Download Section R.1 Fractions

Section R.1 Fractions:
Fractions are things we have all learned, but often leave students a
bit confused and just not very confident about how to handle them.
I think this is because they just don’t seem to work like “regular”
numbers. In this section we will do a quick review of how fractions
work as a warm up to more complicated topics in this course
In this section we will discuss:
• Writing fractions in lowest terms
• Multiplying and dividing fractions
• Adding and subtracting fractions
• Solving applied problems that involve fractions
Basics about fractions
Fractions are really one number divided by another. The top
number which is called the numerator and a bottom number called
the denominator.
An important concept that will help us in simplifying fractions will be
the idea of prime and composite numbers.
A product is the result of multiplying two numbers together.
When a number is written as two numbers multiplied together, then
those numbers are called the factors of the number.
For example: 15 = 5 x 3 so this is a way to write 15 as a product of
two numbers so 5 and 3 are factors of 15.
Important things about the number 1
The most basic operation we can perform on fractions is writing
them in lowest terms. To do that we need to be able to write the
numerator and a denominator in factored form.
The first thing we need to know is the fact that any number divided
by itself is equal to 1. For example
4
=1
4
The next thing is that multiplying a number by 1 leaves the number
unchanged. For example: 4 ⋅ 1=4
And finally, dividing by a 1 leaves the number unchanged
4
= 4.
1
Taking it a step further, any whole number can be written as itself
over 1.
Armed with this information, we can now discuss how to write a
fraction in lowest terms:
Step 1: Look for the largest common factor in the numerator
and denominator, and write the numerator as a product of that
factor and another number. Do the same for the denominator.
Step 2: Replace each pair of factors common to the
numerator and denominator with a 1
Step 3: you may have to repeat steps 1 and 2 if you didn’t
pick the largest factor common to numerator and denominator.
Examples: Write each fraction in lowest terms
1.
8
14
2.
35
42
3.
72
120
Multiplying and Dividing Fractions
To multiply two fractions multiply across the numerator and across
the denominator. Then simplify by writing in lowest terms.
Examples:
1.
5 2
⋅
8 10
2.
1 12
⋅
10 5
3.
7 12
⋅
9 14
Dividing Fractions
Division and multiplications are very similar processes. They are
reverse processes of one another which really means in some
sense they are the same. Technically division is defined as
. When you multiply reciprocals together they always multiply to
give 1 as its product.
7 9 63
⋅ =
=1
9 7 63
And
5⋅
1 5 1 5
= ⋅ = =1
5 1 5 5
To divide fractions:
Step 1: Multiply the first fraction by the reciprocal of the second
fraction.
Step 2. The result, called the quotient should be written in lowest
form.
Examples:
1.
3 2
÷
10 7
2.
3 7
÷
4 16
3.
4
÷6
3
Adding Fractions:
Just as multiplying and dividing related to one another, so are
adding and subtracting related to one another.
Subtraction is defined as adding
. So if you know
how to add fractions then you know how to subtract them as well.
The process for adding and subtracting is however quite different
from multiplication and division.
To add/subtract fractions, they
To add/subtract fractions:
1. Find a common denominator. The easiest way to do this is to
just multiply the two denominators together. That is
guaranteed to give you a common denominator..
2. Next you will convert each original fraction in the problem into
a fraction with this new denominator. .
3. Once the same denominator is achieved, add/subtract across
the top and leave the denominator unchanged!
An example is really necessary here.
Examples:
1.
3 4
+
5 5
2.
7
2
+
30 45
3.
13 5
−
15 6
Applied problems with fractions
If 4 serving of Quaker grits requires 3/4 tbsp of grits, 3 cups of
water, and ¼ tsp of salt, how much of each does 2 servings
require?
Section R.2 Decimals and Percents
In this section we will discuss
• How to convert from decimals to fractions
• How to convert from fractions to decimals
• How to convert percents to decimals and decimals to
percents.
Decimals and Place Value
They key to converting decimals to fractions is understanding place
values in decimals
For example: 0.1567
There are 4 numbers after the decimal which in order are the 10ths,
100ths, 1000ths and 10,000ths place, so you would just put
1567/10,000
Write each decimal as a fraction:
1. 0.8
2. .431
3. 20.58
Converting Percents to Decimals
All you have to do here is move the decimal point 2 places to the
left.
Examples
Convert to a decimal:
1. 310%
2. 0.03%
3. 4.5%
To convert a decimal to a percent:
You do the opposite: Move the decimal place two places to the
right
1. .71 as a percent
2. 1.32 as a percent
3. 0.0685 as a percent
An application:
Where Lan lives, a sales tax of 5% is added to all purchases.
If he buys a DVD player that costs $80, how much will he pay
for the DVD player, including the tax?
Section 1.1 Exponents, Order of Operations and
Inequality
In this section we will discuss
• Exponents
• Rules for Order of Operations
• Use more than one grouping symbol
• Know the meanings of inequality symbols
• Translate word statements to symbols
Exponents:
Exponents are shorthand expressions for multiplication.
For example
2 5 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 = 32
Examples:
1. 6 2
2.
35
3.
⎛ 3 ⎞
⎜ ⎟
⎝ 4 ⎠
2
4.
⎛ 1 ⎞
⎜ ⎟
⎝ 2 ⎠
4
Order of Operations:
There is an easy way to remember the order of operations:
Please Excuse My Dear Aunt Sally
P:
E:
M:
D:
(Note: Mult/Division are on the same level. You do them after
exponents, in order from left to right. You do not necessarily do
multiplication before division)
A:
S:
(Note: just like mult/division, add/sub are on the same level. You
do them in order from left to right. You do not necessarily do
addition before subtraction)
Examples:
1. 7 + 3 ⋅ 8
2. 7 ⋅ 6 − 3(8 + 1)
3.
2 + 32 − 5
4. 9[(4 + 8) − 3]
5. 2(7 + 8) + 2
3⋅5 +1
Using Inequality Symbols:
Less than: <
Less than or equal to: ≤
Greater than: >
Greater than or equal to ≥
Not equal to ≠
For example All of the following statements are true:
4<5
4≤4
4≤5
5>4
5≠ 4
The mathematical symbols listed above, in addition to the = sign are
used to write mathematical sentences. When none of those
symbols are present, you have what is called a mathematical
expression
Examples:
Write each word statement in symbols:
1. Fifteen is equal to five plus ten
2. Five is less than or equal to nine.
Write each statement in words and decide if it is true or false.
1. 7 < 19
2.
10 3
≠
7 2
Section 1.2 Variables, Expressions and Equations
In this section we will discuss
• Evaluate algebraic expressions, given values for the variables
• Translate phrases from words to algebraic expressions
• Identify solutions to equations
• Translate sentences to equations
• Distinguish between expressions and equations
A few definitions:
A variable is a symbol, usually represented by a letter used as a
place holder for an unknown number.
For example, if you learned the formula for the slope-intercept form
of a line: y = mx + b à All of those letters are variable, which are all
standing in for a number.
An algebraic expression is a collection of numbers, variables,
operation symbols, and grouping symbols such as parentheses .
The following are examples of algebraic expressions:
x+5
p+ 12
8 p 2 − 12
Evaluating Expressions:
If you know the number (or numbers) that your variable is standing
in for, you can replace the the variable with that number and then
“evaluate” the expression.
Evaluate just means find the value of the expression if you plug in
some particular number.
Examples:
If p =3, find
1. 2 p 2 − 12
2. (2 p )2 − 12
If x = 6 and y = 9
3. 4x + 7y
4.
4x − 2 y
x +1
Using Variables to write word phrases as algebraic expressions
Key words:
Sum: +
Minus: Subtracted from: Product: x
Divided by: ÷
The difference between: Examples:
Write each word phrase as an algebraic expression/ Us x as the
variable.
1. The sum of 5 and a number
2. A number minus 4
3. The difference between 48 and a number
4. The product of 6 and a number
5. 9 multiplied by the sum of a number and 5
Determining if a number is a solution to an equation
An equation is a statement that two expressions are equal.
Equations always include the equality symbol, =.
If a number is a solution to an equation, when you substitute that
number in for the variable, you get a true statement. Otherwise it is
not a solution.
Examples:
1. p -1 = 3; 2
2. 8k – 11 = 5; 2
Translating Sentences to equations
We already saw how to convert an English phrase into a
mathematical expression, and now we will convert full sentences
into mathematical sentences, which are equations.
The (in)equality part of the sentence is also denoted by a verb, for
examples:
Is à =
Gives à =
Is less than à <
Examples:
1. Three times the sum of a number and 13 is 19.
2. Five times a number is subtracted from 21 giving 15
Distinguishing between expressions and equations
An equation is a sentence (with an =)
An expression is a phrase whose result represents a number
Expression or Equation
1.
3x − 1
5
2. 2x + 5 = 7
Section 1.3 Adding Real Numbers
Section 1.4 Subtracting Real Numbers
In these sections we will discuss
•
•
•
•
•
Adding two numbers with the same sign
Adding two numbers with different signs
Subtracting Real Numbers and using the definition of sub.
Using order of operations with real numbers
Translate words and phrases that indicate addition
Adding and Subtracting Real Numbers
Real numbers are all the numbers on the number line, including
positive and negative numbers as well as 0.
To add two positive numbers: Just add them!
To add a negative and a positive number: Take the sign of the
greater and subtract.
To add two negative numbers: find the sum of the numbers, but
keep the negative sign.
Subtraction: By definition, subtracting is defined as adding the
opposite, so for example: 5 – 4 = 5 + (-4)=1
Or to generalize: a – b = a + (- b)
To subtract two numbers: Change the subtraction symbol to
addition and then change the sign of the number that follows the –
sign, and then add as above.
This may help: your book says to add two numbers of the same
sign, add the absolute value of the numbers and keep the sign.
Examples:
1. -7 + (-3)
2. 6 + (-3)
3. -5 + 1
4. 42 – 42
5. 3 – ( - 5)
6.
1 1
−
5 3
Use the rules for order of operations with real numbers:
The rules are exactly the same. Things just start getting more
complicated when you toss a bunch of signed numbers in your
parentheses. Again, remember to PEMDAS!
Examples:
1. -9 + [-4 + (-8 + 6)]
2. 2 – [(-3) – (4 + 6)]
Translating words and phrases that indicate addition/subtraction
Words that mean addition:
Words that mean subtraction:
Examples:
Write a numerical expression for each phrase and simplify the
expression.
1. 4 more than -12
2. -12 added to -31
3. 7 increased by the sum of 8 and -3
4. The difference between -5 and -12
5. 7 less than -2
Applications
1. Bonika’s checking account balance is $54. She then takes a
gamble by writing a check for $89. What is her new balance?
2. A female polar bear weighed 660 lb when she entered her
winter den. She lost 45 pounds during each of the first two
months of hibernation and another 205 lb before leaving the
den with her two cubs in March. How much did she weigh
when she left the den?
3. In 1998, undergraduate college students had an average
credit card balance of 1879. The average balance increased
by $869 in 2000 and then dropped by $579 in 2004. What
was the average credit card debt of college students in 2004?