The unitary method for finding mark-ups and other increases is similar to the unitary method for discounts. The difference is that for discounts, the key is subtracting the discount rate from 100, while for mark-ups, the mark-up rate is added to 100. Other than that, the proportion equation is identical. And, as the next lesson will show, these same proportions are used to find the original amount prior to discount or mark-ups.
In this lesson, the examples used are be a mixture of discount and mark-up problems. This interleaving of the problem types builds flexibility so that students readily recognize which proportion to use for which type of problem.
Warm-Ups. Some of the problems in this set are mark-up problems which students have learned how to solve. Others include percentage type problems.
1. The population of Eatontown was 157,000 in 2019 and by 2020 had grown 15.2%. What was the population in 2020? Answer: 157,000 +(157,000 × 0.152) = 180,864
2. 70 is 8% of what number? Answer: 70 = 0.08x; x = 875; OR 8/100 = 70/x; 8x=7000; x =875
3. The usual price of a pair of shoes was $45. It was sold at a discount. The reduced price was $36. What was the percent discount rate? Answer: (45-36)/45 = 0.2 or 20% discount.
4. A pair of shoes normally sells for $215. It is discounted 30%. Find the reduced price. (Use either unitary method). Answer: (100-30)/100 = x/215; 100x = 78 ∙ 215; x= $167.70 OR x = 215∙0.7 = $167.70
5. Solve. 3(x-2) = 7; Answer: 3x-6 = 7; 3x = 13; x = 13/3
Problems 1 and 3 are both percent of change problems. It is not unusual for students to have difficulty with these problems as if they have never seen them before. For that reason it is necessary to keep students exposed to these type of problems through interleaving in Warm-Ups and homework problems. In particular students may have difficulty seeing that Problem 3 is a percent of change problem. Problem 4 asks for the problem to be solved using either unitary method. Seventh graders are generally more comfortable with the proportional method, though some will prefer the direct translation method.
Unitary Method for Increases, Method 1. For discounts, students learned that a 40% discount of an item means that the buyer pays 100 -40 or 60% of the original price. Similarly for increases such as mark-ups, tips, tax and so forth, the percent of increase is added to 100.
After this reminder, I offer another reminder of how they initially learned to calculate values that are increased. I start with a problem: A tire dealer buys tires at $40 and marks up the cost by 50%. What is the cost of the marked-up tire?
“Tell me how to do this,” I’ll say and hope that my inclusion of mark-up, tip and tax problems in homework and Warm-Ups has paid off. I might remind them that Problem 1 of the Warm-Ups is similar: 50% of 40 is 20, which is the mark-up and is added to the original price to get the selling price of $60.
“In this problem, the cost of the tire to the dealer is $40. This is 100% of the price he pays for the tire. In working with discounts, we subtracted the discount rate from 100%. What do you think we do with mark-ups?”
Students see where this is going. “The mark-up is 50% so what is our total?”
Hearing 150%, I say “How do we represent this as a decimal.” There may be some hesitation; if so, I refresh their memories: “How do I represent 95% as a decimal?” They recall the decimal point is shifted two places to the left. “So 150% is what?”
Hearing “1.5”, I write on the board:
100%+50% = 150% of original cost = marked up cost
“The marked-up price is 1.5 times the original cost. So 40 x 1.5 = $60. “Let’s do another one.”
Worked Example:
You leave a 15% tip on $35 restaurant bill. What is the final price?
“The $35 bill represents what percent?” I’m waiting to hear “100%”
“And what percent are we adding to 100%?” Answer: 15%
“What is the total percent?” 115%
“And what is that as a decimal?” 1.15
“What do we multiply by 1.15?” The restaurant bill of $35: 35 × 1.15 = $40.25
Examples:
These are done in their notebooks, and I walk around to check progress, answer questions, provide guidance.
1. Helen makes $1,200 per week and receives a 6% raise. What is her new salary?
Answer: 1.06 × 1200 = $1,272
2. A 2 lb can of coffee cost the store $8.00 per can. The manager marks it up 50%. What is the selling price?
Answer: 1.5 × 8 = $12.00
3. A radio sells for $150. The store buys it for $60. What is the percent mark-up?
This is a “percent change” problem. Students need to shift gears to see that this is not the same as the previous two examples. The key prompt is: “What is the problem asking for? What type of problem is that?”
Answer: (150-60)/60 = 90/60 = 3/2 = 1.5 or 150% mark-up
Unitary Method for Increases, Method 2. This method uses a proportional equation like that used for discounts:
Worked Example:
You want to leave a 15% tip on a meal that cost $50. Find the total cost of the meal
“What is the rate of increase?” 15%
“What is the original cost?” $50
“What are we trying to find?” The total cost of the meal.
“We now have everything we need to plug in to the proportion:”
These should be familiar enough to students from their work with discounts in the previous lesson. The examples to follow the worked example should include both discount and mark-up problems so students are attentive to when they should add or subtract the percent increase or decrease.
Examples:
I usually have students do these at the board by first having them work in their notebooks, and then selecting those students who have it correct. I work with students to ensure they understand how to solve the problems by providing hints and guidance. Generally students use the proportion approach.
1.The population of Snailtown was 110,000 in 2010 and increased 25% by 2011. What is the population in 2011? Answer: 1.25 × 110,000 = 137,500 OR 125/100 = x/110000; 100x = 125 × 110,000; x = 125 × 1100; x = 137,500
2. A radio sells normally for $250. It is discounted 30%. What is the sales price? Answer: 0.7 × 250 = $175; OR 70/100 = x/250; 100x = 17,500; x = 175
One short-cut that students may find helpful when using the proportion method is to convert the fraction to a decimal; e.g., 70/100 = 0.7. Then the equation becomes 0.7 = x/250. Seventh graders continue to have difficulty with equations in that form, so to alleviate the problem I have them write the simplification of 70/100 as 0.7/1. Then the proportion becomes 0.7/1 = x/250. Solving it leads to 0.7∙250 = x, which is the same form as Method 1 which some students notice.
3. A set of furniture costs a dealer $3,000. He marks it up by 20%. What is the selling price of the furniture set? Answer: 1.2 × 3,000 = $3,600; OR 120/100 = x/3000; 1.2/1 = x/3000; x = 1.2 ∙3,000 = $3,600
Homework. The homework should be a mix of discount and mark-up type problems. Some variation at the end should include a mixture of final price with tax.
For example, a radio normally selling for $200 is discounted 25%. The sales tax is 8%. What is the sales price including tax? The discounted price of $150 is then multiplied by 1.08 to obtain $162. (Although this can be done in one step by finding 200(0.75)(1.08), that method is best shown in an algebra course when students have more facility with understanding and representing numerical procedures algebraically.)