Directly related to the percent of change are discounts and mark-ups. I approach this topic in three different lessons. In this, the first part, I cover the basics in which the sales price after a discount in a two-step process: the amount of discount is calculated first, and then subtracted from the original price. Mark-ups, tips and taxes are also calculated in two parts, but are added to the original amount.
In the second part, I teach the unitary method which can be done by a direct algebraic equation, or by proportion. The third part extends the unitary method using proportion to allow original amounts to be found, given the after-discount, or after-mark-up prices.
These lessons can be taught sequentially, or spaced out. I have had the most success by spacing out the lessons, so that the unitary approach and finding original amounts are taught after the next chapter on functions. Teaching it later then serves as both a review as well as presenting a new way of solving discount/mark-up problems.
The algebraic approach tends to be difficult at this grade. For example, finding the original cost of an item that has been discounted 15% and sells for $102, can be represented algebraically letting x be the original cost. Then the equation becomes 0.85x = 102 and x = $120. This equation can be derived by translating the original word problem into “85% of what number is 102?” But seventh graders have difficulty making that initial translation. Again, my experience is that seventh graders who I then taught in algebra the next year were able to do this and found what was difficult in seventh grade to now be easy.
Warm-Ups. Some of the questions are direct segues into the day’s lesson.
1. Write as an equation or proportion. What is 25% of 2000? Answer: 500
2. A watch that normally sells for $2,000 is on sale for $1,500. What is the percent decrease? Answer: Change in amount/original amount = 500/2000 = 0.25 or 25%.
3. Write as an equation or proportion and solve. 60 is 75% of what number? Answer: 60 = 0.75x; x = 80; OR 75/100 = 60/x; 75x = 6000; x = 80
4. Green paint is made with 2 parts blue and 3 parts yellow. If you have 24 gallons of yellow, how much blue do you need, using this ratio? Answer: 2/3 = x/24; x = 16
5. Write as an equation or proportion and solve. 18 is what percent of 25? Answer: 18 = 25x; x = 0.72 or 72%; OR x/100 = 18/25; 25x = 1800, x = 25, so 25%.
Problems 1 and 2 relate to the portion of the day’s lesson on discounts. Problem 1 asks for an equation or proportion. It is also a good idea to follow through and remind students that prior to learning about the equation or proportion method, they simply multiplied the decimal form of percent by the number; i.e., 0.25 × 2000. For the day’s lesson it will be simpler if they use that approach rather than using a proportion or algebraic equation each time.
Calculating Discounts. The discussion of the Warm-Ups gives us entry into the day’s lesson as students at this point in the school year have caught on (and I hope as readers of this book have as well). Specifically, Problem 1 asks what is 25% of 2000. Problem 2 shows that a $500 decrease in price represents a 25% decrease.
“This is an example of a discount.” I then ask students if they’ve seen advertisements that say “25% off”. Most everyone has. The following definition is written on the board:
A discount is the amount by which the regular price of an item is reduced. The sale price=original price – discounted amount.
Worked Example: A bicycle that normally sells for $1400 is discounted by 20%. What is the discounted selling price?
Step 1: Find amount of decrease by finding the percent of the full price. In this example it is $1400 × 0.2 = $280.
Step 2: Subtract amount of decrease from original amount: 1400-280 = $1120.
(Again, for purposes of this lesson students should be reminded that finding a percent of a value means multiplying the decimal amount of the percent by the value—it is not necessary to write an equation or proportion though students may do so if they wish.)
Examples:
A bicycle normally sells for $200. It is discounted 25%. What is the sales price? Answer: 200 × 0.25 = 50; 200-50 = $150
Jenny bought a fan that was 30% off the original price of $200. What was the amount of the discount? Answer: We want the amount by which the price is reduced not the sales price. 200 × 0.3 = $60 reduction.
A sofa priced at $900 is discounted 75%. What is the sales price? Answer: 900 × 0.75 = 675; 900-675 = $225
Now a problem that recalls the previous lesson.
What is the discount rate for a radio that originally sold for $150 and now sells for $120? Answer: (150-120)150 = 30/150 = 0.2 or 20% discount.
Calculating Mark-Ups, Tips, Taxes. Most students know what tips and taxes are. They may not know what a mark-up is, so I explain how that works. The main point is that the final value of a marked-up item, or the cost of something with tips or taxes, are all calculated in the same way.
“For discounts, we took the value of the reduction and subtracted it from the original price. How do you think we calculate the price of a $100 meal with a 15% tip added?”
Usually students will make the connection, that if discounts involved subtraction, then mark-ups, tips and taxes involve addition. I’ll work out the example:
Step 1: Find the amount of increase by finding the percent of the original price. (100 × 0.15 = $15)
Step 2: Add the amount of increase to original price. (100 + 15 = $115)
Examples:
Amount of 15% tip on a $35 meal. Answer is $5.25
What would be the total price for a $1200 bicycle with 8% sales tax? Answer: $1,296.00
A 2 lb can of coffee costs $12.00 at the Star Market. If the manager buys the coffee for $8.00 a can, what percent markup does he allow on each can? Answer: (12-8)/8 = 0.5 or 50% markup
Mix of discount and tax: Suppose a calculator costs $75 originally and is discounted by 20%. What is the final price if the sales tax is 6%?
First, I have students find the discounted price: 75 × 0.2 = 15; 75-15 = $60
Now, the tax is calculated and added on: 60 × 0.08 = $4.80; 60 + 4.8= $64.80
Homework. My experience is that students find these problems straightforward. Homework should be a mix of discount and mark-up, tax and tip problems. Also there should be problems that require finding the amount by which an item is reduced, as well as problems asking for the amount of a mark-up. Problems requiring finding what the discount, tip, tax or mark-up rates are should also be included.