Maths

Order of Operations and Evaluating Expressions

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Order of Operations and Evaluating Expressions

Order of Operations
When a numerical expression involves two or more operations, there is a specific order in which these operations must be performed.
Order of Operations and Evaluating Expressions 4

When evaluating an expression, proceed in this order:

  1. parentheses are done first.
  2. exponents are done next.
  3. multiplication and division are done as they are encountered from left to right.
  4. addition and subtraction are done as they are encountered from left to right.

The proper application of “order of operations” is needed when working with such mathematical topics as evaluating formulas, solving equations, evaluating algebraic expressions, and simplifying monomials and polynomials.

There is a phrase that may help you to remember this order: PEMDAS
Parenthesis, Exponents, (Multiplication/Division), (Add/Subtract)

While PEMDAS lists M before D, remember that multiplication and division are done as they are read from left to right. It may not always be the case that multiplication is done “before” division.
The expression 16 ÷ 4 x 2 = 8 (not 2).
The same is true of addition and subtraction: 8 – 4 + 2 = 6 (not 2).

The reason (multiplication & division – MD) and (add & subtract – AS) are “grouped” in sets of parentheses is that when those operations are next to each other you do the math from left to right. You do not always do multiplication or addition first. It may be the case where division will be done BEFORE multiplication or subtraction will be done BEFORE addition.

 

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It may be helpful to build a PEMDAS table. Check off the operation after it has been performed. For operations that are not part of the problem, place a hyphen.

  1. Simplify any parenthesis first, starting with the inner most group, and check off the “P” box.
  2. Simplify any powers ( exponents) and check off the “E” box.
  3. Perform the multiplication and division in order from left to right and check off the “M” & “D” boxes.
  4. Do the addition and subtraction last. Remember, if the operations are written next to each other work from left to right and check off the last two boxes.

When there are two or more parenthesis, or grouping symbols, perform the inner most grouping symbol first.

Example 1: Simplify 40 – 2(6 – 4)2
Order of Operations and Evaluating Expressions 1

Example 2: Simplify 30 – (8 – 15 ÷ 3) × 2
Order of Operations and Evaluating Expressions 2

Example 3: Simplify: 2(20 – 32 + 1) – (42 ÷ 2 × 3)
Order of Operations and Evaluating Expressions 3

It is very important to understand that it DOES make a difference if the order is not performed correctly!!!

 

Maths

Proportions

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Proportions

  1. A proportion is a comparison of ratios.
  2. A proportion is an equation that states that two ratios are equal, such as
    Proportions 1
  3. Proportions always have an EQUAL sign!
  4. A proportion can be written in two ways:
    Proportions 2In each proportion the first and last terms (4 and 2) are called the extremes. The second and third terms (8 and 1) are called the means.
Determine if a Proportion is TRUE:

You can tell if a simple proportion is true by just examining the fractions. If the fractions both reduce to the same value, the proportion is true.

Proportions 3
This is a true proportion, since both fractions reduce to 1/3.
Find a Missing Part (a Variable) in a Proportion: 

You can often use this same approach when solving for a missing part of a simple proportion. Remember that both fractions must represent the same value. Notice how we solve this problem by getting a common denominator for the two fractions.

Proportions 4
To find x, use the common denominator of 30.
To change 5 to 30, multiply by 6.
The SAME must be done to the top,
(multiply 2 by 6) to keep the fractions equal.
ANSWER: x = 12

This simple approach may not be sufficient when working with more complex proportions.

You need a rule:
Some people call this rule Cross Multiply!!

Universal Rule:
There is a rule (or algorithm) that can be followed to determine if two fractions are equal. The process multiplies from the top “across” to the bottom between the fractions in the proportion. If these products are equal, the fractions are equal. This method is very handy for setting up an equation to solve for a variable.
A more precise statement of the rule is:
RULE: In a true proportion, the product of the means equals the product of the extremes.
Proportions 5
Proportions can also be solved by multiplying each side of the proportion by the common denominator for both fractions.

Proportions 6

A proportion can be rewritten in different ways, yet remain true.
The following proportions are all equivalent (mathematically the same).
Check that this is true using “cross multiply”.
Proportions 7

Example 1: Solve for x algebraically in this proportion:
Proportions 10Solution:
Proportions 11

Example 2:
Proportions 8
Solution:
Proportions 9

Example 3: The length of a stadium is 100 yards and its width is 75 yards. If 1 inch represents 25 yards, what would be the dimensions of the stadium drawn on a sheet of paper?
Solution: This problem can be solved by an intuitive approach, such as:
100 yards by 75 yards
100 yards = 4 inches (HINT: 100 / 25)
75 yards = 3 inches (HINT: 75 / 25)
Therefore, the dimensions would be 4 inches by 3 inches.
Solution by proportion: (Notice that the inches are all on the top and the yards are all on the bottom for this solution. Other combinations are possible.)
Proportions 12

Example 4: The ratio of boys to girls in Spanish club is 4 to 5. If there are 25 girls in the club, how many boys are in the club?
Solution:
Method 1:
Use the common denominator of 25:
Proportions 13
x = 20 means there are 20 boys in the club.
Method 2:
Using the rule:
Proportions 14
4•25 = 5•x
100 = 5x
x = 20 means there are 20 boys in the club.

Example 5: Find the missing term:
Proportions 15
Solution:
Method 1:
Getting a common denominator is not as easy in
this problem as it was in Example 2. The common denominator will be 11x (where x represents the empty box). You can, in this problem, notice that
8 x 4 = 32. Multiplying the denominator by this same value will create a TRUE proportion: 11 x 4 gives the answer 44.
If you want to use 11x as the common denominator,
you will get:
Proportions 16

Method 2: (using RULE)
Proportions 17

Maths

Percents

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Percents

Percents are used to describe parts of a whole base amount. When one of the parts of the relationship is unknown, we can solve an algebraic equation for the unknown quantity.
(The solution methods shown on this page are of an algebraic, sentence translation nature. Of course, other methods of solution are also possible.)

There are two main types of problems dealing with percents:

1. In the first type of problem, the percent is given.
In these problems, you will change the percent to a decimal.
To change a percent to a decimal, divide the number by 100.
This will move the decimal point two places to the left.

Example 1: Find 7% of 250.
(This can also be read “What number is 7% of 250?”)
Solution: Let x = the answer
x = 7% • 250
x = 0.07 • 250    changing 7% to a decimal
x = 17.5
Remember that the word “of ” means to multiply!

Example 2: 30 is 15% of what number?
Solution: Let n = the answer (“what number”)
30 = 15% • n
30 = 0.15 • n      changing 15% to a decimal
30/0.15 = n       dividing both sides by 0.15
200 = n
n = 200
Remember that the word “of ” means to multiply!

2. In the second type of problem, you are looking for the percent.
In these problems, you will represent the % as a fraction.

Example 1: 3 is what percent of 12?
Percents 1
To change a percent to a fraction, divide the percent value by 100.

Application Problems

Example 1: If 120 million roses were sold on Valentine’s Day, and 75% of the roses were red, how many red roses were sold on Valentine’s Day?
Solution: Let x = the number of red roses sold
x = 75% of 120
x = 0.75 • 120
x = 90 million red roses sold

Example 2: Juan missed 6 out of 92 questions on a test. To the nearest percent, what percent of the questions did he solve correctly?
Solution:
If he missed 6 questions, he got 86 questions correct.
∴ 86 is what percent of 92
86 = x% of 92
⇒ 86 = \(\frac { x }{ 100 } \) • 92
⇒ 86 = \(\frac { 92x }{ 100 } \)
⇒ 8600 = 92x
⇒ x = 93.47
To the nearest percent, he got 93% correct.

Maths

Rational and Irrational Numbers

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Rational and Irrational Numbers

Both rational and irrational numbers are real numbers.

Rational and Irrational Numbers 1
This Venn Diagram shows the relationships between sets of numbers. Notice that rational and irrational numbers are contained in the large blue rectangle representing the set of Real Numbers.

Rational Numbers

A rational number is a number that can be expressed as a fraction or ratio.
The numerator and the denominator of the fraction are both integers.
Examples of rational numbers are:
Rational and Irrational Numbers 2

A rational number can be expressed as a ratio (fraction) with integers in both the top and the bottom of the fraction.
When the fraction is divided out, it becomes a terminating or repeating decimal. (The repeating decimal portion may be one number or a billion numbers.)
Rational and Irrational Numbers 3

Rational numbers on  number line:
A number line is a straight line diagram on which every point corresponds to a real number.
Since rational numbers are real numbers, they have a specific location on a number line.
Rational and Irrational Numbers 6

To convert a repeating decimal to a fraction:
Rational and Irrational Numbers 4

To show that the rational numbers are “dense”:
(The term “dense” means that between any two rational numbers there is another rational number.)
Rational and Irrational Numbers 5

Irrational Numbers

An irrational number cannot be expressed as a fraction.

  1. Irrational numbers cannot be represented as terminating or repeating decimals.
  2. Irrational numbers are non-terminating, non-repeating decimals.
  3. Examples of irrational numbers are:
    Rational and Irrational Numbers 7

Note: Many students think that π is the terminating decimal, 3.14, but it is not. Yes, certain math problems ask you to use π as 3.14, but that problem is rounding the value of to make your calculations easier. π is actually a non-ending decimal and is an irrational number.

There are certain radical values which fall into the irrational number category.
For example, √2 cannot be written as a “simple fraction”
which has integers in the numerator and the denominator.
As a decimal, √2 = 1.414213562373095048801688624 …
which is a non-ending and non-repeating decimal, making √2 irrational.

Irrational Numbers on a Number Line:
By definition, a number line is a straight line diagram on which every point corresponds to a real number.
Since irrational numbers are a subset of the real numbers, and real numbers can be represented on a number line, one might assume that each irrational number has a “specific” location on the number line.
“Estimates” of the locations of irrational numbers on number line:
Rational and Irrational Numbers 8
Maths

Hints for Remembering the Properties of Real Numbers

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Hints for Remembering the Properties of Real Numbers

Commutative Property – interchange or switch the elements
Example shows commutative property for addition:
X + Y = Y + X
Think of the elements as “commuting” from one location to another. “They get in their cars and drive to their new locations.” This explanation will help you to remember that the elements are “moving” (physically changing places).
Hints for Remembering the Properties of Real Numbers 1

Associative Property – regroup the elements
Example shows associative property for addition:
(X + Y) + Z = X + (Y + Z)
The associative property can be thought of as illustrating “friendships” (associations). The parentheses show the grouping of two friends. In the example below, the red girl (y) decides to change from the blue boyfriend (x) to the green boyfriend (z). “I don’t want to associate with you any longer!” Notice that the elements do not physically move, they simply change the person with whom they are “holding hands” (illustrated by the parentheses.)
Hints for Remembering the Properties of Real Numbers 2

Identity Property – What returns the input unchanged?
X + 0 = X       Additive Identity
X • 1 = X        Multiplicative Identity
Try to remember the “I” in the word identity. Variables can often times have an “attitude”. “I am the most important thing in the world and I do not want to change!” The identity element allows the variable to maintain this attitude.
Hints for Remembering the Properties of Real Numbers 3

Inverse Property – What brings you back to the identity element using that operation?
X + -X = 0        Additive Inverse
X • 1/X = 1        Multiplicative Inverse
Think of the inverse as “inventing” an identity element. What would you need to add (multiply) to this element to turn it into an identity element?
Hints for Remembering the Properties of Real Numbers 4

Distributive Property – multiply across the parentheses. Each element inside the parentheses is multiplied by the element outside the parentheses.
a(b + c) = a•b + a•c
Let’s consider the problem 3(x + 6). The number in front of the parentheses is “looking” to distribute (multiply) its value with all of the terms inside the parentheses.
Hints for Remembering the Properties of Real Numbers 5

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Maths