U3 - Percents Notes 1

Introduction We recommend that you actually play this presentation to get the full value You can also print these slides if you prefer to work on paper 2

Percent Review – Part 1 Notes with Solutions 3

Explanation Recall that “per” is the operation of division (or out of) and “cent” means 100 in French. So, when we write x%, this is written as a fraction out of 100 as x/100. Percents can also be written using decimals. For example, if we write 6.25%, write this as a fraction and also as a decimal. 4

Answer Fraction:= 6.25/100 Decimal: or just

Explanation (con’t) Remember that if we get an answer for a percent problem in decimal form, we should state the final answer as a percent. ex the income tax rate in Quebec is 0.16 for people who make between $ and $ The tax rate in Quebec, written as a percent is 16%. Feel free to use either decimals or fractions when working with percents, but remember that in an online test, you will enter the answer as a decimal. 6

Percent Review – Part 2 Applications of Percents 7

Explanation In Ontario, the provincial sales tax (PST) is 8%_ and the goods and services Tax (GST) is 5%. So normally a tax rate of 13%_ is added to the purchase price. Discounts are a certain percent of the original price which is removed, before tax. Commission is a certain percent which is paid to a sales person, based on the amount sold. Mark up is the amount above cost paid for an item. 8

Example 1 Caitlin goes into the GAP to look for a present for her mom. Surprisingly, Daniel is working at the GAP. a)Caitlin purchases socks which cost the GAP $12, but have been marked up by 45%. Find the mark-up, before taxes. Mark up = 0.45x$12 = $5.40 9

Example 1 (con’t) b) Find the final price after all taxes. Total Cost=$17.40(1.13)=$19.66 Note: The “1” in front of the “.13” gets the original amount back so 113% including tax. - it is actually 1.13 =

Example 1 (con’t) Daniel tells Caitlin that she has done very well this week. Daniel makes $9.80 per hour and a further 5% in commission. Daniel has sold $1300 worth of clothes this week and has worked 15 hours. How much did Daniel make in total? Total Salary = hourly + commission = $9.80x x$1300=$212 11

Example 1 (con’t) d) The following week, Daniel’s sales averaged of $63/h of clothing. If his total earnings, including commission and hourly wages totaled $271.95, find the number of hours he must have worked. 12

Example 1 (con’t) d) Solution: The $ comes from both commission and hourly sales. If we let h= the number of hours, then the total commission is 5% of the total sales, which are 63h (h hours at $63/h), so commission =.05(63h) His hourly wages are $9.80 per hour (and we let h be the number of hours) So, $271.95= hourly wages + commission = 9.8h (63h) =9.8h+3.15h =12.95h h=21 So, Daniel worked for 21 hours in all. 13

Example 1 (con’t) e) What percentage of his total pay each week came from clothing sales? Comment on this, from the company’s point of view. % from clothing = 14

Example 2 Jamie buys a pair of amazing running shoes on sale (discounted) at 30% off for $90, before taxes. a) How much did Jamie pay, including tax? Cost of pants=$90(1.13)=$

Example 2 (con’t) b) How much were the shoes before they went on sale? In general, think of the formula for discount as (1-r) where r is the discount written as a decimal. Let x be the price before they went on sale. So, she actually paid (1- 0.3)x= 0.7x or 70% of the original price. 0.7x=90 x= x=$ before they went on sale. (Note: when going backwards, you can’t simply add 30% to what she paid, as the 30% was taken off a different (higher) amount. 16

Example 3 The employment figures for Canada for the past three years were 8,000,000 employed in 2006, 8,400,000 in The population of adults eligible for work was 8,500,000 in 2006 and 8,550,000 in Find the increase or decrease in the unemployment rate between 2006 and 2007 in Canada. 17

Example 3 (con’t) Solution: We know there were 100,000 unemployed in 2006 % unemployed in 2006 = % unemployed in 2007 = The decrease in the unemployment rate = = 4.07% 18

Example 4 45% of a certain number is 290. Find this number. Method 1: Let x be the number 0.45x=290 x=290/.45 x=

Example 4 Solution (cont.) Here is an alternative solution using the definition of percent. 20

Example 5 21 Year Unemployed Students Total Number of Students

Example 5 continued a)Find the % of students who were unemployed in 2005 b)If the % unemployed remained the same in 2006, find the number of unemployed students in 2006 c) If the % of unemployed students increased by 5% in 2007, find the total number of students in

Example 5 Solution 23

Percent Review Extra Percent Problems Problem Set A-Answers only 24

Problem 1 Calvin is 150 cm tall, which is 75% of Darryl’s height. How many centimeters tall is Darryl? 25

Problem 1 Answer 1) 200 cm 26

Problem 2 A toy-store manager received a large order of Mr. Slinky’s just in time for the holidays. The manager places 20% of them on the shelves, leaving the other 120 Mr. Slinky’s in storage. How many Mr. Slinky’s were there in all? 27

Answer #2 2)

Problem 3 a)Find 35% of b)35% of what number is 7742 ? c)What percent is 35 of 7742 Each of the parts below has a different answer. Read each carefully before beginning. 29

Answer #3 a)$ b)$22120 c)0.45% 30

Problem 4 Jamila bought a pair of pants for $90 plus taxes of 15%. Find the overall price of the pants. 31

Answer #4 $

Problem 4 I bought a shirt for $90 including 15% tax. Find the original price of the shirt before tax. $

Problem 5 There are 26,000 potential voters in Ward 9 of the City of Toronto. In the election in the year 2000, 14,500 of these actually voted. This year, 16,060 voted. Find the percentage increase in voters. 6% increase 34

6% increase 35

Problem 6 It costs $120 for excess baggage today. Suppose there is a discount of 20% offered and then a fuel surcharge of 15% is applied. Find the final price of the excess baggage. $

Problem 7 Petrona got paid $6 per hour plus a commission of 8% on all her sales at a music store. She sold $1200 worth of CD’s in a month where she worked 96 hours. Find her earnings. 37

Answer #6 $672 38

Problem 7 I got into a heated discussion with my son last night. I asked him which is the better deal. Deal 1: A hockey stick costing $150 has its price increased by 40% and then it was decreased by 40%. You then buy the hockey stick. Deal 2: The same hockey stick costing $150 has its price decreased by 20%, decreased by 20%, increased by 20% and finally increased by 20% again. You then buy the hockey stick. He claimed it was a silly question as you end up paying $150 (before tax) in each case. I claimed that you didn’t end up paying $150 (before tax) in either case and that one final price was better than the other. Who was right? 39

Answer #7 Deal 1: $126 Deal 2: $

Problem #8 Year Monthly Salary $4800$5100$

Problem 8 cont. a) Find the percentage increase in the monthly salary from 2005 to b) If the monthly salary increased by 6% between 2007 and 2008, find the missing monthly salary in

Answer #8 43 Solution: