lecture 5 ima 101: basic math 6/17/2010 1 ima101: basic mathematics

IMA101: Basic Mathematics

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LECTURE 5

IMA 101: Basic Math

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Lecture Outline

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Turn in HW1 and JournalWhere we’ve been and where we are goingPercentsBreakRatios


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JUST SO YOU KNOW HOW THIS WEEK CONNECTS TO LAST WEEK

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A little signposting


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Review: What we’ve covered so far

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Different sets of numbers: Whole numbers Integers Rational numbers Real numbers

Different operations on numbers: Addition and subtraction Division Multiplication & exponents Square roots

Key methods: Prime factorization Using reciprocals Rounding The order of operations


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This week

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Discuss interpretations of these numbers Percents (decimals) Ratios (fractions) Proportions (fractions) Unit conversion (multiplication/division) Estimation (rounding) Descriptive statistics (NEW!)


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Percents


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Percent: Definition

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A percent means parts per one hundredWe write it using the % symbolExamples:

27 % is the same as 27 parts per 100 8% is the same as 8 parts per 100

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100 square grid

When all the squares in the grid are shaded we say that 100% of the of the grid is shaded.

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100 square grid

If only 83 of the 100 squares are shaded, then we have 83% of the grid shaded.

What percent of the grid is NOT shaded?

100% - 83% = 17%

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Percents as Fractions

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We can convert a percent into a fraction by dropping the % symbol and writing the percent over 100, then simplifying

Examples

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9

100

45%45

3

2

100

1

3

200100

3

266

10032

66%3

266

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100 square grid

2/3 of the grid is shaded, so we have

66 2/3% of the grid shaded.

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Percents as Decimals

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Recall: a percent is just parts out of 100To change a percent to a decimal, drop the %

symbol, and divide the number by 100, leaving the answer in decimal notation

Examples65% = 65/100 = 0.6514.25% = 14.25/100 = 0.1425100% = 100/100 = 1.0


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Caution: Small and Large Percents

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Write 0.6% as a decimal

Write 354% as a decimal

Important: Make sure that the 1’s place in a percent is in the 100th place in the decimal!


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Decimals as Percents

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Recall: decimals can be written as a sum of fractions of different powers of 10, as well as one fraction over the largest power of ten

Decimals are fractions over 100. How can we write this as a fraction over 100, and then as a % ?

10000

8743

10000

3

1000

4

100

7

10

88743.0


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Fractions as Percents

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Write the following as a percent

4/5

3/4

Note: these are easy because they multiply into 100 neatly


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Fractions as Percents

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Recall: sometime we have a denominator that does not multiply neatly into 100, such as 3, 7, 11, etc. These are examples that give us repeating decimals

To write a fraction as a percent first convert to decimal notation, then convert to a percent

Example4/3 as a percent(approximate and exact answers)


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Some numbers to know and love

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Fraction

Decimal

Percent Fraction

Decimal

Percent

0.5 50% ~33.3%

0.25 25% ~66.6%

0.75 75% 0.1 10%

0.2 20% 0.01 1%

2

1

4

1

4

3

5

1

3

1

3

2

10

1

100

1

3.0

6.0


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Solving Percent problems

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Recall: a percent is the number of parts out of 100But really this is just a conversion of a different

fraction into one with a denominator of 100

Using the conversion techniques we have just discussed, we can write any amount out of a base as a percent

Find the percent of female students in this class

percentbase

amount


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Percent Problems

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Student Practice

Amir International got a score 0f 81 out of a possible 150 on HW1*. Did Amir get a passing grade (at least 60%) on his first homework?

If Annie Globetrotter got a score of 126. Did she get a passing grade on her first homework?

*yay for those of you who got the bonus!


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Percent Problems

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We can also solve for the amount necessary to get a certain percent

ExampleWhat minimum score does Chetan have to

get in order to get a passing grade on his first homework?

amountbasepercent


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Percent Problems

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You are starting an intramural ultimate Frisbee team. The league rules require that at least 25% of the players be female. If you need at least 30 players, how many female players do you need?


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Percent Problems

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Given a certain amount and a known associated percent, we can also find the base

ExampleIf 18% of students at Tonedeaf University are

enrolled as music majors and there are 270 students in the music program, how big is the student body?

basepercent

amount


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Percent Problems

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Student PracticeAn apartment building has 42 rooms

currently rented out. If the occupancy rate is at 60%, how many rooms are currently empty?

If there are 360 motorcycles outside the international college and ~90% of students drive motorcycles, approximately how many students are in the international college? (assume all the motorcycles belong to students)


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Percentage Increase and Decrease

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Sometimes we are interested in finding how much something changes as a percent of the original amount

ExampleIf the population of Imaginaria is 450,000 in

2008 and 486,000 in 2009, what was the rate of population growth?

changeoriginal

originalnew%100


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Percentage Increase and Decrease

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Student Practice

You originally go to the store and see a pair of fly kicks for 5200, upon your return the next day, you see that the price has been changed to 5850. What was the percentage change in the price of the shoes?

Your dog used to weight 5.2 kg but it now weighs 3.64. What percent of the original weight did your dog lose?


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SECTION 6 .1

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Ratios and Proportions


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Ratio: Introduction

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Definition:A ratio is a relationship between two

numbers that have the same units

Examples:4:1 5:2Equivalent to fractions


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Rates: Introduction

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A ratio of two different types of quantities

ExampleIf you drive 450 km in 9 hours then the rate

of your driving is:


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Rates

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Student Practice

If you are paid a salary of 443,700 for at total of 255 work days with 8 hours of work each day, what is your hourly rate of pay?

If the fastest-growing flower plant on record grew 12 feet in 14 days, what was its daily growth rate?

lecture 5 ima 101: basic math 6/17/2010 1 ima101: basic mathematics

Documents

basic mathematics slide

percents slide

week ima101

square grid

percent example

little signposting slide

large percents

sum of fractions