Basic_Math

1

A. BASIC ARITHMETIC

• Foundation of modern day life.
• Simplest form of mathematics.

Four Basic Operations :

• Addition plus sign
• Subtraction minus sign
• Multiplication x multiplication sign
• Division division sign

Equal or Even Values equal sign

2

1. Beginning Terminology

• NNuummbbeerrss - Symbol or word used to express value or quantity.

 Arabic number system - 0,1,2,3,4,5,6,7,8,9

• DDiiggiittss - Name given to place or position of each numeral.

Number Sequence

2. Kinds of numbers

• Whole NNuummbbeerrss - Complete units , no fractional parts. (43)

 May be written in form of words. (forty-three)

• Fraction - Part of a whole unit or quantity. (1/2)

3

2. Kinds of numbers (con’t)

• DDeecciimmaall NNuummbbeerrss - Fraction written on one line as whole no.

 Position of period determines power of decimal.

4

B. WHOLE NUMBERS
1. Addition

• NNuummbbeerr LLiinnee - Shows numerals in order of value

• AAddddiinngg oonn tthhee NNuummbbeerr LLiinnee (2 + 3 = 5)

• AAddddiinngg wwiitthh ppiiccttuurreess

5

1. Addition (con’t)

• Adding in columns - Uses no equal sign

5 897 Answer is called “sum”.
+5 + 368
10
1265
Simple
Complex

Table of Digits

6

ADDITION PRACTICE EXERCISES

1. a. 222 b. 318 c. 611 d. 1021
+ 222 + 421 + 116 + 1210
444 739 727 2231

2. a. 813 b. 924 c. 618 d. 411
+ 267 + 429 + 861 + 946
1479 1357
1080 1353

3. a. 813 b. 1021 c. 611 d. 1021
222 611 96 1621

+ 318 + 421 + 861 + 6211
1353 2053 1568 8853

7

2. Subtraction

• Number Line - Can show subtraction.

Number Line Subtraction with pictures

Position larger numbers above smaller numbers.

If subtracting larger digits from smaller digits, borrow from
next column.

4513 8
-397

141

8

SUBTRACTION PRACTICE EXERCISES

1. a. 6 b. 8 c. 5 d. 9 e. 7
-3 -4
3 4 -2 -5 -3
3 4 4

2. a. 11 b. 12 c. 28 d. 33 e. 41
-7 -8
-6 -4 -9 26 33
5 8 19

3. a. 27 b. 23 c. 86 d. 99 e. 72
- 19 - 14 - 57 - 65
8 9 29 - 33 7
66

9

SUBTRACTION PRACTICE EXERCISES (con’t)

4. a. 387 b. 399 c. 847 d. 732

- 241 - 299 - 659 - 687
146 100 188 45

5. a. 3472 b. 312 c. 419 d. 3268

- 495 - 186 - 210 - 3168
2977 126 209 100

6. a. 47 b. 63 c. 47 d. 59

- 38 -8 - 32 - 48
9 55 15 11

7. a. 372 b. 385 c. 219 d. 368

- 192 - 246 - 191 - 29
180 139 28 339

10

3. Checking Addition and Subtraction

• Check Addition - Subtract one of added numbers from sum.

Result should produce other added number.

2 5 73
+ 8 + 3 + 48
10 8 121
- 8 - 3 - 48

2 5 73

• Check Three or more #s - Add from bottom to top.

927
318
426
183
927

• Check SSuubbttrraaccttiioonn - Add subtracted number back.

5 62 103

- 4 - 37 - 87

1 25 16

+4 + 37 + 87

5 62 103 11

CHECKING ADDITION & SUBTRACTION PRACTICE EXERCISES

1. a. 6 b. 9 c. 18 d. 109
+8 +5 + 18 + 236
14 26
13 335

2. a. 87 b. 291 c. 367 d. 28
- 87 - 192 - 212 -5
1 99 55 24

3. a. 34 b. 87 c. 87 d. 21
+ 12 13 13 - 83
46 81 81 104

+ 14 + 14
195 746

4. a. 28 b. 361 c. 2793142
- 16 - 361 - 1361101
22 0
1432141

Check these answers using the method discussed.

12

CHECKING ADDITION & SUBTRACTION PRACTICE EXERCISES

1. a. 6 b. 9 c. 18 d. 109
+8 +5 + 18 + 236
13 14 26 335
-8 -5 - 18 - 236

5 9 8 99

2. a. 87 b. 291 c. 367 d. 28
- 87 - 192 - 212 -5
1 99 55 24
+ 87 + 192 + 212 +5

88 291 267 29

3. a. 34 b. 195 c. 949 d. 21
+ 12 + 83
46 87 103 104
- 12 13 212 - 83
81 439
34 + 14 + 195 21
195 746

4. a. 28 b. 361 c. 2793142 # = Right
- 16 - 361 - 1361101 # = Wrong
22 0 1432141
+ 16 + 361 + 1361101

38 361 2793242

13

4. Multiplication

• In Arithmetic - Indicated by “times” sign (x).

Learn “Times” Table

6 x 8 = 48

14

4. Multiplication (con’t)

• Complex Multiplication - Carry result to next column.

Problem: 48 x 23

+ 2 48 + 2 48 + 1 48 + 1 48

X 23 X 23 X 23 X 23
4 144 144 144
6 960
1104

Same process is used when multiplying
three or four-digit problems.

15

MULTIPLICATION PRACTICE EXERCISES

1. a. 21 b. 81 c. 64 d. 36
x4
84 x9 x5 x3
729 320 108

2. a. 87 b. 43 c. 56 d. 99
x6
x7 x2 x0 594
609 86 0

3. a. 24 b. 53 c. 49 d. 55
x 13 x 15 x 26
312 795 1274 x 37
2035

16

MULTIPLICATION PRACTICE EXERCISES (con’t)

4. a. 94 b. 99 c. 34 d. 83
x 73 x 27
6862 2673 x 32 x 69
1088 5727

5. a. 347 b. 843 c. 966

x 21 x 34 x 46
7287 28,662 44,436

6. a. 360 b. 884 c. 111
x 37 x 63
x 19
13,320 55,692 2109

7. a. 493 b. 568 c. 987
x 432 x 654
x 216
106,488 245,376 645,498

17

5. Division

• Finding out how many times a divider “goes into” a
whole number.

15 5 = 3 15 3 = 5

18

5. Division (con’t) ““ ssiiggnn..

• SShhoowwnn bbyy uussiinngg aa ssttrraaiigghhtt bbaarr ““ ““ oorr ““

105 48 “goes into” 50 one time.
48 5040
1 times 48 = 48
48 50 minus 48 = 2 & bring down the 4
48 goes into 24 zero times.
2 40 Bring down other 0.
48 goes into 240, five times
240

0

5 times 48 = 240

240 minus 240 = 0 remainder

So, 5040 divided by 48 = 105 w/no remainder.

Or it can be stated:

48 “goes into” 5040, “105 times”

19

DIVISION PRACTICE EXERCISES

211 62 92
1. a. 48 5040 b. 7 434 c. 9 828

13 310 101
2. a. 9 117 b. 12 3720 c. 10 1010

256 687
3. a. 23 5888 b. 56 38472

98 67
4. a. 98 9604 b. 13 871

50 123
5. a. 50 2500 b. 789 97047

20

DIVISION PRACTICE EXERCISES (con’t)

7 9000
6. a. 21 147 b. 3 27000

61 101
7. a. 32 1952 b. 88 8888

67 r 19 858 r 13
8. a. 87 5848 b. 15 12883

12 r 955 22 r 329

9. a. 994 12883 b. 352 8073

21

C. FRACTIONS - A smaller part of a whole number.

Written with one number over the other, divided by a line.

3 11 or 3 11
8 16 8 16

Any number smaller than 1, must be a fraction.

Try thinking of the fraction as “so many of a specified number of parts”.

For example: Think of 3/8 as “three of eight parts” or...
Think of 11/16 as “eleven of sixteen parts”.

1. Changing whole numbers to fractions.

Multiply the whole number times the number of parts being
considered.

Changing the whole number 4 to “sixths”:

4 = 4 x 6 = 24 or 24
66 6

22

CHANGING WHOLE NUMBERS TO FRACTIONS EXERCISES

1. 49 to sevenths = 49 x 7 = 343 or 343
2. 40 to eighths 7 7 7
3. 54 to ninths 320 or
4. 27 to thirds = 40 x 8 = 8 320
5. 12 to fourths 8 486 or 8
9
= 54 x 9 = 81 or 486
9 3 9
48 or
= 27 x 3 = 4 81
3 3

= 12 x 4 = 48
4 4

6. 130 to fifths = 130 x 5 = 650 or 650
55 5

23

2. Proper and improper fractions.

Proper Fraction - Numerator is smaller number than denominator.
3/4

Improper Fraction - Numerator is greater than or equal to denominator.
15/9

3. Mixed numbers.

Combination of a whole number and a proper fraction.

4. Changing mixed numbers to fractions.

Change 3 7/8 into an improper fraction.

• Change whole number (3) to match fraction (eighths).

3 = 3x8 = 24 or 24
8 8
8

• Add both fractions together.

24 + 7 = 31
8
88

24

CHANGING MIXED NUMBERS TO FRACTIONS EXERCISES

1. 4 1/2 4x2 8 1 =9
2 2 22
+= =

2. 8 3/4 8x4 24 3 = 27
4 4 44
+= =

19 x 16 304 7 = 311
16 16 16 16
3. 19 +7/16= =

7 x 12 84 11 = 95
12 12 12 12
4. 7 +11/12= =

6 x 14 = 84 9 = 93
14 14 14 14
5. 6 9/14 +=

5 x 64 = 320 1 = 321
64 64 64 64
6. 5 1/64 +=

25

5. Changing improper fractions to whole/mixed
numbers.

Change 19/3 into whole/mixed number..
19/3 = 19 3 = 6, remainder 1 = 6 1/3 (a mixed number)

CHANGING IMPROPER FRACTIONS TO WHOLE/MIXED NUMBERS EXERCISES

1. 37/7 = = 37 7 = 5, remainder 2 = 5 2/7 (a mixed number)
2. 44/4 = = 44 4 = 11, no remainder = 11 (a whole number)
3. 23/5 = = 23 5 = 4, remainder 3 = 4 3/5 (a mixed number)
4. 43/9 = = 43 9 = 4, remainder 7 = 4 7/9 (a mixed number)
5. 240/8 = = 240 8 = 30, no remainder = 30 (a whole number)
6. 191/6 = = 191 6 = 31, remainder 5 = 31 5/6 (a mixed number)

26

6. Reducing Fractions

Reducing - Changing to different terms.

Terms - The name for numerator and denominator of a fraction.
Reducing does not change value of original fraction.

7. Reducing to Lower Terms

Divide both numerator and denominator by same number.

Example: 3 9 = 3 .. 3= 1 3 9 & 1 3 Have same value.
9 .. 3= 3

8. Reducing to Lowest Terms

Lowest Terms - 1 is only number which evenly divides both numerator
and denominator.

Example: 16 32 =

a. 16 .. 2= 8 b. 8 .. 2= 4 c. 4 .. 2= 2 d. 2 .. 2= 1
32 .. 2= 16 .. 2= 8 8 .. 2= 4 4 .. 2= 2
16

27

REDUCING TO LOWER/LOWEST TERMS EXERCISES

1. Reduce the following fractions to LOWER terms:

a. 15 20 to 4ths = 15 .. 5 = 3
20 .. 5 = 4

• Divide the original denominator (20) by the desired denominator (4) = 5..

• Then divide both parts of original fraction by that number (5).

b. 36 40 to 10ths = 36 .. 4 = 9
40 .. 4 = 10

c. 24 36 to 6ths = 24 .. 6 = 4
36 .. 6 = 6

d. 12 36 to 9ths = 12 .. 4 = 3
36 .. 4 = 9

e. 30 45 to 15ths = 30 .. 3 = 10
45 .. 3 = 15

f. 16 76 to 19ths = 16 .. 4 = 4
76 .. 4 = 19

28

REDUCING TO LOWER/LOWEST TERMS EXERCISES (con’t)

2. Reduce the following fractions to LOWEST terms:

a. 6 10 = a. 6 .. 2 = 3
10 .. 2 = 5

b. 3 9 = a. 3 .. 3 = 1
9 .. 3 = 3

c. 6 64 = a. 6 .. 2 = 3
64 .. 2 = 32

d. 13 32 = Cannot be reduced.

e. 32 48 = a. 32 .. 2 = 16 b. 16 .. 2 = 8 c. 8 .. 8 = 1
64 .. 2 = 32 32 .. 2 = 16 16 .. 8 = 2

f. 16 76 = a. 16 .. 2 = 8 b. 8 .. 2 = 4
76 .. 2 = 38 38 .. 2 = 19

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9. Common Denominator

Two or more fractions with the same denominator.

1 2 6 7
8 8 8 8

When denominators are not the same, a common denominator is
found by multiplying each denominator together.

1 3 2 55 7 1
6 8 9 12 18 24 36

6 x 8 x 9 x 12 x 18 x 24 x 36 = 80,621,568

80,621,568 is only one possible common denominator ...
but certainly not the best, or easiest to work with.

10. Least Common Denominator (LCD)

Smallest number into which denominators of a group of two or
more fractions will divide evenly.

30

10. Least Common Denominator (LCD) con’t.

To find the LCD, find the “lowest prime factors” of each denominator.

1 3 2 5 5 7 1
6 8 9 12 18 24 36

2x3 2x2x2 3x3 2x3x2 2x3x3 3x2x2x2 2x2x3x3

The most number of times any single factors appears in a set is
multiplied by the most number of time any other factor appears.

(2 x 2 x 2) x (3 x 3) = 72

Remember: If a denominator is a “prime number”, it can’t be
factored except by itself and 1.

LCD Exercises (Find the LCD’s)

1 1 1 11 1 3 4 7
6 8 12 12 16 24 10 15 20
2x3 2x5
2x2x2 2x3x2 2 x 2 x3 2 x 2 x 2 x 2 3 x 2 x 2 x 2 3x5 2x2 x5

2 x 2 x 2 x 3 = 24 2 x 2 x 2 x 2 x 3 = 48 2 x 2 x 3 x 5 = 60

31

11. Reducing to LCD

Reducing to LCD can only be done after the LCD itself is known.

1 3 2 5 5 7 1
6 8 9 12 18 24 36

2x3 2x2x2 3x3 2x3x2 2x3x3 3x2x2x2 2x2x3x3

LCD = 72

Divide the LCD by each of the other denominators, then multiply
both the numerator and denominator of the fraction by that result.

1 3 2 5
6 8 9 12

72 .. 6 = 12 72 .. 8 = 9 72 .. 9 = 8 72 .. 12 = 6

1 x 12 = 12 3 x 9 = 27 2 x 8 = 16 5 x 6 = 30
6 x 12 = 72 8 x 9 = 72 9 x 8 = 72 12 x 6 = 72

Remaining fractions are handled in same way.

32

Reducing to LCD Exercises

Reduce each set of fractions to their LCD.

1 1 1 11 1 3 4 7
6 8 12 12 16 24 10 15 20
2x3 2x5
2x2x2 2x3x2 2 x 2 x3 2 x 2 x 2 x 2 3 x 2 x 2 x 2 3x5 2x2 x5

2 x 2 x 2 x 3 = 24 2 x 2 x 2 x 2 x 3 = 48 2 x 2 x 3 x 5 = 60

24 .. 6 = 4 1 48 .. 12 = 4 3 60 .. 10 = 6
1 12 1 x4= 4 10 3 x 6 = 18
12 x 4 = 48 10 x 6 = 60
6 1x4= 4 1 4
6 x 4 = 24 16 48 .. 16 = 3 15 60 .. 15 = 4
1 x3= 3 4 x 4 = 16
24 .. 8 = 3 16 x 3 = 48 15 x 4 = 60
1
1 48 .. 24 = 2 7 60 .. 20 = 3
8 1x3= 3 24 20
8 x 3 = 24 1 x2= 2 7 x 3 = 21
24 x 2 = 48 20 x 3 = 60
24 .. 12 = 2
1

12 1 x 2 = 2
12 x 2 = 24

33

12. Addition of Fractions

All fractions must have same denominator.
Determine common denominator according to previous process.
Then add fractions.

+ + 112 1
4 2
4
3 =6 =

44

Always reduce to lowest terms.

13. Addition of Mixed Numbers

Mixed number consists of a whole number and a fraction. (3 1/3)

• Whole numbers are added together first.
• Then determine LCD for fractions.
• Reduce fractions to their LCD.
• Add numerators together and reduce answer to lowest terms.
• Add sum of fractions to the sum of whole numbers.

34

Adding Fractions and Mixed Numbers Exercises

Add the following fractions and mixed numbers, reducing answers to lowest terms.

3+1. 3 = +2.2 7 =
4 4 5 10

16 = 1 +4 7= 11
42 10
10 10

11

= 10

9 15 4. 5 2 5 + 13 4 =
32 16
+3. =

+9 30 = 39 5+1=6
32 32 32
+8 15 = 23
17 20
20 20
= 32
=13 20 + 6 73

= 20

35

14. Subtraction of Fractions

Similar to adding, in that a common denominator must be found first.
Then subtract one numerator from the other.

-20 14 = 6
24
24 24

To subtract fractions with different denominators: 5-( 1 )
16 4
• Find the LCD...

-5 1

16 4

2x2x2x2 2x2

2 x 2 x 2 x 2 = 16

• Change the fractions to the LCD...

5- 4
16
16

• Subtract the numerators...

-5 4= 1
16
16 16

36

15. Subtraction of Mixed Numbers

• Subtract the fractions first. (Determine LCD)

10 2 3 - 4 1 2

3 x 2 = 6 (LCD)

• Divide the LCD by denominator of each fraction.

6 .. 3 = 2 6 .. 2 = 3

• Multiply numerator and denominator by their respective numbers.

2 x 2 = 4 1 x 3 = 3
3 2 6 2 3 6

• Subtract the fractions.

4 - 3 =1
6 6 6

• Subtract the whole numbers.

10 - 4 = 6

• Add whole number and fraction together to form complete answer.
1
6 + 6 = 6 1
6

37

15. Subtraction of Mixed Numbers (con’t)

Borrowing

• Subtract the fractions first. (Determine LCD)

5 1 16 - 3 3 8 5 - 3becomes1 6

(LCD) = 16 16 16

• Six-sixteenths cannot be subtracted from one-sixteenth, so
1 unit ( 16 ) is borrowed from the 5 units, leaving 4.

16

• Add 16 to 1 and problem becomes:

16 16

4 - 317 6
16 16

• Subtract the fractions.

17 - 6 = 11
16 16 16

• Subtract the whole numbers.

4-3=1

• Add whole number and fraction together to form complete answer.
11
1 + 16 = 1 11
16

38

Subtracting Fractions and Mixed Numbers Exercises

Subtract the following fractions and mixed numbers, reducing answers to lowest
terms.

2-1. 1 = -4. 331 3 15 2 5 =
5 3

6 - 5 = 1 -33 5 15 6 =
15 15
15 15 15

20 6
15 15
-32 15 =171415

-2.5 3 -5. 1 57 1516=
8 12 = 101 4

15 - 6 = 9 =3 -101 4 57 15 =
16 16
24 24 24 8

20 15
16 16
-100 57 =43 5
16

-3. 2 28 1 =
47 5 3 -6. 3 5
14 4 10 12 =

-47 6 28 5 = 19 1 -14 9 10 5 = 4 4 = 4 1
15 15 15 12 12 12 3

39

16. MULTIPLYING FRACTIONS

• Common denominator not required for multiplication.

3 X 4
4 16

1. First, multiply the numerators.

3 X 4 = 12 =
4 16

2. Then, multiply the denominators.

3 X 4 = 12 =
4 16 64

3. Reduce answer to its lowest terms.

..12 4 = 3
64 4 16

40

17. Multiplying Fractions & Whole/Mixed Numbers

• Change to an improper fraction before multiplication.

43 X
4

1. First, the whole number (4) is changed to improper fraction.

4
1

2. Then, multiply the numerators and denominators.

3 X 4 = 12
4 1 4

3. Reduce answer to its lowest terms.

..12 4 = 3 =3
44 1

41

18. Cancellation

• Makes multiplying fractions easier.

• If numerator of one of fractions and denominator of other
fraction can be evenly divided by the same number, they can be
reduced, or cancelled.

Example:

8 X 5 = 18 X 5 =
3 16 3 16

2

1 X 5 = 5
3 2 6

Cancellation can be done on both parts of a fraction.

11
12
21 X 3 = 1 X 1 = 1
24 7 2 14

72

42

Multiplying Fractions and Mixed Numbers Exercises

Multiply the following fraction, whole & mixed numbers.
Reduce to lowest terms.

1. 3 X 4 = 3 2. 26 X 1 = 1
4 16 16 26

3. 4 X 3 = 2 2 4. 9 X 2 = 1 1
5 5 5 3 5

5. 35 X 4 = 1 6. 9 X 3 = 27
4 35 10 5 50

7. 1 X 7 = 7 8. 2 X 5 = 10
6 12 72 3 11 33

9. 5 X 77 = 25 2
15 3

43

19. Division of Fractions

• Actually done by multiplication, by inverting divisors.

• The sign “ “ means “divided by” and the fraction to the
right of the sign is always the divisor.

Example: 1 becomes 3 X 5 = 15 = 3 3
5 4 1 4 4
3
4

20. Division of Fractions and Whole/Mixed Numbers

• Whole and mixed numbers must be changed to improper fractions.

Example:

3 3 2 1 becomes 16 X 3 + 3 = 51 and 2 X8 + 1 = 17
16 8 16 16 8 8

51 17 Inverts to 51 X 8 31
16 8 17
16 = 51 X 8 = 3 X 1
16 17 2 1
2 1

3 X 1 = 3 = 1 1 Double
2 1 2 2 Cancellation

44

Dividing Fractions,Whole/Mixed Numbers Exercises

Divide the following fraction, whole & mixed numbers. Reduce

to lowest terms.

1. 5 3 = 1 1 2. 51 3 = 8 1
8 6 4 16 8 2

3. 18 1 = 144 4. 15 7 25 5
8 12 = 7

5. 14 7 = 2 2
3 4 3

45

D. DECIMAL NUMBERS

1. Decimal System

• System of numbers based on ten (10).

• Decimal fraction has a denominator of 10, 100, 1000, etc.

Written on one line as a whole number, with a period (decimal
point) in front.

5 = .5 5 = .05 5 = .005
10 100 1000

3 digits

.999 is the same as 999
1000
(1+
same number of zeros
as digits in numerator)

46

2. Reading and Writing Decimals

5 7 is written 5.7
10

Whole Number Decimal Fraction (Tenths)

55 7 is written 55.07
100
Decimal Fraction (Hundredths)
Whole Number Decimal Fraction (Tenths)

555 77 is written 555.077
1000

Whole Number Decimal Fraction (Thousandths)
Decimal Fraction (Hundredths)

Decimal Fraction (Tenths)

47

2. Reading and Writing Decimals (con’t)

• Decimals are read to the right of the decimal point.

.63 is read as “sixty-three hundredths.”
.136 is read as “one hundred thirty-six thousandths.”
.5625 is read as “five thousand six hundred twenty-five

ten-thousandths.”
3.5 is read “three and five tenths.”

• Whole numbers and decimals are abbreviated.
6.625 is spoken as “six, point six two five.”

One place .0 tenths
Two places .00 hundredths
Three places .000 thousandths
Four places .0000 ten-thousandths
Five places .00000 hundred-thousandths

48

3. Addition of Decimals

• Addition of decimals is same as addition of whole
numbers except for the location of the decimal point.

Add .865 + 1.3 + 375.006 + 71.1357 + 735

• Align numbers so all decimal points are in a vertical column.
• Add each column same as regular addition of whole numbers.
• Place decimal point in same column as it appears with each number.

.8650 “Add zeros to help eliminate errors.”
1.3000 “Then, add each column.”
375.0060
71.1357
+ 735.0000

1183.3067

49

4. Subtraction of Decimals

• Subtraction of decimals is same as subtraction of whole
numbers except for the location of the decimal point.

Solve: 62.1251 - 24.102

• Write the numbers so the decimal points are under each other.
• Subtract each column same as regular subtraction of whole numbers.
• Place decimal point in same column as it appears with each number.

62.1251 “Add zeros to help eliminate errors.”
- 24.1020 “Then, subtract each column.”

38.0231

50