
3 ÷ 1 3 or
Or
3
1
3
1
1
1
3
÷
=
3
1
4
3
3
3
9
1
= 1 𝑥 4 = 4 = 24
4
3
3 3 9
1
𝑥 = =2
1 4 4
4
45
Dividing Mixed Numbers.
A.
B.
C.
D.
Convert all mixed numbers to improper fractions.
Invert the divisor and change the division sign to multiply.
Cancel where possible.
Multiply the fractions and give the results in lowest terms.
Example 1:
1
1
3
1
1
2
1
1
1 3 ÷ 1 2 or
4
Or
÷
3
=
4
3
3
2
4
2
8
= 3𝑥3 = 9
3
2
4 2 8
𝑥 =
3 3 9
Example 2:
3
2
18 ÷ 45
11
8
÷
22
5
or
3
8
2
4
5
1
11
8
22
5
=
=
11 5
5
x = 16
8 22
1
11
Or
8
5
𝑥
22
2
1 5
5
𝑥 =
8 2 16
Example 3:
1
3
4 ÷1
5
4
21
5
7
÷ 4 or
1
5
3
1
4
4
=
21
5
7
4
=
21
5
4
𝑥7=
12
5
2
= 25
3
Or
21
5
𝑥
1
4
7
3 4 12
2
𝑥 =
=2
5 1
5
5
46
ASSIGMENT SHEET #6—DIVIDING FRACTIONS
Directions: Divide the following fractions and mixed numbers and give the result in the lowest terms.
1
1
2
4
1. ÷
11.
÷
2
5
3
5
2.
3.
4.
5.
6.
5
÷1
6
9
10
7
3
16
3
9
13.
3
2
5
9
83
1
22
14. 5 ÷ 6
3
÷1
÷
7
2
1
5
15.
1
÷3
1
4
1
5
4
12
16. 6 ÷ 3
1
3
4
10
7. 2 ÷ 1
8.
12. 3 ÷
÷3
8
2
÷
1
17.
÷ 10
1
3
2
10. 1
7
10
2
1
1
4
3
÷
1
5
1
2
9. 2 ÷ 7
1
3
÷1
18.
13
1
42
1
3
Answers:
1. 2
2.
3.
4.
5.
6.
1
2
3
8.
4
2
7
10
7
24
15
128
1
7.
3
5
9.
1
26
1
25
16
45
10. 1
11.
19
11
40
5
6
12. 3
6
7
47
13. 3
14.
15.
7
16. 1
15
7
8
1
4
17. 22
5
8
18. 27
3
40
TEST #5—DIVIDING FRACTIONS AND MIXED NUMBERS
Directions: Divide the following fractions and mixed numbers. Be sure your results are in the lowest
terms.
7
1
7
3
1. ÷
7. 6 ÷ 1
9
3
8
8
2. 12 ÷ 2
3.
9
16
÷8
5
8.
8
1
1
4
3
1
5
2
8
16
Answers:
1
1. 2
3
2.
3.
4.
5.
4
4
÷
1
9
24
5
1
4
1
1
4
2
12
6.
7.
8.
136
17
6
5
7
9
5
10. 6 ÷ 12
4. 10 ÷ 6
6.
8
÷
9. 25 ÷
2
5. 25 ÷ 2
5
9.
10.
9
20
5
3
4
100
1
2
7
48
ADDITION OF FRACTIONS
Addition of like fractions—those which have the same denominators.
a. Add the numerators—these tell you how many parts of the divided object you are concerned
with.
b. Write the common denominator. Do not add the denominators—these tell you into how
many equal parts an object or objects have been divided.
c. Reduce the answer or express it as a mixed number.
Examples:
1
1
2
1. + =
3
3
3
3
2
5
2. + =
7
7
7
4
7
11
2
3. + = = 1
9
9
9
9
Addition of unlike fractions—those which do not have the same denominators.
A. Change the unlike fractions to like fractions by determining the lowest common
1
denominator. This is the same thing as “raising” a fraction to its equivalent (for example 2 =
2
). You do this by determining a new denominator into which you can divide both
denominators evenly. Sometimes you can use one of those you already have; sometimes you
have to find a new one.
1
1
Example
+
2
4
1
B. Raise to a fraction with a denominator of 4.
2
1 2
=
(Four is the least common denominator.)
2 4
4
1
4
=
1
4
C. Add as like fractions.
1
2
=
2
4
+
1 1
=
4 4
3
4
Addition of mixed numbers.
A. Turn the fractions into like fractions if necessary.
B. Add them. Be sure the resulting fraction is a proper fraction reduced to its lowest terms. If
the result is an improper fraction, turn it into a mixed number.
C. Add the whole numbers.
D. Combine the results you obtained in B and C.
49
1
5
36 + 26
Example 1:
1
36
5
+2 6
6
56 = 5+1=6
2
3
Example 2: 2 5 + 1 10
2
4
2 5 = 2 10
3
3
+ 1 10 = 1 10
7
3 10
2
3
Example 3: 4 3 + 1 4
2
8
4 =4
3
12
3
9
+ 1 4 = 1 12
17
5
5
5 12 = 5 + 1 12 = 6 12
ASSIGNMENT SHEET #7—ADDING FRACTIONS
Directions: Add the following fractions and mixed numbers. Be sure the results are in the lowest terms.
2
1
7
1
1. +
6. 6 + 1
5
5
8
8
2.
4
5
+
3
4
2
7
3
9
3. 2 + 7
4. 2
5.
3
7
3
16
+
+4
1
7
2
1
3
5
7. 9 + 5
1
4
8.
1
2
+
1
3
5
1
6
4
9. 6 + 5
10.
2
5
+
4
5
50
7
11.
8
3
12.
3
13.
5
+
3
20
+
6
1
16.
7
+2
3
9
4
2
18.
7
5
1
1
6
2
4
1
2
2
3
14.
5 +1 +5
15.
5 +6
8
9
7
2
8
3
3 +4
17.
4
+
3
+
1
2
2
3
5
10
19.
10 + 12
20.
20 + 1
1
1
3
11
Answers:
1.
3
11. 1
5
11
2.
1
3.
10
4.
6
20
4
9
7
16
13. 1
56
9
10
6
35
7
14. 12
12
7
1
15. 12
6
6.
8
16. 1
7.
14
5.
8.
9.
4
12. 5
41
13
15
5
18. 1
6
12
10. 1
17. 8
1
12
1
5
13
24
1
6
7
19. 22
10
14
20. 21
33
51
Test #6—ADDING FRACTIONS
Directions: Add the following fractions and mixed numbers. Be sure the results are in the lowest terms.
1
1
1
6
5
1. 7 + 5 + 2
7.
8
+
8
9
2
3
7
8
2
5
7
7
1
1
5
2
2. 9 + 1
3. 5 + 6
4. 5
5
+9
16
5
1
8
1
1
1
2
3
4
+
7
1
3
1
1
3
6
4
4
16
7
21
9. 4 + 2
5. 4 + 6 + 4
6.
2
8. 7 + 2 + 8
1
1
2
16
10.
5 +2
11.
2
12.
2 +4
3
10
+1
2
3
3
5
7
8
Answers:
17
1. 14
18
2.
3.
4.
5.
6.
11
11
18
9.
7
10. 7
11. 4
1
21
1
12
1
12
1
8.
56
7
16
15
17
7
10
14
27
7.
12. 7
1
3
9
16
7
40
4
15
SUBTRACTION OF FRACTIONS
Subtraction of like fractions—those which have the same denominators.
A. Subtract the numerators—these tell you how many parts of the divided objects you are concerned
with.
B. Write the common denominator. Do no subtract the common denominators—this tells you into
how many equal parts an object or objects have been divided.
C. Reduce if necessary.
52
Examples:
1.
2.
3.
7
11
5
8
11
−
−
−
1
=
6
11
11
3
2
1
=
8
9
8
=
=
2
4
=
1
16
16
16
8
Subtraction of unlike fractions—those which do not have the same denominators.
A. Change the unlike fractions to like fractions by determining the lowest common denominator.
You do this by determining a new denominator into which you can divide both denominators
evenly. Sometimes you can use those you already have; sometimes you have to find a new one.
2
6
(This is similar to “raising” a fraction to its equivalent. Example: = . )
3
9
5
1
Example:
−
8
4
5
Since four will divide evenly into 8, we can use 8 as our common denominator. Therefore,
8
5
can remain .
8
1
However, needs to be raised to 8ths.
4
1
4
𝑥
2
2
=
2
8
B. Subtract as like fractions.
5
5
=
8
8
1 2
=
4 8
=
3
8
Subtraction of mixed numbers.
A. Borrowing not necessary
1. First turn the fractions to like fractions if necessary; then subtract them. Be sure the result is
reduced to its lowest terms.
2. Subtract the whole numbers.
3. Combine these two results.
53
Example 1:
5
1
8
2
4 −1
5
5
8
8
4 =4
1
2
=
Example 2:
4
1 =1
-
3
7
1
9
9
3 −1
3
−1
8
1
8
7
9
1
9
6
2
=2 =2
9
3
B. Borrowing necessary
1. Turn the fractions into like fractions if necessary.
2. If the number to be subtracted is larger than the number to be subtracted from (if the lower
numerator is larger than the upper numerator), you must borrow one from the whole number.
Change this “1” into an equivalent fraction with the same denominator as the converted
fractions. Remember that a fraction with the same numerator and denominator is equal to
6
3
“1”. (Example: 6 = 1, 3 = 1
3. Add this borrowed fraction to the converted fraction.
4. Now you can subtract—fractions first, then whole numbers. Be sure the result is in the
lowest term.
1
3
Example: 1. 12 3 − 8 4
1
3
12 = 12
3
- 84 =
=
4
12
12
4
16
= 11 + 12 + 12 = 11 12
- 8
3
9
12
7
12
54
ASSIGNMENT SHEET #8—SUBTRACTING FRACTIONS
Directions: Subtract the following fractions and mixed numbers. Be sure the answers are in the lowest
terms.
7
3
7
3
1. −
11.
42 − 9
8
8
8
16
2.
3.
4.
5.
6.
7.
8.
3
4
−
7
16
7
11
2
3
7
8
7
9
4
7
−
−
−
−
−
−
1
4
1
16
3
11
1
2
3
4
2
3
1
3
5
1
6
3
9. 8 − 6
11
10.
11
12
−8
1
4
9
12.
13
13.
10 − 4
14.
24
15.
10 − 10
16.
8 − 7
17.
4 − 3
18.
8 − 5
19.
11 − 2
20.
25 − 21
10
− 9
3
4
1
1
3
2
1
16
− 8
3
4
5
7
6
12
7
15
8
16
1
2
6
3
3
1
4
6
1
1
2
3
3
4
4
5
Answers:
1.
2.
3.
4.
5.
6.
1
2
1
2
3
8
4
11
1
6
1
8
7.
8.
9.
1
9
5
21
2
10. 3
1
2
2
3
11
11. 33
16
12. 4
3
20
55
13. 5
5
17.
6
5
14. 15
16
15.
16.
1
2
18. 3
1
19. 9
4
15
20. 3
16
7
12
1
6
19
20
TEST #7—SUBTRACTING FRACTIONS
Directions: Subtract the following fractions and mixed numbers. Be sure the results is in the lowest
terms.
7
2
1
1
1.
−
6. 7 − 4
11
11
3
5
1
2
7
3
5
5
8
8
2.
9 − 3
3.
3 − 2
4.
5.
5
6
−
1
2
3
3
4
7 −1
9− 6
8.
8 −6
9.
3
3
7.
3
4
4
7
1
9
3
−
1
5
10. 11 −
7
5
8
Answers:
1.
5
11
2.
5
3.
1
4.
5.
10
21
6.
3
7.
2
8.
2
1
2
5
9.
11
12
2
15
1
4
4
9
11
20
10. 3
3
8
56
FINAL TEST—FRACTIONS
1. Identify the following as either proper or improper.
2
_______a.
_______d.
3
3
_______b.
_______e.
2
9
_______c.
8
7
8
4
5
2. Reduce these fractions to the lowest terms.
a.
b.
c.
4
=__________
8
5
15
3
21
d.
= _________
e.
15
75
6
9
= _________
=__________
=_________
3. Write the following improper fractions as mixed numbers. Reduce the results if necessary.
7
15
a. =___________
d.
=__________
5
2
b.
c.
32
31
17
14
=__________
e.
12
10
=___________
=_____________
4. Write the following mixed numbers as improper fractions.
1
1
a. 3 2 = __________
d. 1 4 =
__________
1
b. 2 3 =
__________
c.15
1
e. 9 2 =
__________
4
= _____________
7
5. Write equivalent fractions by filling in the blanks.
1
21
2
c. =
a. =
3
3
9
b.
5
=
10
20
d.
10
12
=
e.
4
5
=
40
6
57
6. Add the following fractions and mixed numbers. Reduce the answers.
1
1
a. + =
3
4
b.
c.
1
2
3
4
2
3
+ 15 =
4
1
d.3
e.
1
+ =
4
1
+2 =
6
1
3
2
8
5 +4 =
7. Subtract the following fractions and mixed numbers. Reduce the answers.
5
1
a.
−
12
3
b.
3
4
−
1
c. 2
d.
4
1
2
−
5
6
7
1
8
4
1 −1
5
7
e. 10 16 − 3 8
8. Multiply the following fractions and mixed numbers. Reduce the answers.
1 1
a. x
5 3
1
1
b. 2 2 x3 4
c. 1
d.
1
2
5
7
x2
x3
1
4
1
2
1 1
e. 2 5 x 3
9. Divide the following fractions and mixed numbers. Reduce the answers.
1
1
a. ÷
3
3
58
1
b.2
2
1
c. 3
d.
e.
3
5
7
7
8
1
÷3
4
1
÷2
÷3
4
1
2
÷3
Answers:
Final Test
1. a. P
e. 1
b. I
c. I
4. a.
d. P
b.
e. P
2. a.
b.
c.
d.
e.
1
c.
2
1
d.
3
1
e.
7
1
7
2
7
3
109
7
5
4
19
2
b. 10
2
c. 63
3
d. 5
2
3. a. 1
5
c. 1
5
5. a. 6
5
b. 1
1
1
31
3
e. 50
6.
a.
7
12
b. 1
14
d. 7
1
c.16
2
1
2
5
d. 5
12
59
7
e. 9
8
7. a.
b.
c. 3
1
e.
1
12
2
1
e.
4
8. a.
11
15
9. a. 1
b.
5
10
13
8
6
1
7
c. 1
16
d.
15
1
b. 8
8
8
d. 2
5
c. 1
12
d.
3
e.
13
27
10
49
7
24
60
Decimals
A decimal number is related to a fraction in that it also is a statement that an object has been divided
into parts. However, decimal notation is based on the number “10,” all parts of a decimal number
indicating a division by 10 or a multiple of 10, such as 10, 100, 1000, etc. Therefore, decimal
numbers are sometimes simpler for students to understand than are fractions.
Place Value
In the decimal number 0.576143 every digit has a place name.
1. the 0 indicates that there is no whole number.
2. The 5 means tenths. An object has been divided into 10 parts, and we are concerned with 5
5
of them. It is read “five tenths.” Its fraction equivalent is 10.
3. The 7 means hundredths. Combined with the preceding 5, this indicates an object is divided
into 100 parts, and we are concerned with 57 of them. IT is read “fifty-seven hundredths.”
57
It fraction equivalent is 100.
4. The 6 means thousandths. Combined with the preceding 5 and 7 this indicates an object has
been divided into (a) ________parts and we are concerned with (b) ______ of them. It is
read (c) _____________. Its fraction equivalent is (d) _____________.
5. The 1 means ten thousandths. Combined with the preceding 5, 7 and 6 this indicates an
object has been divided into (e)_________________parts, and we are concerned with (f)
_____________ of them. It is read (g) _____________. Its fraction equivalent is
(d)____________.
6. The 4 means hundred thousandths. Combined with the preceding 5,7,6 and 1, this indicates
an object has been divided into (i) ___________parts, and we are concerned with (j)
____________ of them. It is read (k) ______________. Its fraction equivalents is
(l)______________.
7. The 3 means millionths. Combined with the preceding 5, 7, 6, 1, and 4 this indicates that an
object has been divided into (m) ________parts, and we are concerned with (n) __________
of them. It is read (o) _________. Its fraction equivalent is (p) ________________.
Answers:
4) a. 1000
b. 576
c. five hundred seventy-six thousandths
5) e. 10,000
f. 5,761
g. five thousand seven hundred sixty-one ten thousandths
5761
h. 10,000
6) i. 100,000
j. 57,614
61
k. fifty-seven thousand six hundred fourteen hundred thousandths
57,614
l. 100,000
7) m. 1,000,000
n. 576,143
o. five hundred seventy-six thousand one hundred forty-three millionths
576,143
p. 1,000,000
Decimal notation for fractions
Fractions with a denominator of 10 or a multiple of 10 – such as 100, 1000, 10,000, etc. –
may be easily written as decimals.
 There will be as many decimal places as there are zeros in the denominator.
 Place a decimal point before the numerator and eliminate the denominator.
Examples:
3
= 0.3
10
3
1000
27
100
= 0.27
= 0.003
(Zeros may be needed as place holders. They are important. Do not omit them!)
 Mixed numbers (whole numbers with fractional parts) are written with the whole number
first, then the decimal point, and then the fractional part.
Examples:
75.175
6.03
29.5
Reading and writing decimal numbers
 Fractions only
1. Write the numbers first, no decimal point yet.
2. Determine the number of places necessary for the place value written at the end of the
number word.
Example: Nine hundred thirty-five ten thousandths
a. Write 935
b. Determine 4 places for ten thousandths, counting them from right to left.
←4 ←3←2←1
9
3 5
c. Place the decimal point where the 4th arrow reached. (Remember to add a zero as a
place holder. You must have 4 places.)
.0 9 3 5
62
d. Write a 0 to indicate there is no whole number.

1.
2.
3.
0.0935
Mixed numbers
In a mixed number the word “and” or the word “point” indicates the decimal point.
Number words before “and” or “point” are whole numbers.
Number word after “and” or “point” are decimal fractions.
Example:
(Whole number)
(Decimal point)
Fraction (2 places)
Sixty-five
and
thirty-nine hundredths
65
.
39
65.39
Try it! Three hundred five and ninety-six thousandths _____________
Answer: 305.096
Assignment sheet #1—Decimals
1. Write equivalent decimal fractions for the following.
7
_______a. 5 10
_______b. 71
91
100
3
_______c. 1000
27
_______d. 483 100
3
_______e. 7 10
85
_______f. 1000
1
_______g.23 1000
_______h.645
45
100
913
_______i. 1000
71
_______j.9 100
2. Write each of the following as a decimal number.
______a. Five and seven tenths
______b. Five and seven hundred fifty-six thousandths
63
______c. Seven hundred and fifty-six hundredths
______d. Forty-one thousand and forty-one thousandths
______e. Fourteen thousand six and sixty-six hundredths
______f. One thousand twenty-five ten thousandths
______g. Sixty-five hundred thousandths
______h. Six hundred sixty-six hundred thousandths
Answers:
1. a. 5.7
b. 71.91
c. 0.003
d. 483.27
e. 7.3
f. 0.085
g. 23.001
h. 645.45
i. 0.913
j. 9.71
2. a. 5.7
b. 5.756
c. 700.56
d. 41,000.041
e. 14,006.66
f. 0.1025
g. 0.00065
h. 0.00666
Test #1 Decimals
1. write the name of the place value of each of the digits in this number—0.671985
64
2. Write these numbers in decimal notation.
3
________________a. 93 100
67
________________b. 8 100
9
________________c. 56 10
71
________________d. 41000
17
________________e. 38 10,000
3. Write each of the following decimal numbers in words.
a. 81.3 ______________________________________________________
b. 7.55 ______________________________________________________
c. 0.009 _____________________________________________________
d. 0.0083 ____________________________________________________
e. 0.00056 ___________________________________________________
Answers
1. a. millionths
b. hundred thousandths
c. ten thousandths
d. thousandths
e. hundreths
f. tenths
2. a. 93.03
b. 8.67
c. 56.9
d. 4.071
e. 38.0017
3. a. eighty-one and three tenths
b. seven and fifty-five hundredths
c. nine thousandths
d. eighty-three ten thousandths
e. fifty-six hundred thousandths
65
If you are not satisfied with your results or if there is something you do not understand, ask the
instructor for more explanation and for more exercises.
Rounding decimals—When complete accuracy is not essential, decimal numbers are rounded to a
specified place.
 Look at the digit in the place immediately to the right of the specified place.
 If the digit to the right of the specified place is less than 5, drop it and all following digits.
Leave the other digits unchanged.
 If the digit to the right of the specified place is 5 or larger, drop it and all following digits,
and increase the preceding digit by 1.
Examples:
1. 6.7 to the nearest whole number is 7
2. 0.73 to the nearest tenth is 0.7
3. 0.925 to the nearest hundredth is 0.93
4. 0.4399 to the nearest thousandth is 0.440 (the zero to the far right is necessary because the
instruction said to the nearest “1000”)
5. 2.03456 to the nearest hundredth is 2.03
6. 14.39652 to the nearest thousandth is 14.397
Assignment sheet #2—Rounding decimals
1. Round the following numbers to the nearest whole number.
___________a. 75.63
___________b. 3.1274
___________c. 0.98931
2. Round the following numbers to the nearest tenth.
__________a. 43.333
__________b. 7.787
__________c. 0.7531
3. Round the following numbers to the nearest hundredth
__________a. 63.632
__________b. 123.935
__________c. 0.678
66
4. Round the following numbers to the nearest thousandth
__________a. 231.3875
__________b. 0.5802
__________c. 44.4444
Answers:
1. a. 76
b. 3
c. 1
2. a. 43.3
b. 7.8
c. 0.8
3. a. 63.63
b. 123.94
c. 0.68
4. a. 231.388
b. 0.580
c. 44.444
II. Arithmetic operations
 Adding decimals
1. Write the given numbers in a column.
2. Be sure the decimal points are directly under each other. To avoid confusion you may add
zeros so that each number has the same number of places, but be sure that the decimal points
are in a straight line.
3. Then add the columns as for regular numbers. Be sure the decimal point in the answer is
directly under the decimal points in the columns.
Example: add 96.3 + 7.62 + 14.076 + 381
96.3
7.62
14.076
381.
498.996
or
96.300
7.620
14.076
381.000
498.996
 Subtracting decimals
1. Write the given numbers in a column
2. Be sure the decimal points are directly under each other. To avoid confusion you may add
zeros so that each number has the same number of places, but be sure that the decimal points
are in a straight line.
3. Then subtract the columns as for regular numbers. Be sure that the decimal point in the
answer is directly under the decimal points in the columns.
67
Example: Subtract: 79.36 – 62.134
79.36
-62.134
17.226
or
79.360
-62.134
17.226
4. To check your result add your answer to the number that you subtracted. The sum should be
the same as the top number. Check:
17.226
+ 62.134
79.360
 Multiplying decimals
1. Write the given numbers the same as you do with whole numbers. In multiplying, the
decimal points need not be directly beneath each other.
2. Multiply as with regular numbers.
3. Count the number of decimal places—those numbers to the right of the decimal point—
in both factors being multiplied.
4. Locate the decimal point in the answer by starting at the extreme right digit in the
answer and counting off as many places to the left as the total you determined in Step
#3. Zeros may be added at the left if there are not enough digits to fulfill the
requirements of Step #4.
Examples:
a. Multiply 32.61 x 0.06
32.641
x
0.06
(5 places)
195846
000000
1.95846
(answer)
b. Multiply 1.23 x 0.018
1.23
x 0.018
984
1230
00000
0.02214
(5 places)
(answer)
 Multiplying by 10, 100, or 1,000: A short-cut method when multiplying by a power (or
multiple) of 10 is simply to move the decimal point in the number to be multiplied as
many places to the right as there are zeros in the multiplier (10, 100, etc.)
68
Examples:
75.6 x 10
75.6
x 10
000
7560
756.0
(1 place)
or 75.6 x 10 = 756
(answer)
39.16 x 100
39.16 (2 places)
x 100
0000
00000
391600
3916.00 (answer)
or 39.16 x 100 = 3916
40.5 x 1000
40.5 (1 place)
x 1000
000
0000
00000
405000
40,500.0 (answer)
or 40.5 x 1000 = 40,5000
 Dividing decimals
1. Dividing a decimal by a whole number (This is the easiest division.)
a. Place the decimal point in the quotient (the answer) directly above its location in
the dividend (the number to be divided).
b. Divide as with whole numbers. Continue to divide until the answer comes out
even or until you reach the number of decimal places required by the instruction.
Round to the place required by the instruction. (See rounding decimals).
c. Check your result by multiplying the divisor and the answer.
Example: 2.35 ÷ 5
0.47
5)2.35
20
35
35
0
Answer
Check 0.47
x 5
2.35
0.963 ÷ 7 (round answer to nearest hundredth)
69
0.137 ÷ 7 = 0.14 answer
7)0.963
7
26
21
53
49
4 (remainder)
Check 0.137
x
7
0.959
+
4 (remainder)
0.963
2. Dividing a decimal by a decimal. It is not possible to divide a decimal by another
decimal until some preliminary work has been done.
a. Move the decimal point in the divisor to the end of the divisor.
b. Move the decimal point in the number to be divided the same number of places
that you moved the point in the divisor. Sometimes you will need to add zeros to
the dividend in order to move the point the required number of places. Be sure to
do this. (In effect you have multiplied each number by 10, 100, etc., and thus
have not changed the value of your result; you have simply made it possible to
solve the problem.)
c. Immediately place the point in the answer directly above the new point in the
number to be divided (the dividend).
d. Divide as with whole numbers.
Examples:
3.6 ÷ 0.6
6. Answer
0.6)3.6
36
0
Check:
6
x 0.6
3.6
0.36 ÷ 0.06
6
Answer
0.06)0.36
36
0
Check 0.06
x 6
0.36
0.36 ÷ 0.006
70
60
0.006)0.360
Answer
36
00
Check 0.006
x 60
0.360
3. Dividing a whole number by a decimal (This is a combination of the other two
operations.)
a. Place a decimal point at the end of the whole number in the divisor.
b. Move the decimal point in the divisor to the right of the divisor.
c. Count the number of places you moved the decimal point.
d. Move the point in the dividend the same number of places, adding zeros
as necessary.
e. Place the decimal point in the answer immediately above the newlylocated point in the dividend.
f. Divide as with whole numbers.
Examples:
36 ÷ 0.6
60
0.6)36.0
Answer
36
00
Check
60
x 0.6
36.0
36 ÷ 0.06
6 00
0.06)36.00
Answer
36
000
Check:
600
x 0.06
36
71
36 ÷ 0.006
6 000 Answer
0.006)36.000
36
0 000
Check:
x
6 000
0.006
36.000
4. Dividing by 10, 100, 1000, etc. A short-cut method when dividing by a power (or multiple)
of 10 is simply to move the decimal point of the number to be divided as many places to the
left as there are zeros in the divisor (10, 100, 1000, etc.).
Examples:
75.6 ÷ 10
7.56 Answer
or 75.6 ÷ 10 = 7.56
10)75.60
70
(move decimal point in 75.6 one place to the left)
56
50
60
60
0
39.16 ÷ 100
0.3916 Answer
or 39.16 ÷ 100 = 0.3916
100)39.1600 (move the decimal point in 39.16 two places to the left)
30 0
9 16
9 00
160
100
600
600
0
Assignment sheet #3-Review of decimals
A. Number words and rounding
1. Write the decimal numbers for the word numbers.
________a. Six and nine hundred sixteen thousandths
________b. Five and six-tenths
________c. Ninety-three ten-thousandths
72
2. Round to the nearest whole number.
________a. 53.1
________b. 97.8
3. Round to the nearest tenth.
________ a. 101.87
________ b. 213.09
4. Round to the nearest hundredth.
________ a. 0.049
________ b. 1.173
5. Round to the nearest thousandth.
________ a. 08743
________ b. 0.0706
B. Add:
1) .62
43.7
9.584
2) 843.794
0.9041
85.
4) 394 + 6.72 + 0.183
C. Subtract:
1) 89.745
- 31.231
4) 74 – 57.93
D. Multiply:
1) 6.435
x 0.82
3) 684.1
74
9.037
5) 0.89 + 6.043 + 0.0791
2) 15.97
- 8.894
3) 645.7
- 69.3
5) 100 – 76.52
2) 0.376
x 0.04
4) 43.61 x 100
3) 187.2
x 0.354
5) 0.8976 x 1000
E. Divide
1) 0.12)144
2) 12)0.144
4) 7 ÷ 8
5) 6.205 ÷0.25
3) 1.2)0.144
73
6) 76 ÷ 100
7) 0.693 ÷3.5
8) 96.7 ÷ 10
Answers:
A.
1. a. 6.916
2. a. 53
3. a. 101.9
4. a. 0.05
5. a. 0.874
B.
C.
D.
E.
1) 53.904
1) 58.514
1) 5.2767
1) 1,200
7) 0.198
b. 5.6
b. 98
b. 213.1
b. 1.17
b. 0.071
c. 0.0093
2) 929.6981 3) 767.137
2) 7.076
3) 576.4
2) 0.01504
3) 66.2688
2) 0.012
3) 0.12
8) 9.67
4) 400.903
4) 16.07
4) 4361
4) 0.875
5) 7.0121
5) 23.48
5) 897.6
5) 24.82
6) 0.76
Test #2 Decimals
1. Write this word number as a decimal numeral.
seven and one hundred twenty-five thousandths______________________
2. Round each of the following as indicated.
a. To the nearest whole number
________1) 27.3
________2) 8.72
b.To the nearest tenth
________ 1) 0.51
________ 2) 6.97
c. To the nearest hundredth
________ 1) 0.008
________ 2) 1.284
d.To the nearest thousandth
_________1) 48.14876
_________2) 0.00521
3. Add:
a. 16.7 + 0.03 + 9
4. Subtract
a. 867 – 721.42
b. 0.875 + 214.6 + 23
b. 16.931 – 10.37
74
5. Multiply
a. 6.834 x 0.25
b. 0.1073 x 89.1
6. Divide:
a. 169 ÷ 0.13
b. 1.69 ÷ 1.3 c. 0.21 ÷ 14
d. 7 ÷ 16 (round to nearest hundredth.)
Answers:
1. 7.125
2.
a.
b.
c.
d.
3.
a.
b.
4.
a.
b.
5.
a.
b.
6.
a.
b.
c.
d.
1) 27
1) 0.5
1) 0.01
1) 48.149
2) 9
2) 7.0
2) 1.28
2) 0.005
25.73
238.475
145.58
6.561
1.7085
9.56043
1300
1.3
0.015
0.44
Percent and Proportion
I.
Numbers with percent signs are also related to common fractions and to decimal
numbers in that they, too, make a statement that an object has been divided into parts.
However, percents differ significantly from fractions and decimals in that the objects to
which they refer have always been divided into 100 parts. (The word itself means “by
the hundred.”) This fact gives percents special meaning and really makes them simpler
to deal with than fractions and decimals. Because percents are based on “100,” the
numbers always represent parts of 100. The percent sign replaces the decimal point in a
two-place, or hundredths, decimal number.
II.
There are three types of percent problems. All of them can be solved by one method, the
proportion. Of course, there are other methods, but why bother with three different
methods when one will do the job?
a. Find a certain percent of a number.
i. Example: what is 15% of 95?
b. Find what percent a certain number is of a given number.
i. Example: 6 is what % of 92?
c. Find the number when a percent of it is given.
i. Example: 6 is 20% of what number?
75
III.
One of the simplest methods of solution of percent problems is the proportion method.
a. A proportion is a comparison of two equal ratios.
Example 1:
1
2
1. 2 = 4
1
2
and 4 are ratios. They tell us that 1 has the same relationship to 2 that 2 has to
2
4. We can prove this by multiplying 1 x 4 and 2 x 2. Both multiplications give
us the same result – 4. That proves our proportion is true.
Example 2:
2
6
=
3
9
2 x 9 = 18
3 x 6 = 18
Example 3:
5
10
=
10 20
5 x 20 = 100
10 x 10 = 100
Assignment sheet #1 Proportions
Directions: Supply the missing numbers in these proportions. Multiply the number to prove your
answer.
2
?
1. 4 = 8 This is like “raising” fractions to their equivalents.)
2.
3.
4.
5.
6.
7.
8.
1
3
5
6
9
8
?
6
=
=
=
=
?
=
7
15
?
3
?
=
=
?
6
10
?
18
?
10
30
2
14
3
5
9
12
76
Answers:
1. 4
2. 2
3. 12
4. 16
5. 2
6. 1
7. 25
8. 4
b. Replace the “?” with an “x.” (An “x” represents what we do not know.)
49
𝑥
=
98
7
1) Multiply
98 ● x = 98x
49 ● 7 = 343
2) Set up the equation—an equation is a statement that says two things are equal to each
other.
98x = 343
This equation says that 98 multiplied by some number, which we do not know yet is
equal to 343.
Because we had to multiply to write this equation, we will divide to solve it.
98𝑥
98
=
343
98
x = 3.5
Answer
3) Try another one
1.5
𝑥
=
4
9
a. Multiply
4 ● x = 4x
1.5 ● 9 = 13.5
b. Write the equation
4x = 13.5
c. Solve by dividing both sides by 4
4𝑥 13.5
=
4
4
𝑥 = 3.375 Answer
77
4) Try this one
60
15
=
10
𝑥
a. Multiply
60 ● x = 60x
10 ● 15 = 150
b. Write the equation
60x = 150
c. Solve (divide both sides by 60)
60𝑥
60
=
150
60
x = 2.5 Answer
Notice that if we reduce the first ratio, we save ourselves some work, and the result is the same.
When you are solving proportions, always check to see if you are able to reduce one of the ratios.
6
60
15
=
10
𝑥
1
6x = 15
x = 2.5 Answer
Assignment Sheet #2 Proportions
Directions: Solve for “x” in the following proportions.
1.
2.
3.
4.
1
2
=
3
4.5
9
7
5
5.
𝑥
=
6
𝑥
𝑥
= 14
2.3
4
=
6.
7.
𝑥
8
8.
𝑥
21
𝑥
18
17
𝑥
3
𝑥
13.5
=
27
65
=
90
153
=
=
382.5
19
57
Answers:
1)
2)
3)
4)
10
9
18
4.6
5)
6)
7)
8)
10.5
13
42.5
9
78
C. Solving story problems using proportions.
Example 1: The readout on an exercise machine says if you exercise for 24 minutes, you
will burn 356 calories. How many calories will you burn if you exercise for 30 minutes?
Solution:
Step 1: write a proportion with labels
24 𝑚𝑖𝑛
356 𝑐𝑎𝑙𝑜𝑟𝑖𝑒𝑠
=
30 𝑚𝑖𝑛
𝑥 𝑐𝑎𝑙𝑜𝑟𝑖𝑒𝑠
Notice like quantities are in the numerator and like quantities are in the
denominator.
Step 2: Solve by cross multiplying (drop the labels at this point)
24x = 10680
Step 3: Divide both sides by 24
24𝑥
10680
=
24
24
x = 445
Step 4: Answer the question
445 minutes
Example 2: Jason bought 8 tickets to an international food festival for $52. How many
tickets can he purchase with $90?
Step 1: Write a labeled proportion
8 𝑡𝑖𝑐𝑘𝑒𝑡𝑠
$52
=
𝑥 𝑡𝑖𝑐𝑘𝑒𝑡𝑠
$90
Step 2: Cross multiply and drop labels
52x = 720
Step 3: Solve by dividing
52𝑥
52
=
720
52
x = 13.8
79
Step 4: Answer question
Since we can’t buy a part of a ticket and don’t have enough money for
another whole ticket, we must round down.
Answer: 13 tickets
Example 3: Mary bought a new car. In the first 8 months, it was driven 10,000 miles. At
this rate, how many miles will the car be driven in 1 year?
Step 1: Write the labeled proportion
8 𝑚𝑜𝑛𝑡ℎ𝑠
10,000 𝑚𝑖𝑙𝑒𝑠
=
1 𝑦𝑒𝑎𝑟
𝑥 𝑚𝑖𝑙𝑒𝑠
Note: we do not have like quantities in the numerator. Therefore, we must
write
8 𝑚𝑜𝑛𝑡ℎ𝑠
12 𝑚𝑜𝑛𝑡ℎ𝑠
=
10,000 𝑚𝑖𝑙𝑒𝑠
𝑥 𝑚𝑖𝑙𝑒𝑠
Step 2: cross multiply and drop labels
8x = 120,000
Step 3: Solve by dividing
8𝑥
120,000
=
8
8
Step 4: Answer question
15,000 miles
Problems:
1. An 8 lb. turkey breast contains 36 services of meat. How many pounds of turkey breast
would be needed for 54 servings?
2. A bookstore manager knows that 24 books weight 37 lbs. How much do 40 books
weigh?
3. Bill uses 3 gallons of paint to cover 1275 ft2 of siding. How much siding can Bill paint
with 7 gallons of paint?
4. A doctor orders 225 mg of a drug. The drug is available in 75 mg per ml vials. How
many ml will the nurse administer?
5. A dog is to receive 20 units of insulin twice a day. The insulin comes in a 1000u/10ml
container. How many days will the container last?
80
Answer:
1.
2.
3.
4.
5.
12 lbs
61 2/3 lbs
2975 ft2
3 ml
25 days
D. It’s time to apply this knowledge to percent problems. All three types of percent problems
can be solved by means of a proportion. In a percent proportion the first ratio is the number
ratio; the second ratio is the percent ratio.
100, which represents 100% of the whole object, is always the fourth term of the proportion.
(There, one-fourth of the work is already done for you!)
Number ratio
% ratio
_____________=
_____________
100
1) Look at the first type (II-A) of percent problems.
Find a certain percent of a given number.
What is 15% of 95?
a. We know what percent we are dealing with, so 15 is the third term of the proportion.
It goes above the 100.
=
15
100
b. Now you must match the first ratio to the second. Because “100,” which is on the
bottom, is the whole object, the bottom of the first ratio must also represent the
whole object. In this problem that number is 95. (whatever comes after the word
“of” is the whole)
95
=
15
100
c. We don’t know the number that is 15% of 95, so “x” (which represents what we
don’t know) goes on the top part of the ratio.
𝑥
15
=
95
100
d. Solve the proportion.
81
100𝑥 1425
=
100
100
x = 14.25
15% of 95 is 14.25
Answer
2) The second type (II-B) of percent problem.
Find what % a certain number is of a given number.
8 is what % of 80?
= 100
a)This time we don’t know the % of “x” goes above the 100.
𝑥
= 100
b) 80 represents the whole (note it comes after of); it must match the 100, so it goes
on the bottom of the number ratio.
𝑥
80
= 100
c) 8 represents the part; it goes above the 80.
8
80
𝑥
= 100
d) Now solve:
80𝑥
800
=
80
80
x = 10%
(remember we’re looking for a percent, so don’t forget to put the %
sign after the 10)
8 is 10% of 80 Answer
3) The third type (IIC) of percent problem.
Find the number when a percent of it is given.
6 is 20% of what number?
= 100
82
a)We know the percent.
20
= 100
b) We know that 6 is part of the whole object.
6
20
= 100
c) We do not know the whole number since what comes after “of” is not known.
6
𝑥
20
= 100
d) Solve:
20𝑥 600
=
20
20
x = 30
6 is 20% of 30 Answer
Assignment sheet #3 Percent and proportions
Direction: Solve each of the following by means of a proportion.
1. 25% of 40 is _____________.
2. 80% of 45 is _____________.
3. 28% of 92 is _____________ to the nearest tenth.
4. 48 is ____________% of 80.
5. _________% of 60 is 30.
6. 64 is ____________% of 80.
7. ___________% of 65 is 13.
8. 175% of 30 is _____________.
9. ¾% of 100 is ______________.
10. 1 is ______% of 1,000.
83
11. 70% of ______________is 28.
12. 140% of ____________is 35.
13. 219 is 12% of _________.
14. 5 is ½% of ____________.
Answer:
1. 10
9. .75
2. 36
10. .1%
3. 25.76
11. 40
4. 60%
12. 25
5. 50%
13. 1825
6. 80%
14. 1000
7. 20%
8. 52.5
E. In another type of percent problem you are required to find a percent of increase or decrease.
These are always set up as follows:
𝑇ℎ𝑒 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑜𝑟 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑒
𝑥
=
𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡
100
Examples: 1. Last year your income was $10,000. This year it is $12,000. What is the percent
of increase? This requires a two-step solution. First find the amount of increase. Then set up a
proportion to find the percent of increase, putting the figure for last year in the whole number
place in the number ratio.
-
$12,000
10,000
$ 2,000—amount of increase
(𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒)2,000
(𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙)10,000
10,000𝑥
10,000
𝑥
= 100
200,000
= 10,000
X = 20%
Income increased by 20%
Answer
2. Last year your income was $10,000. This year it is $9,000. What is the percent of
decrease?
-
$10,000
9,500
500—amount of decrease
84
1
(𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑒)500
(original)10,000
𝑥
= 100
20
(see how much simpler it is when you reduce one of the
ratios)
20x =100
x=5%
Income decreased by 5%.
Answer
Story Problems
1. A restaurant typically sells 250 desserts in an evening. On an especially busy evening
they sold 350 desserts. What was the percent increase?
2. Joe’s September electric bill was $105. In October it dropped to $73.50. Find the
percent decrease in his electric bill.
3. A nurse calculated the flow rate on an IV to be 25gtt/min. Later she discovered it had to
be increased to 33gtt/min. Find the percent increase.
4. By using only cold water in the washing machine, a family with a montly fuel bill of $78
can reduce their bill to $74.88. Find the percent decrease.
5. A part-time salesperson earns $9800 one year and gets a 9% increase the next year.
What’s the new salary?
6. The population of Stanton increased from 1500 to 3600. Find the percent increase in
population.
Answers:
1.
2.
3.
4.
5.
6.
40%
30%
32%
4%
$10,682
140%
Test—Percent and Proportion
Directions: Solve the following percent problems.
1. 50 is what percent of 200?
2. 143 is 130% of what number?
3. 45% of 200 is ___________?
4. 37% of 86 is ____________?
85
5. 25% of what number is 6?
6. What percent of 720 is 180?
7. 37 is what percent of 60? (to the nearest whole number)
8. 65% of what number is 19.5?
9. 6% of 24 is_________?
10. What percent is four hundred twenty of one hundred twenty?
Answer:
1. 25%
6. 25%
2. 110
7. 62%
3. 90
8. 30
4. 31.82
9. 1.44
5. 24
10. 350%
86
The following pagers are general information for you to be familiar with.
Common fraction, decimal, and percent equivalents.
Fraction
Decimal
Percent
½
.50
50%
¼
.25
25%
¾
.75
75%
1/8
.125
12 ½% or 12.5%
3/8
.375
37 ½% or 37.5%
5/8
.625
62 ½% or 62.5%
7/8
.875
87 ½% or 87.5%
1/16
.0625
6 ¼ % or 6.25%
3/16
.1875
18 ¾% or 18.75%
5/16
.3125
31 ¼% or 31.25%
7/16
.4375
43 ¾% or 43.75%
9/16
.5625
56 ¼% or 56.25%
11/16
.6875
68 ¾% or 68.75%
13/16
.8125
81 ¼% or 81.25%
15/16
.9375
93 ¾% or 93.75%
1/3
.33 1/3
33 1/3% or 33.33%
2/3
.66 2/3
66 2/3% or 66.67%
1/6
.16 2/3
16 2/3% or 16.67%
5/6
.83 1/3
83 1/3% or 83.33%
1/12
.08 1/3
8 1/3% or 8.33%
5/12
.41 2/3
41 2/3% or 41.67%
7/12
.58 1/3
58 1/3% or 58.33%
11/12
.91 2/3
91 2/3% or 91.67%
1/7
.14 2/7
14 2/7% or 14.29%
2/7
.28 4/7
28 4/7% or 28.57%
3/7
.42 6/7
42 6/7% or 42.86%
4/7
.57 1/7
51 1/7%
5/7
.71 3/7
71 3/7%
6/7
.85 5/7
85 5/7%
87
Fractions, Decimals, and Percents
Fractions, decimals, and percent all represent parts of a complete quantity. They are
interchangeable. Sometimes it is more convenient to use one form rather than either of the other
two.
Because all three are interchangeable, it is necessary to know a few rules for changing from one to
another.
I.
To change any decimal to a percent, move the decimal point two places to the right and
add the percent sign.
Example: .25
1.04
.005
II.
=
=
=
25%
104%
.5% or ½%
To change a percent to decimals or whole numbers, move the decimal point two places
to the left and drop the percent sign.
Example: 32%
110%
.5%
½%
=
=
=
=
.32
1.10 or 1.1
.005
.5% or .005
III.
To change a percent to its fractional equivalent, change it directly to a decimal, and then
write the decimal fraction as a common fraction. Reduce it to its lowest terms. A
decimal fraction will have a multiple of ten as a denominator, such as 10,100, 1000, etc.
IV.
To change a fraction to a decimal, divide the denominator into the numerator; carry the
quotient to as many places as necessary to come out even, or as required by the problem
being solved.
.1875
3
Example: 16
= what as a decimal? 16)3.0000
88
Divisibility of integers
You can divide by
2
If
The last digit is 0, 2, 4, 6, 8
3
The sum of the digits is divisible by 3
4
The integer named by the last two digits is divisible by 4
5
The last digit is 0 or 5
7
From the right, group the digits by threes, and mark these
groups alternately positive and negative; then total the signed
groups. Is the sum divisible by 7?
9
The sum of the digits is divisible by 9.
10
The last digit is 0.
11
Mark the digits alternately positive and negative fro the right;
then total the signed digits. Is this sum divisible by 11?
13
Compute the sum as in the test for 7. Is this sum divisible by
13?
89