Basic Principles 0.2 Item Pricing: Markup vs. Margin0.1 Percentages: Percent Increase/DecreaseThe word percent literally means per hundred. A number followed by a percent sign is tobe interpreted as a ratio or fraction per hundred. For example, 4% stands for 4 out of100.Example 1: Compute 6% of 200.Solution: 6% of 200 = (6/100)×200 = 6×200 / 100 = 12 ■Note in the above example that 6% means 6 per hundred, which written a fraction is6/100. This fraction can be written as the decimal .06, so 6% can be ought of as thedecimal .06, or 6% =.06. Example 2: What percent is 18 out of 250? Solution: r= (18/250×100) %.So, r = 6.2 % ■Example 3: What percent is 56 out of 140? Solution: r=(56/140×100) %.So, r = 40 % ■Calculating Percentages of a Number( r % of N ) = (r/100)×N = r×N / 100Calculating Percentage rTo find the percentage r that A is of Br = (A/B × 100) %Calculating Percentage IncreaseTo find percentage r that an initial amount A increased to B:r = ( net increase / initial amount × 100 ) %r = ( (B–A) / A× 100 )%r = ( (B/A–1) × 100 )%r = ( ( (ending amount/initial amount) –1) × 100 )%Basic Principles 0.2 Item Pricing: Markup vs. MarginExample 4: Standardized test scores at Happy Valley High School have increased from840 to 920. Find the percentage increase. Solution: r= ( (920/840–1)×100 )%So, r= 9.52380952% ■Example 5: John Henry’s salary increased from $45,810 to $46,974. Find the percentageincrease in his salary. Solution: r= ( (46974/45810–1)×100 )%r=2.54092993%So, r= 2.54092993% ■Example 6: In one year, gas prices rose 40%. If the current price is $3.39 per gallon, findthe price per gallon one year ago. Solution: In this problem, we are given the percentage increase and B, we need to find A.Since r = 40%, and B= 3.39 we have40 % = ( (3.39/A–1) × 100 )%or40 = ( (3.39/A–1) × 100 )Basic Principles 0.2 Item Pricing: Markup vs. MarginWe first divide by 100,40 /100 = 3.39/A–1Then add 1, 40/100 + 1 = 3.39/ANext, multiply by AA(40/100 + 1) = 3.39We simplify the term in the parenthesesA(1.4) = 3.39Finally, we divide by 1.4.A= 3.39/1.4 =2.421 142857Since A is a dollar figure, we round: A=$2.42 ■The formula for percentage increase also works for decreases, the formula simply gives a negative number in the event of a decrease. (A negative increase is a decrease.)Example 7: James’ net federal income tax was lowered from $3,742.32 in 2008 to$3,512.17 in 2009. Find the percentage change from 2008 to 2009. Solution: In this problem, A=$3,742.32 and B = $3,512.17. So,r = ( (B/A–1) × 100 )%r = ( (3512.17/3742.32–1) × 100 )%r= –6.14992839%So James’ taxes decreased by 6.14992839% (in other words, they increased by – 6.14992839%).■The next example illustrates a common misconception that people have about percentages.Example 8: A retailer took 5% off as suit that originally was priced at $100. A weeklater, they marked the price back up by 5%. What is the new price? Solution: The answer this problem is not $100. The first 5% discount amounted to a $5reduction of the price.Basic Principles 0.2 Item Pricing: Markup vs. Margin5% of $100 is $5So the price after the discount was $95.00.When the vendor increased the price by 5%, they increased it by5% of $95, which is $4.75 So the final price of the suit is $99.75 ■Example 9: Clarice bought stock in XYZPDQ Corp. Her investment lost 10% in the pastyear. By what percentage must XYZPDQ Corp.’s stock increase so that Clarice breakseven?Solution: Let us label the initial investment as A and the amount after the 10% loss as B.We actually do not know A or B. But we do know that Clarice lost 10% in the first yearfrom A, so we know she lost 0.1A. So the amount that she had after the loss must havebeen B= A – .1 A= 0.9 ANow we need to find the percentage increase needed to get her from 0.9A back to A. We can actually do this using the formula: r = ( (ending amount/ initial amount–1) × 100 )%So r = ( (A/0.9A–1) × 100 )%which simplifies tor = ( ( (1/ 0.9) –1) × 100 )%r = ( ( 1.111111 –1) × 100 )%r = ( ( 0.111111 ) × 100 )%r = 11.1111% ■Problems1. Compute 11% of 60.2. Compute 92% of 160.3. Compute 34% of 560.Basic Principles 0.2 Item Pricing: Markup vs. Margin4. The Knittany Lion Corp makes knit stocking caps that sell for $50.00. If they make aprofit of 5% per hat, how much profit do the make per hat?5. Find the percentage increase from 50 to 75.6. The population of Jackson, Wyoming went from 8,647 in 2000 to 9,577 in 2010. Whatpercentage increase is this?7. If a $150,000 home increases in value by 5% in one year, how much will it be worth inone year?8. After an 8% raise, Keesha is making $35,000 per year. What was her salary last year?9. Shady Day Trading Co. lost 45% of its assets after day trading on the stock market.What percentage increase is needed to recover their losses?10. Unregulated Motors had its net worth increase by 100%, the next year its net worthdecreased by 100%. What is the net worth of Unregulated Motors?Basic Principles 0.2 Item Pricing: Markup vs. Margin0.2 Item Pricing: Markup vs. MarginVendor MarkupWhen determining the final price of an item, a vendor will often markup their own cost inproducing or obtaining the item by a fixed percentage in order to arrive at the final price.If the vendor did not mark up, they would make no profit by selling the item. We define afew terms to help us in our computations. Example 1: Kenco makes women’s sweaters. It costs Kenco $46.00 to produce onesweater. If have a markup rate of 25% , what will they charge for a sweater? What is theirprofit per sweater?Revenue = Cost + ProfitC = cost to vendor to obtain an itemR= final price of itemP = net profit to the vendor thenP = R – CandR = C + PVendor MarkupC = cost to vendor to obtain an itemR= final price of itemP = net profit to the vendor Markup percentage rate r = percentage increase from C to R = ( (R/C-1)×100 )%Basic Principles 0.2 Item Pricing: Markup vs. MarginSolution: In this problem, we are given C = 46.00 and r. We can compute the net priceincrease by taking 25% of 46 (which is $11.50). This is the amount that they will profit persweater. The final price R = C + P = 46.00 + 11.50 = $57.50