Introduction to Percentages

Introduction to Percentage

• Percentages are a crucial topic in mathematics and are frequently tested.
• They are vital for direct questions and support understanding to simplify solutions.
• They have usefulness in profit and loss calculations, determining simple and compound interest, data interpretation, and simplification problems.
• Percentages can be used with time and work, speed, and distance problems, where relationships are often expressed as percentage differences.

Session Overview

• The session covers a variety of topics, from basic percentage understanding to practical applications.
• Introduction: Meaning and purpose of percentages will be covered.
• Percent to Fraction Conversion: Techniques for conversion will be discussed.
• Application: practical uses across different types of math problem
• Examples: will be used to help understand the material taught during this session.

Understanding Percentages

• "Percent" is derived from "per" (out of) and "cent" (hundred).
• Percentages represent a quantity or ratio in relation to 100.
• "Out of 100" signifies the amount of something per 100 units.

Example

• 300 out of 600 marks illustrates the concept.
• Conversion of the fraction results in 50 out of 100.
• 50% and "50 out of 100" represent equivalent values.

The Importance of Percentages

• Real-world decision-making and comparisons are made easier with percentages.
• For example, annual employee evaluations during increment cycles determine efficiency, and also the time saved with that efficiency.

Real-World applications

• Percentages are used as a differentiator when evaluating job applicants
• Applicants above a certain percentage are selected for interviews
• Car performance is based on percentages from safety standards to "fun" standards

Percent to Fraction Conversion

• Simplification when using Percent to fraction conversion helps with many math problems
• p/q form is a way of representing fraction is simple terms

How to Convert From Percents into fractions

• Percent means "one by a hundred" (1/100).
• Remove the percent sign by dividing by 100 or multiplying by 1/100.
• 100% = 1.
• Many relationships can be derived from 1 = 100%.
• A method to finding relationships is to half a value and its fraction

Halving Percentages and Fractions with 100%

• Equivalence is maintained when halving value and fraction.
• If 1 = 100%, then 1/2 = 50%, 1/4 = 25%, 1/8 = 12.5%, 1/16 = 6.25%.
• Quick recall of equivalent percentages and values aids.
• This information should be recalled on demand and not memorized
• Understanding relationships is better for difficult problems

Dividing by Other Numbers

• Equivalent fractions with percents can be created by dividing both.
• If 1 = 100%, then 1/3 is equal to 100/3.
• 100/3 = 33%

Breaking down 100/3 and simplifying calculation

• 3 x 33 is 99, so 0.33 is left over where the answer is 33.3

More halving of 1/3

• Half of 1/3 is 1/6, half of the percent is 16/3
• This simplifies to 16.6%

Dividing Again for additional simplification example

• Dividing 1/6 means the result is 1/12, halving the percent occurs if the relationship exists
• 16 over 2 is 8 with thirds
• The value is 8.33%

Percents as fifths

• You can divide and derive more relationships
• if 1 = 100%, 1/5 is 20percent
• From 1/5 you can half divide the numbers again creating a 1/10 relationship

Other relationships with 1/7

• 1/7 is about 14%.
• Other relationships can be made with addition, multiplication, or division of this approximation.

Nines and Elevens

• There’s an inverse relationship between nines and elevens.

11-to-9 relationship

• 1/9 creates a repeating decimal of 11.1
• 1/11 creates a repeating decimal of .09.09.
• Remember one goes to 11 when using nines but goes to 9 when using elevens
• Shown as .09.09 in repeating decimals
• Shown as .11.11 in repeating decimals

One to Twenty relationships

• Memorizing 1/1 to 1/20 is useful.
• Consistent practice is more effective than memorization.

Values from multiple

• Other values can be derived from multiplication with existing facts

Example using 20 Percent and Multiplying

• Multiplying known relationship is faster
• 20% is 1/5, so to find 40 multiply.
• Therefore, 20 to 40 is x2, then 1/5 x2 = 2/5 which means 2/5 = 40percent
• 20 to 60 is x3, then 1/5 x3 = 3/5 which means 3/5 = 60percent
• 20 to 80 is x4, then 1/5 x4 = 4/5 which means 4/5 = 80percent
• 20 to 100 is x5, then 1/5 x5 = 5/5 which means 5/5 = 100percent
• This method can be repeated again

Derives from 33.3% for simple memorization

• 33.3% = 1/3

Other Multiples for 37.5 Percent and above

• 37. 5% can be seen as three times 12.5 where you derive the decimal by multiplying the percent
• 87. 5% can be seen as seven times 12.5 by deriving the same process of multiplying previous knowledge

Breaking it down for simple derivation

• Break down large percentages
• 137.5% = 100% + 37.5% = 1 + 5/8 = 11/8.

Other examples include

• 216. 6 percent can be seen as 200 percent + 16%.
• 406 Percent can be seen as 400 percent + 6.25.

Comparing Data using Percentages

• Understanding "what is a percent of" can be confusing.
• Asking “A” is what percent of “B?” it’s important to remember what has “of” so you do not mess up this simple question

Explanation of A being a percent of B

• “B” contains “of” meaning “B” is the denominator, and A is compared to it

Explanation of A comparing to B in more or less terms.

• You have the variables of A and B with the scenario of A being more then B, how do you find the percent where A is more, if you don’t have data to work with
• Steps:
• 1: You must derive the difference from each A and B.
• 2: What has the “than” to know who the reference point it.
• The “A - B”/“B” is formula to derive A being more then B.

Analogy Break down

• Since, you are the point of reference as you are the one being compared too it’s important understand from who
• Since, you are the point of reference between the comparison from I, you go down.

Same explanation but less

• It is important to know where to make the reference and base from.
• This information could help solve future, possibly more difficult math equations.

Application

• Application includes calculating everything from profit to to efficiency
• Skill and calculating and using relationships will be maximized
• The percent to fraction value needs to become second nature

Deriving data with profit gain

• To maximize effectiveness, use 20% is equal to 1/5
• 5 stands for cost price and is relevant depending on the scenario
• The top number is the profit which one will need to know more details on

In a scenario where you sell items for profit,

• Cost is relative as a 100 percent and can be useful
• variables represent a data
• 5 is the cost
• 1 is the profit
• 6 is the selling price of the value
• If 6 = 60, then what is 1, can be easily calculated with this data.

Calculating loss based on percentage

• If you know how to calculate profit then you can do loss
• Since 25 Percent is loss. Where cost is four can be derive to find the loss.

Speed Data from Percentage

• Apply these methods to any type of calculation such as speed
• If 3- percent is given then 3/10 is the identified as how efficient
• Initial value is the ten, and increase is the three
• Meaning the relations ship is initial vs the speed that has since changed

Working with an Increased Number

• When working with percentage. Work from an initial.
• When using 37.5 you can use know both the before and know after
• Remember fractions may help you remember the workload

Practice Problems

• The problems highlight the relationship that was stated
• Receive percent questions which you can use to convert to fraction to simplify the math

Practice Problem 1

• The question states “42.84 percent of x, equals to - 210. What is x?”
• It is important to recall the basic value when seeing something like this
• Remember that when doing percentages it is important to see how the fractions and percentage interrelate in the question.
• Multiply that fraction from the existing data to get answer of 90

Practice Problem 2

• The question give “What is 33.3+ percent of 540 + (3/10) + 1/8
• This answer equals 665

Practice Problem 3

• Solving needs simple derivation, especially with 5.08 What is 5.08+ percent of 1230 equals to x

Relationship Data From Two People

• The scenario is John Salary = to “x Percent” of Martin
• Question to ask when someone then another, such as John Salary is 25. To what does their data change

Using Ratios vs Values to measure

• To ease the work load by making common understanding of data relation
• Use what seems right for you, even what answer maybe
• Cutting out data is value when using ratios

Using Ratio Examples

• The question to help ease the workload uses ratios
• Martin is base at 4 if John is 25 or 5 percent with easy values the question eases itself

Percentage Relationship with Variables

• With the data; x is 40 Percent of y = Y, 70 percent of z=Z
• Getting fraction from multiplying from Y by x and “see” what relationship that exist may ease the problem.

Finding Values in Variables

• If z = a, and X is z percentile, of .28 is 50% of Z
• . 72 or 72%.

Multiples with 5 Percent

• With 5 Percent is of, is EQUAL TO 15 Percent of B, if c = 10. “What R”
• Derive percent of one from the relationship between both.
• Easy calculation that can be done using previous methods or simply cutting each variables out.

15+ Percent

• Derivation and Question “What Is the number 25 percentile” is 1985 and 304 1/2 Percent and 5 or and 1.
• See is there number that aligns close to previous information
• What comes to ease or simplify, may ease more than you could known

Success Rate Problems

• Successive Increase decrease is some thing and can continue to happen there are variables that increase or decrease
• Interest or the depreciation for items over some time

Increment Example

• One Example is store items.
• With those variable with data change. Product rate x is
• If a store items is at “x then is “Y”, then derive new point after single point is
• Do same technique with other increase and what would it “mean” after total for total

Another option

• If a is increased and B is an increase then what is point
• Increased/Decrease plus Increased/Decrease Then is multiply what has changed after what point.

Formula Breakdown

• It it the plus is. Add or sub the equation will what the variable entails.
• Equation only useful with what it is, variable + or -

Another Option

• Short codes are complex, fractions can help where need
• Change is “how value from x variable” that can be compared
• Same with another with those values

Ratio Breakdown

• What it can be cut to ease the long way for process of it.

Combining Skills of Fractions + Multiplications

• The what each the data is changing by x
• Recalling value what it “means”. By simply knowing to keep you organized

Practice problem

• The Population of what’s that it rate, is x…what is, or was…y.
• Starting what that changing the product with what its rate? Or with other technique (the previous what.)

Decreasing population

• What “opposite what” is “there doing what’s that rate” or doing what process. “It” can relate back to “the other”

Increasing vs decreasing

• And “or” process how do the state by changing, to
• Keeping values to show as they progress the what over

Word Problem

• With A “plus or more” or can what “less or than” what what equals

• Start what to what or how set the it for. What what to, what can can work

Application to Real world use

• Prices value can “inc more value from before”, or less, but by what

Volume with percentages.

• You had a with to what what
• The is and side with the rate as for with the volume?

More data from volumes with percentages

• If the volume is “what what”. You may have what variable “may needs” to what may or what is “said rate or volume”.

Sales Person/commission

• He sale sales what
• What rate is his sales or value what.

Two Paths that could be taken from salesman

• Can be how to or what process with what

One more type that may be stated

• What with the what. It for may with What be easy .

If you want to know the smart person may take there is one technique

• What the values are that are there with
• He it as with the It’s by. The rate

More on tax and Income

Tax on Groceries

• What will and what The what will with for the

Multi Problem in text

• A what be has
• The set with they is that point.
• Be what for for all it

Multi variable

• What and they it for’t it

• Here is’t is

• A: what data I.

• B: “B” what or do

• C: is “it” what or”what.

• D: Is 4:It” is it For you from the

Variable Increase Problems

When you read the words increase, decrease it make lead how it has to:

• Be may set the one may. And what of with set,

Next

Can you hear “them” If not why, what is there for?

After The process, one should ask that are those number real and therefore what would change to make it so.

Multi Step Questions

These are some questions that test math skills not one thing but multi that happen in order what to take

Most important with anything related or not make sure the problem is under Stable