IMA 101: Basic Math Lecture 5 IMA101: Basic Mathematics 6/17/2010

Lecture Outline Turn in HW1 and Journal Where we’ve been and where we are going Percents Break Ratios IMA101: Basic Mathematics 6/17/2010

Just so you know How this week connects to Last week A little signposting Just so you know How this week connects to Last week IMA101: Basic Mathematics 6/17/2010

Review: What we’ve covered so far 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 IMA101: Basic Mathematics 6/17/2010

This week Discuss interpretations of these numbers Percents (decimals) Ratios (fractions) Proportions (fractions) Unit conversion (multiplication/division) Estimation (rounding) Descriptive statistics (NEW!) IMA101: Basic Mathematics 6/17/2010

Percents IMA101: Basic Mathematics 6/17/2010

Percent: Definition A percent means parts per one hundred We write it using the % symbol Examples: 27 % is the same as 27 parts per 100 8% is the same as 8 parts per 100 IMA101: Basic Mathematics 6/17/2010

100 square grid When all the squares in the grid are shaded we say that 100% of the of the grid is shaded. IMA101: Basic Mathematics 6/17/2010

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% IMA101: Basic Mathematics 6/17/2010

Percents as Fractions We can convert a percent into a fraction by dropping the % symbol and writing the percent over 100, then simplifying Examples IMA101: Basic Mathematics 6/17/2010

100 square grid 2/3 of the grid is shaded, so we have 66 2/3% of the grid shaded. IMA101: Basic Mathematics 6/17/2010

Percents as Decimals Recall: a percent is just parts out of 100 To change a percent to a decimal, drop the % symbol, and divide the number by 100, leaving the answer in decimal notation Examples 65% = 65/100 = 0.65 14.25% = 14.25/100 = 0.1425 100% = 100/100 = 1.0 IMA101: Basic Mathematics 6/17/2010

Caution: Small and Large Percents 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! IMA101: Basic Mathematics 6/17/2010

Decimals as Percents 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 % ? IMA101: Basic Mathematics 6/17/2010

Fractions as Percents Write the following as a percent 4/5 3/4 Note: these are easy because they multiply into 100 neatly IMA101: Basic Mathematics 6/17/2010

Fractions as Percents 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 Example 4/3 as a percent (approximate and exact answers) IMA101: Basic Mathematics 6/17/2010

Some numbers to know and love 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% IMA101: Basic Mathematics 6/17/2010

Solving Percent problems Recall: a percent is the number of parts out of 100 But 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 IMA101: Basic Mathematics 6/17/2010

Percent Problems 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! IMA101: Basic Mathematics 6/17/2010

Percent Problems We can also solve for the amount necessary to get a certain percent Example What minimum score does Chetan have to get in order to get a passing grade on his first homework? IMA101: Basic Mathematics 6/17/2010

Percent Problems 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? IMA101: Basic Mathematics 6/17/2010

Percent Problems Given a certain amount and a known associated percent, we can also find the base Example If 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? IMA101: Basic Mathematics 6/17/2010

Percent Problems Student Practice An 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) IMA101: Basic Mathematics 6/17/2010

Percentage Increase and Decrease Sometimes we are interested in finding how much something changes as a percent of the original amount Example If the population of Imaginaria is 450,000 in 2008 and 486,000 in 2009, what was the rate of population growth? IMA101: Basic Mathematics 6/17/2010

Percentage Increase and Decrease 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? IMA101: Basic Mathematics 6/17/2010

Ratios and Proportions Section 6.1 IMA101: Basic Mathematics 6/17/2010

Ratio: Introduction Definition: A ratio is a relationship between two numbers that have the same units Examples: 4:1 5:2 Equivalent to fractions IMA101: Basic Mathematics 6/17/2010

Rates: Introduction A ratio of two different types of quantities Example If you drive 450 km in 9 hours then the rate of your driving is: IMA101: Basic Mathematics 6/17/2010

Rates 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? IMA101: Basic Mathematics 6/17/2010