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= 0.3961

• In 1999 39.61% 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)/y1Y1: never married femalesY2: never married males(32,444,000-26,655,000)/26,655,000= 0.2172

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)/y1Y1: never married malesY2: never married females(26,655,000-32,444,000)/32,444,000= 0.1784

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 valueY2: the new/ different valueFormula: (Y2-Y1)/Y1

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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 $40.00. If you have a coupon, you can receive an additional 20% off the sale price. What is the overall percentage savings?

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

40.00

45.00

Retail Price Sale Price Final Price withCoupon

Overall PercentageChange

Price Amount Saved

$40.00 - $16.00 = $24.00

$24.00 x.20 = $4.80

$24.00 - $4.80 = $19.20

$40.00 x.52 = $20.80

$40.00 - $20.80 = $19.20

$40.00 x .40 = $16.00

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Example: Jeans are on sale for 40% off the retail price. The retail price is $40.00. 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:

40 - 40 (0.40) = 24∙

Determine the final price with coupon:24 - 24 (0.20) = 19.20∙

Determine the overall percentage change: (19.20-40)/40 = -0.52– which is an overall savings of 52%.

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The Formula: (1 ± P1) (1 ± P∙ 2) – 1 = % (where the % is written as a decimal) P1 = First percentage increase/decrease

P2 = Second percentage increase/decrease

Example: Jeans are on sale for 40% off the retail price. The retail price is $40.00. If you have a coupon, you can receive an additional 20% off the sale price. What is the overall percentage savings?

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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 ValueI = Intermediate ValueF = Final ValueP1 = First percent decrease

P2 = Second percent decrease

Determining the process:With Variables With Numbers (this is considered the “long way”)First Equation B - B P∙ 1 = I (40 – 40 0.40) = 24∙

Second Equation I - I P∙ 2 = F (24 – 24 0.20) = 19.20∙

Final Equation (F – B) / B (19.20 – 40) / 40 = -0.52 or 52% savingsRewrite 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, P1 and P2.

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 – P1) (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 ± P1) (1 ± P∙ 2) – 1

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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?