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LSP 120: Quantitative Reasoning and Technological Literacy Topic 6: Percentages Prepared by Ozlem Elgun 1 REVIEW!!! REMEMBER FROM TOPIC 4…. 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. Prepared by Ozlem Elgun 2 REVIEW!!! REMEMBER FROM TOPIC 4…. Absolute and Relative Change • 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 = = Prepared by Ozlem Elgun 3 REVIEW!!! REMEMBER FROM TOPIC 4…. • 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) Prepared by Ozlem Elgun 4 Percentage of… • Understanding “Percentage of” in 3 ways: I. The Formula: part % whole (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: part % whole part Prepared by Ozlem Elgun whole % * whole part % 5 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: Prepared by Ozlem Elgun 6 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. by Ozlem Elgun • For all problems, remember toPrepared use the decimal version of the %. 7 VI. Number Drills • 2 is what percentage of 10? 2/10= 1/5= 0.2 = 20% • 20% of what number is 2? 20%= 0.2 2/0.2= 10 • What is 20% of 10? 20% = 0.2 0.2*10= 2 Prepared by Ozlem Elgun 8 • 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%=2.33 2.33*71= 165.43 • What is .7% of 50? 0.7% = 0.007 0.007*50= 0.35 • 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 million one billion= 1000 million 1.3/1000= 0.0013 = 0.13% • one thousand is what percentage of two thousand three hundred and six? • 35 is 9% of what number? 0.433651= 43.37% Prepared by • Ozlem 1000/2306= Elgun 9 35/0.09= 388.8889 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 million 0.754/2.895 = 0.260449 = 26% Prepared by Ozlem Elgun 10 • DePaul’s undergraduate student body is approximately 21,000 students. 54% of the student body is female. Approximate how many females attend DePaul? 54% = 0.54 0.54* 21,000= 11,340 Prepared by Ozlem Elgun 11 • 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 = 268.5 19.8/268.5 = 0.073743 = 7.37% Prepared by Ozlem Elgun 12 • 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%= 0.0875 0.0875*15000= $1312.5 Prepared by Ozlem Elgun 13 • 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=56.94 56.94/949=0.06= 6% Prepared by Ozlem Elgun 14 • 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 Prepared by Ozlem Elgun 15 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: 45.00 40.00 $40.00 x .40 = $16.00 35.00 $40.00 x.52 = $20.80 30.00 25.00 $24.00 x.20 = $4.80 20.00 $40.00 - $16.00 = $24.00 15.00 $24.00 - $4.80 = $19.20 10.00 $40.00 - $20.80 = $19.20 5.00 0.00 Retail Price Sale Price Price Final Price with Coupon Overall Percentage Change Amount Saved Prepared by Ozlem Elgun 16 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%. Prepared by Ozlem Elgun 17 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? IV. The Formula: (1 ± P1) ∙(1 ± P2) – 1 = % (where the % is written as a decimal) P1 = First percentage increase/decrease P2 = Second percentage increase/decrease Prepared by Ozlem Elgun 18 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 P1 = First percent decrease P2 = Second percent decrease Determining the process: With Variables First Equation Second Equation Final Equation Rewrite the first equation: With Numbers (this is considered the “long way”) B - B∙P1 = I (40 – 40 ∙ 0.40) = 24 I - I∙P2 = F (24 – 24 ∙ 0.20) = 19.20 (F – B) / B (19.20 – 40) / 40 = -0.52 or 52% savings B - B∙P1 = I as: B∙(1 – P1) = I Rewrite the second equation as: I - I∙P2 = F as: I∙(1 - P2) = F Using the final equation, the goal is to get the entire equation in terms of B, P 1 and P2. Substitute F with I∙(1 - P2) to get: Substitute I with B∙(1 – P1) to get: The B’s cancel out to arrive at: (1 – P1) ∙(1 - P2) – 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 ± P2) – 1 Prepared by Ozlem Elgun 19 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 Elgun 20 Also see: • New ways of thinking about Percentage Change.doc • Size Comparisons Using Percentages.doc Prepared by Ozlem Elgun 21