Percents Practice Makes Perfect!

Where did we encounter percents so far? Exponential models: – Percent change (growth/decay) Relative quantities – Percentages/fractions – Relative change (percent change) – Percent of- part/whole

First identify the type of problem: Is it about – Percent of Formula  part/whole=percent of – Percent change, percent more/less, percent growth/decay, percent greater/smaller Formula  y=P(1+r) or r=(y2-y1)/y1 (depending on what the question is asking) – Successive percents We will do this today. The answer depends on the type of question.

In 1999 there were 108,000,000 coffee consumers in the United States. If the US population was 272,690,813 in 1999, what percent of the US population were coffee consumers? Formula: Percent of  part/whole=percent 108,000,000/272,690,813= In % of the US population were coffee consumers.

In 2009, there were 32,444,000 males and 26,655,000 females in the US that were never married. By what percent were the number of never-married males larger than never-married females? Percent change(difference) formula: (y2-y1)/y1 Y1: never married females Y2: never married males (32,444,000-26,655,000)/26,655,000= The number of never-married males was 21.72% larger than never-married females.

In 2009, there were 32,444,000 males and 26,655,000 females in the US that were never married. By what percent were the number of never-married females less than never-married males? Percent change(difference) formula: (y2-y1)/y1 Y1: never married males Y2: never married females (26,655,000-32,444,000)/32,444,000= The number of never-married females was 17.84% less than never-married males.

Percent change Percent larger/smaller By what percent greater/ less? What is the percent increase/decrease? Identify the direction of change (the change is from what value to what value) Y1: the reference/previous value Y2: the new/ different value Formula: (Y2-Y1)/Y1

Successive Percents The process: Goal: Our goal is to calculate the overall percentage change between the Final Value (in this example, Final Price with Coupon) and the Beginning Value (in this example, Retail Price) when you are given two intermediate percentage decreases, increases or a mixture. In this example, we are given two intermediate decreases. Example: Jeans are on sale for 40% off the retail price. The retail price is $ If you have a coupon, you can receive an additional 20% off the sale price. What is the overall percentage savings? Prepared by Ozlem Elgun8

Example: Jeans are on sale for 40% off the retail price. The retail price is $ If you have a coupon, you can receive an additional 20% off the sale price. What is the overall percentage savings? Mathematically: Determine the sale price: ∙(0.40) = 24 Determine the final price with coupon: ∙(0.20) = Determine the overall percentage change:( )/40 = – which is an overall savings of 52%. Prepared by Ozlem Elgun9

The Formula: (1 ± P 1 ) ∙(1 ± P 2 ) – 1 = % (where the % is written as a decimal) P 1 = First percentage increase/decrease P 2 = Second percentage increase/decrease Prepared by Ozlem Elgun10 Example: Jeans are on sale for 40% off the retail price. The retail price is $ If you have a coupon, you can receive an additional 20% off the sale price. What is the overall percentage savings?

V. Deriving the Successive Percent Formula: Goal: Our goal is to calculate the overall percentage change between the Final Value (F) and the Beginning Value (B) when you are given the intermediate percentage decreases. Deriving the Formula: The overall percentage change doesn’t depend on the Beginning Value (B). Can we show this by determining a process (formula) that includes just the two percents? Variables: B = Beginning Value I = Intermediate Value F = Final Value P 1 = First percent decrease P 2 = Second percent decrease Determining the process: With VariablesWith Numbers (this is considered the “long way”) First EquationB - B∙P 1 = I(40 – 40 ∙ 0.40) = 24 Second EquationI - I∙P 2 = F(24 – 24 ∙ 0.20) = Final Equation(F – B) / B(19.20 – 40) / 40 = or 52% savings Rewrite the first equation:B - B∙P 1 = I as: B∙(1 – P 1 ) = I Rewrite the second equation as: I - I∙P 2 = F as: I∙(1 - P 2 ) = F Using the final equation, the goal is to get the entire equation in terms of B, P 1 and P 2. Substitute F with I∙(1 - P 2 ) to get: Substitute I with B∙(1 – P 1 ) to get: The B’s cancel out to arrive at:(1 – P 1 ) ∙(1 - P 2 ) – 1 For percents that increase, substitute “+” for “-“. The final formula that works for all successive percent problems is: Overall Percentage Change (Successive Percent) = (1 ± P 1 ) ∙(1 ± P 2 ) – 1 Prepared by Ozlem Elgun11

VI. Solving Problems: 1. Situation to discuss in class: A politician promises, “If elected, I will cut your taxes by 20% for each of the first three years of my term, for a total of 60%.” Evaluate the promise. 2. Solve: Spot prices for crude oil are rather volatile. From 1998 to 1999, spot prices for crude oil decreased by 28%. From 1999 to 2000, they increased by 106%. What was the percentage change over the two year period from 1998 to 2000? Prepared by Ozlem Elgun12