arithmetic chapter 1. arithmetic 1.1 operations with rational numbers 1.2 exponents, base &...
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ARITHMETIC
1.1 Operations with Rational Numbers1.2 Exponents, Base &
Decimals1.3 Estimation & Decimal Operations1.4 Equivalence, Order & Sequences1.5 Percents 1.6 Word Problems
1.1 Rational Numbers
Types of Numbers:
natural, whole, integers, rational, prime, composite, fractions, mixed
Addition Sign Rules: If same signs, add & keep the sign. If different signs, subtract smaller from larger and give sign of the larger.
Rational Numbers
Change mixed numbers to fractions.
Find Least Common Denominators
Addition continued
1.1 Adding & Subtracting
1.
=+9
1
2
12
18
2
18
92 +
Remember to find common denominators first.Did you forget the 2
1.1 Adding & Subtracting
=+−4
112
4
11- D.
4
3- C.
4
3 B.
4
13 .A
4
11
4
41 +−3.
It is subtraction! Subtract smaller from larger and give same sign as larger. (Thus result is negative)
We need to get 4/4 from 2: 2 = 1 and 4/4
1.1 Adding & Subtracting
)1(3
2- −−
3
1- D.
3
1 C. 1- B.
3
5- .A
13
2+−=4.
First let us change the -(-1) to a +1Remember: bigger minus smaller, sign bigger! (result must be positive)
1.1 Multiplying & Dividing
Multiplication & Division Rules of Signed Numbers:
Multiplication of Fractions db
a
d
c
b
a
x
c x x =
Division of Fractions c
d
b
a
d
c
b
ax=÷
If same signs, result is positive. If different signs, result is negative.
1.1 Multiplication & Division
=⎟⎠
⎞⎜⎝
⎛−÷⎟⎠
⎞⎜⎝
⎛3
2
5
1-
3
10- D.
10
3- C.
10
3 B.
3
10 .A
€
1
5 ×
2
3
10
3=7.
Same signs means positive result!!
Remember to invert the second fraction!
1.2 Exponent; Base; Decimal
A. Definition of Exponents
B. Place Value & Base
Place value increases moving left of units place, and decreases moving right of units place.
...10
1
10
1
10
1
10
1.10101010...
43211234 U
n times) used is a(
x...xx aaaaan =
1.2 Examples
5. Select the place value associated with the underlined digit 83,584.02
2323
10 D. 10 C. 10
1 B.
10
1 .A
1.3 Estimation & Operations
A. Estimating Sums, Averages or Products:
An estimate of the average is between the highest and lowest.
1.3 Estimation & Operations
B. Operations with Decimals: To add or subtract: line up dec. pts. To multiply: number of dec. places in the product is the sum of the number of dec. places in the factors. To divide: if divisor is whole number, bring decimal pt. up. If divisor is not, move decimal point as needed.
1.3 Estimation Examples 1. If a unit of water costs $1.82 and 40.435 units were used, which is a reasonable estimate? (Water is sold…)
A. $80,000 B. $800 C. $8000 D.$80
1.3 Estimation Examples 4. 500 students took an algebra test. All scored less than 92 but more than 63. Which of the following could be a reasonable estimate of the avg. score?
A. 96 B. 63 C. 71 D. 60
1.3 Decimal Examples
7. 14.22 - 1.761=
A.12.459
B.13.459
C.11.459
D.12.261
14.220-1.761
It is smaller than
14.22 - 1.22=13It is larger than
14.22 -2=12.22
C. 7.1344
1.3 Decimal Examples
10. 3.43 x 2.8
A. 0.9604 Estimate 3 x 3 = 9
B. 8.504
D. 9.604
Larger than 3 x 2.8 = 8.24
1.3 Decimal Examples
=÷0.0536.75 .
75.3605.0735
A. 735
12.
B. 73.5
C. 7.35D. 0.0735
Dividing by a number between0 and 1 will cause the result to be larger than original number
1.4 Equivalence; Order; Seq.
Rational numbers can be written as fractions, mixed numbers, dec. or %
%2525.4
1 ==Example
To compare two rational numbers, express them in the same way
A sequence of numbers is arranged according to some law. Look for the pattern to find the next number.
1.4 Equivalence Examples
1. 0.19=10019
100
19 D.
10
19 C.
10
91 B.
100
19 .A %
0.19 is not greater than 1
% “means divided by 100”19/100 %=0.19/100=0.0019
1.4 Order Examples
20
11
26
5
100 ~260
sm lgB. <
20
17 0.82
85.0100
85
5 x 20
5 x 17
20
17===
sm lg<
<
5.
8.
A. =
C. >
B. <A. =
C. >
1.4 Sequence Examples
10. Identify the missing term in the following geometric progression
€
−1,1
4,−
1
16,
1
64,−
1
256,_____
PATTERN: Multiply each denom. by 4 to get the nextSigns alternate
4
1 D.
1024
1- C.
1024
1 B.
2048
1 .A
256 x 4 = 1024Thus, positive
1.5 Percents
100#
p
original
difference=
Percent problems100
""
""
"" p
of
is=
Real-world problems with percent
R S T U V Method
Percent increase or decrease
1.5 Percent Examples
1. If 30 is decreased to 6, % decrease?
10030 .)(
24 .)( p
orig
diff=
5p = 400 p = 80
A. 8% B. 24% C. 20% D. 80%
5
4
1.5 Percent Examples
5. What is 120% of 30?
100
120
30=
x 10x = 360
x = 36
A. 0.25 B. 25 C. 36 D. 3.6
B. $380
Find the cost of renting this
1.6 Word Problems
1. A car rents for $180 per
A. $280
week plus $0.25 per mile.
car for a two week trip of 400miles for a family of 4.
D. $760C. $460
when divided by 14.
D. 53C. 48B. 18
multiple of 6 which leaves a
1.6 Word Problems
6. Find the smallest positive
A. 36
remainder of 6 when dividedby 10 and a remainder of 8