Basic Principles 0.2 Item Pricing: Markup vs. Margin
0.1 Percentages: Percent Increase/Decrease
The word percent literally means per hundred. A number followed by a percent sign is to be interpreted as a ratio or fraction per hundred. For example, 4% stands for 4 out of 100.
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 is 6/100. This fraction can be written as the decimal .06, so 6% can be ought of as the decimal .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 % ■
Example 4: Standardized test scores at Happy Valley High School have increased from 840 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 percentage increase 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, find the 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 have
40 % = ( (3.39/A–1) × 100 )%
or
40 = ( (3.39/A–1) × 100 )
We first divide by 100,
40 /100 = 3.39/A–1
Then add 1,
40/100 + 1 = 3.39/A
Next, multiply by A
A(40/100 + 1) = 3.39
We simplify the term in the parentheses
A(1.4) = 3.39
Finally, we divide by 1.4.
A= 3.39/1.4 =2.421142857
Since 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 week later, 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 $5 reduction of the price.
5% of $100 is $5
So the price after the discount was $95.00.
When the vendor increased the price by 5%, they increased it by
5% 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 past year. By what percentage must XYZPDQ Corp.’s stock increase so that Clarice breaks even?
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 year from A, so we know she lost 0.1A. So the amount that she had after the loss must have been
B= A – .1 A= 0.9 A
Now 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 to
r = ( ( (1/ 0.9) –1) × 100 )%
r = ( ( 1.111111 –1) × 100 )%
r = ( ( 0.111111 ) × 100 )%
r = 11.1111% ■
Problems
1. Compute 11% of 60.
2. Compute 92% of 160.
3. Compute 34% of 560.
4. The Knittany Lion Corp makes knit stocking caps that sell for $50.00. If they make a profit 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. What percentage increase is this?
7. If a $150,000 home increases in value by 5% in one year, how much will it be worth in one 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 worth decreased by 100%. What is the net worth of Unregulated Motors?
0.2 Item Pricing: Markup vs. Margin
Vendor Markup
When determining the final price of an item, a vendor will often markup their own cost in producing 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 a few terms to help us in our computations.
Example 1:Kenco makes women’s sweaters. It costs Kenco $46.00 to produce one sweater. If have a markup rate of 25% , what will they charge for a sweater? What is their profit per sweater?
Solution: In this problem, we are given C = 46.00 and r. We can compute the net price increase by taking 25% of 46 (which is $11.50). This is the amount that they will profit per sweater. The final price R = C + P = 46.00 + 11.50 = $57.50 ■
Example 2:A retailer marked up a grocery item by 25%. The final price is $1.99. What was the cost to the retailer?
Solution: In this problem, we are given R = 1.99 and r. We can compute the cost by plugging setting 25 % =( (R/C-1)×100 )%
25 = (1.99/C-1)×100
Dividing both sides by 100:
0.25 = 1.99/C - 1
Adding 1 to both sides:
1.25 = 1.99/C
Solving (multiply both sides by C and divide by 1.25):
C= 1.88/1.25 = $1.592 or $1.59 ■
In the previous problem, we note that $1.99-$1.59 = $0.40 is profit. The profit margin is defined to be the percentage that the profit is of the final price. In the previous example, the profit margin is $.40/$1.99 ×100% = 20.1005025%. Often, a vendor wishes to maintain a given profit margin instead of marking up their costs.
Example 3: A hamburger franchise has total sales of $1.6 million dollars with a profit margin of 5.2%. How much is their profit?
Solution: In this problem, we are given R = 1,600,000 and profit margin r=5.2% and are looking for P. Plugging into the first profit margin formula:
Profit margin r = ( (P/R)×100 )%
5.2 % = (P/1,600,000)×100 %
5.2 = (P/1,600,000)×100
0.052 = P/1,600,000
0.052 × 1,600,000 = P
P = $83,200 ■
Example 4: Sunco Farms charges $1.99 per dozen of eggs. It costs them $0.52 to produce each dozen of eggs. What is Sunco Farms’ profit margin rate and markup rate?
Solution: In this problem, we are given C = $0.52 and R = $1.99. The markup is
Markup rate r= ( (R/C-1)×100 )%
= ( (1.99/0.52-1)×100 )%
=( 2.82692308×100)%
= 282.692308%
Profit margin r = ( (1-C/R)×100 )%
= ( (1-0.52/1.99)×100 )%
= 0.738693467×100 %
= 73.8693467 % ■
Example 5: A retailer wants to maintain a 35% profit margin. If their cost to produce an item is $4.72, what should they charge?
Solution: In this problem, we are given C = 4.72 and profit margin r=35% and are looking for R. Plugging into the profit margin formula:
Profit margin r = ( (1-C/R)×100 )%
35% = ( (1-4.72/R) ×100 )%
35 = (1-4.72/R) ×100
0.35 = 1-4.72/R
4.72/R = 0.65
R= 4.72/0.65
R= $7.26■
Problems
1. A book publisher has a markup rate of 225%. It costs them $10.50 to produce a mathematics text. What is the final cost of the text?
2. A sports retailer buys cases of 50 baseball bats for $250. They then sell each bat for $19.99. What is their profit margin and markup for each bat.
3. It costs Linda’s Lemonade Stand $0.29 to produce a 16 ounce bottle of lemonade. If Linda’s operates with a profit margin of 45%, what does Linda’s charge for a 16 ounce bottle of lemonade?
4. A gas station charges $3.49 for a gallon of gas. Their cost is $3.27 per gallon. What is the markup rate and profit margin rate?
5. It costs Primo Motorcycles $12,015.00 to buy each motorcycle from the manufacturer. If they wish to have a profit margin of 10%, what will they need to charge per motorcycle?
0.3 Fixed and Variable Costs
Businesses typically incur two kinds of costs: fixed and variable costs. A fixed cost to a business is a cost that does not change with a change in production or sales. For example, a manufacturer of women’s shoes has to pay certain costs, regardless of the number of shoes that are produced. Such costs include the cost to maintain a production facility, the cost to maintain a web site, the cost of maintaining personnel. These costs are essentially the same, regardless of the number of shoes that are produced. A variable cost does change with the number of items produced. For instance, in order to produce women’s shoes, a producer must first purchase materials to make each of the shoes. These costs vary with the number of shoes produced. (Note that the cost of a machine used to make the shoes would be a fixed cost). A variable cost is typically quoted per unit (or units) of production.
Example 1: A wholesaler