LSP 120: Quantitative Reasoning and Technological Literacy Topic 6: Percentages Prepared by Ozlem Elgun1

Prepared by Ozlem Elgun 1

LSP 120: Quantitative Reasoning and Technological Literacy

Topic 6: Percentages

Different types of Relative Quantities

Fraction or Percent:• Fractions or percents are used when comparing part to total of the same type of

variable. (example: percent of adults with AIDS/HIV) Percents can also be used to show the relative change. Percent change is calculated by dividing the absolute change by the original amount. [Reminder: Percent change (new value –old value)/old value]

Rate:• Rates are used compare different types of variables (example: tickets per person, miles

per hour, or crimes per 1000 people) Ratio:• Ratios are used to compare the same type of variable from two sources. For example:

California’s population is 33,872,000 and Oregon’s population is 3,421,000. Clearly CA’s population is larger but how many times larger? 33,872,000/3,421,000 = 9.90 Calculating the ratio of the populations tells us that CA’s population is almost 10 times as large as OR’s population.

• The type of data you have will determine what type of relative quantity is appropriate.


REVIEW!!! REMEMBER FROM TOPIC 4….

• We use absolute change to describe the actual increase or decrease from a reference (or old/earlier) value to a new (or later) value:

• Absolute Change = new value – reference value

• We use relative change to compare the absolute change to the reference value:

• Relative Change = =

Absolute and Relative Change



• For communication purposes, we convert relative change, which is a fraction (converted to a decimal number) to a percentage (percentage change). The following are three ways to convert a fraction (decimal number) to a percentage:

• Move the decimal place to the right two places• Multiply by 100%• Use the button in Excel

• For this course, we will generally show percentages formatted to two (2) decimal places. (Right click on the cell, format cell)



Percentage of…• Understanding “Percentage of” in 3 ways:

I. The Formula: %whole

part (where the % is written as a decimal)

II. Visually: Whole means the entire pie. Part means one of the shaded regions or pieces of the pie.

III. There are three ways to think about this relationship:

%whole

part whole

part

% partwhole *%


IV. Deriving the formulas: Can you figure out (algebraically) why all three of these are just different versions of the same relationship?

a. Starting with the formula:

Derive:

b. Starting with the formula:

Derive:


V. Solving Problems• There are two approaches to solving the following problems. The first

approach is to identify the two given numbers. Then decide which version of the part/whole relationship will help you answer the question. – If you are given part and whole, then use the first version.

– If you are given part and % then use the second version.

– And, finally, if you are given whole and % then use the third version.

• The second approach is to remember the first formula, fill in the information you are given and then solve for the missing variable.

• For all problems, remember to use the decimal version of the %.Prepared by Ozlem Elgun 7

VI. Number Drills

• 2 is what percentage of 10? 2/10= 1/5= 0.2 = 20%

• 20% of what number is 2? 20%= 0.22/0.2= 10

• What is 20% of 10? 20% = 0.20.2*10= 2


• 17 is 32% of what number?17/0.32 = 53.125

• 67.2 is what percentage of 150? 67.2/150= 0.448= 44.8%

• What is 233% of 71?233%=2.332.33*71= 165.43

• What is .7% of 50?0.7% = 0.0070.007*50= 0.35

• 35 is 9% of what number?35/0.09= 388.8889

• 10,003 is what percentage of 1,762,325?10003/1762325= 0.005676 = 0.5676%

• one million three hundred thousand is what percentage of one billion? one million three hundred thousand =1.3 millionone billion= 1000 million1.3/1000= 0.0013 = 0.13%

• one thousand is what percentage of two thousand three hundred and six?

• 1000/2306= 0.433651= 43.37%Prepared by Ozlem Elgun 9

VII. Applications:

• In Chicago in the year 2000, there were approximately 1.053 million African Americans, 907 thousand whites (non-Hispanic), and 754 thousand Hispanics, and 181 thousand others (other races or two or more races). What percentage of Chicagoans in 2000 were of Hispanic origin?

Total: 1.053 + 0.907 +0.754 + 0.181= 2.895 million0.754/2.895 = 0.260449 = 26%


• DePaul’s undergraduate student body is approximately 21,000 students. 54% of the student body is female. Approximate how many females attend DePaul?

54% = 0.540.54* 21,000= 11,340


• In 1993, 248.7 million people in the United States were born in the United States, and the rest, 19.8 million were foreign born. What percentage of the total population of the US was foreign born?

Total population = 248.7 + 19.8 = 268.519.8/268.5 = 0.073743 = 7.37%


• The sales tax is 8.75% in most counties of Illinois. If you purchase a new car for $15,000, what is the sales tax you will pay?

8.75%= 0.08750.0875*15000= $1312.5


• You are in another state (not Illinois). You are buying a computer at Best Buy. The price before taxes is $949. When the cashier wrings up your purchase you owe $1005.94. What is the sales tax in this state? (You might be in Connecticut or Pennsylvania)

1005.94-949=56.9456.94/949=0.06= 6%


• At one point, the Tribune article refers to a subtotal of murders “with only 10% of the year yet to go.” 10% of the year is how many months?

10% = 0.1 0.1*12= 1.2 months



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?

• II. Visually:

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


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?

• III. 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%.


IV. 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?


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


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?


Also see:• New ways of thinking about Percentage Change.doc

• Size Comparisons Using Percentages.doc

http://qrc.depaul.edu/instructor/InstructorTemplateFall2009/Worksheets/6b%20-%20Percentage%20Change.doc

http://qrc.depaul.edu/instructor/InstructorTemplateFall2009/Worksheets/6c%20-%20Size%20Comparisons%20Using%20Percentages.doc

http://qrc.depaul.edu/instructor/InstructorTemplateFall2009/Worksheets/6c%20-%20Size%20Comparisons%20Using%20Percentages.doc

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LSP 120: Quantitative Reasoning and Technological Literacy Topic 6: Percentages Prepared by Ozlem Elgun1