Document Sample

Uses and Abuses of Percentages Wednesday January 5, 2011 Review • per cent means “divided by 100”, so for example, 50% = 50/100 = .5 and 125% = 125/100 = 1.25. • To convert a percent to a fraction, replace the percent sign with division by 100 (then reduce the fraction). For example, 25 1 25% = = . 100 4 • To convert a percent to a decimal, drop the % symbol and move the decimal two places to the left (i.e. divide by 100). For example, 25% = .25 and 1% = .01 • To convert a decimal to a percentage, move the decimal two places to the right (i.e. multiply by 100) and attach the % symbol. For example, .12 = 12% and .0013 = 1.3% • To convert a fraction to a percentage, ﬁrst convert the fraction to a decimal (using a calculator if necessary) then convert the decimal to a percentage as above. For example 1 = .2 = 20% 5 Three Ways of Using Percentages 1. As Fractions Example 1. Express the following as a percent: Of 305 million Americans, 40.4 million are under the age of 15. 1 Solution Write as a fraction then convert to percentage: 40.4 = .13245902 = 13.245902% 305 So approximately 13.25% of Americans are under the age of 15. 2. To Describe Change • Absolute change refers to the increase or decrease from a reference value to a new value. • Relative change is a fraction that describes the relative change in comparison to the reference value. Example 2. During a 6-month period, Nokia’s stock doubled from $10 a share to $20 a share. What were the absolute and relative changes in the stock price? Solution The reference value is $10, the starting price of the stock. The new value of the stock price is $20. Therefore, absolute change = new value − reference value = $20 − $10 = $10 and absolute change $10 relative change = = =1 reference value $10 Remarks: (a) Although the stock price doubled, the relative increase is only %100 (as opposed to %200). (b) The relative change is unitless. (c) The relative change can be converted to a percent by multiplying by 100 Example 3. The population of the United States increased form 249 million in 1990 to 308 million in 2010. Find the absolute and percentage change in the the U.S. pop- ulation from 1990 to 2010. Solution reference value = 249 million new value = 308 million absolute change (in millions) = 308 − 249 = 59 59 relative change = = .237 249 So the percent change is 23.7%. 3. For Comparisons • absolute diﬀerence = compered value - reference value • relative diﬀerence = absolute diﬀerence/reference value 2 Remark As above, we can convert the relative diﬀerence to a percent diﬀerence by multiplying by 100 and suing the % symbol. Example 4. Suppose we want to compare the price of a $50,000 Mercedes and a $40,000 Lexus. The absolute and percent diﬀerences in the prices of the cars will be diﬀerent depending on whether we use the price of the Mercedes or the price of the Lexus as our reference value. If we use the price of the Lexus as the reference value, we get absolute diﬀerence = 50, 000 = 40, 000 = 10, 000 and as a percentage of the Lexus price, the percent diﬀerence in the car prices is 10, 000 = .25 = 25%. 40, 000 On the other hand if we use the price of the Mercedes as our reference value, we get absolute diﬀerence = 40, 000 − 50, 000 = −10, 000. and as a percentage of the Mercedes price, the percent diﬀerence in the car prices is 10, 000 = .2 = 20% 50, 000 Friday January 7, 2011 “of” vs. “More Than” or “Less Than” Consider the rising prices of gas. In 1997, a gallon of gas cost $1.00. Today a gallon of gas costs $3.00 (if your lucky). The price of gas has tripled since 1997. In terms of percentages, there are two equivalent ways of saying this; one using “of” and one using “more than”. • Using more than: The price of gas today is 200% more than the price of gas in 1997. • Using of : the price of gas today is 300% of the price of gas in 1997. How to convert between of and more than and between of and less than: • If the compared value is P % more than the reference value, it is (100 + P )% of the reference value. • If the compared value is P % less than the reference value, it is (100 − P )% of the reference value. Remark In any particular problem, the compared and reference values should be obvious (or at least attainable) from the given information. In a real life situation, there is sometimes a logical reason to pick a particular value to be the reference value. 3 Example 5. Using the gas price example, and taking the price of gas in 1997 as the reference value, the compared value (i.e. the price of gas today) is 200% more than the reference value. To convert this to a comparison using of, we take P = 200% in the ﬁrst bul- let above. Therefore, the the price today is (100+200)% = 300% of the price of gas in 1997. Example 6. A store is having a “20% oﬀ” sale how does an items sale price compare to its original price? Solution 20% oﬀ means an items sale price is 20% less than its original price. Therefore, the sale price is (100 − 20)% = 80% of its original price. If the original price of the item is $100, the sale price of the item is $80. Example 7.(continued) Suppose an item that originally cost $100 is marked down to 20% oﬀ. Suppose the item is marked down again to 25% oﬀ of the sale price. What is the ﬁnal price of the the item? Solution After the ﬁrst markdown, the item costs $80. After the second markdown, the item costs (100 − 25)% = 75% of $80, so the ﬁnal price of the item is $80 · 75% = $80 · .75 = $60 Notice that the ﬁnal price of the item is not the same price as marking the item oﬀ 45% (in which case the price of the item would be $55). Percentages of Percentages Suppose the interest rate on your student loan gets increased from 10% to 15%. It may be tempting to say that the interest rate increased by 5%. This statement is not true. Although the interest rate increased by 5 percentage points, the relative change in interest rate is new value − old value 15% − 10% 5% = = = 0.5 reference value 10% 10% Converting this to a percent, we see that the interest rate has actually increased by 50%. • When you see a change or diﬀerence expressed in percentage points, you can assume it is an absolute change or diﬀerence. If it is expressed with the % symbol, or the word percent, it is a relative change. Example 8. (Tricky Wording) Suppose 40% of the registered voters in Gainesville are Republicans. Read the following questions carefully and give appropriate answers. 1. The percentage of voters registered as Republicans in 25% higher in Miami than in Gainesville. What percentage of the registered voters in Miami are Republicans? Soluteion Here a change is expressed in percent (using the symbol %), so it is a relative change (relative to the percent of Republicans in Gainesville). Since 25% of 40% is 10% (i.e. .25 · 40% = 10%), and the percent of Republicans is 25% higher in Miami than in Gainesville, we must add 10% to 40% to get the percentage of Republicans in Miami. 4 2. The percentage of voters registered as Republicans is 25 percentage points higher in Miami than in Gainesville. What percentage of the registered voters in Miami are Republicans? Solution Since the change is given in percentage points, we simply add the percents. The percentage of Republicans in Miami is 40% + 25% = 65%. Solving Percentage Problems • If the compared value is P % more than the reference value, then compared value = (100 + P )% · reference value and compared value reference value = (100 + P )% • If the compared value is P % less than the reference value, then compared value = (100 − P )% · reference value and compared value reference value = (100 − P )% Example 9. Suppose you but a shirt with pre-tax price $ 19. If the local sales tax in 7.75%, what is the ﬁnal price of the shirt? Solution Example 10. Suppose you pay $21.00 for a shirt after tax. If the local sales tax is 7.75%, what is the pre-tax price of the shirt? Solution Example 11. Consider the following statement: The rate of smoking in college freshmen is up 44% to 10.4%. What was the previous rate of smoking for college freshmen? Solution(you try it) Monday January 10, 2011 Abuses of Percentages 1. Shifting Reference Values Suppose your employer plans to cut your pay rate by 10% for one month then to raise it again by 10%. Will your original pay rate be restored? No. Observe: Suppose your original pay rate is $200 per week. A pay cut of 10% reduces your pay rate to $200 − $20 = $180 per week 5 A subsequent pay raise of 10% increases your pay rate to $180 + $18 = $198 per week Therefore, your employer wins. Example 12. Suppose your stock broker tells you that your investment lost 60% over his ﬁrst year on the job, but that on his next year on the job, your investment rose 75%. Your stock broker tells you “your investment is now up 15%”. Evaluate your brokers intelligence. Solution Suppose your initial investment was $1,000. Since 60% of $1,000 is $600, after your brokers ﬁrst year on the job, the value of your investment is $1, 000 − $600 = $400. Moreover, since 75% of $400 is $300, a 75% increase in value of your investment (after the 60% decrease in its value) makes your investment worth $400 + $300 = $700. Your actually losing money. Fire your broker. Even if the market changes, he’ll still be incompetent. 2. Less Than Nothing Example 13. A store manager advertises that it will take “150% oﬀ” of the price of all merchandise. What should happen when you attempt to purchase a $500 item? Solution If the discount wer 100% oﬀ, the item would be free. So if the price is 150% oﬀ, the store should pay you $250 (half the price of the item) to take the item home. The manager of the store is clearly FSU alum. 3. Don’t Average Percentages Suppose you take two exams. On the ﬁrst exam you get 70% of the questions correct and on the second exam you get 90% of the questions correct. Is it true that between the two exams you got 80% of the questions correct? No! The problem is that the exams may have a diﬀerent number of questions. Consider an extreme case for the purpose of illustration. Suppose your ﬁrst exam had 1,000 questions and your second exam had 10 questions. Since you scored 70% on your ﬁrst exam you got 700 of the 1,000 questions correct. On your second exam you got 9 of 10 questions correct. In total, you got 709 of 1,010 questions correct. Converting this to a percentage, you got 709 ≈ 0.702 = 70.2% 1010 of the questions correct. 6

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 14 |

posted: | 8/18/2011 |

language: | English |

pages: | 6 |

OTHER DOCS BY ps94506

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.