UNIT SIX: PERCENT RELATIONSHIPS

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Big Idea

Proportional relationships can be used to solve real-world problems.

Enduring Understandings

  • Percent literally means per 100 and can be represented as a ratio with 100 as the denominator. 
  • Understand and communicate information using the relationships of decimals, fractions, integers, and rational/irrational numbers.
  • Make sense of percent problems by modeling the proportional relationship using an equation, a table, a graph, mental math, and factors of 100.
  • Realize when to use algebraic expressions and equations to solve multi-step percent problems.
  • Understand the use of estimation to determine reasonableness, when solving percent word problems.
  • Recognize that when they find a certain percent of a given quantity, the answer must be greater than the given quantity if they found more than 100% of it and less if they found less than 100% of it.
  • Percent decreases and increases are measures of percent change, which is a relative measure based on absolute change.  
  • Understand the use of absolute error to calculate percent error.
  • Simple interest may be calculated by applying the appropriate formula.

Essential Questions

Common Core Standards: Content & Skills

Analyze proportional relationships and use them to solve real-world and mathematical problems.

7.RP.2

Recognize and represent proportional relationships between quantities.

7.RP.3

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Khan Academy

Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

Students should be able to explain or show their work using a representation (numbers, words, pictures, physical objects, or equations) and verify that their answer is reasonable. Models help students to identify the parts of the problem and how the values are related. For percent increase and decrease, students identify the starting value, determine the difference, and compare the difference in the two values to the starting value.

Examples:

  • Gas prices are projected to increase 124% by April 2015. A gallon of gas currently costs $4.17. What is the projected cost of a gallon of gas for April 2015?

A student might say: “The original cost of a gallon of gas is $4.17. An increase of 100% means that the cost will double. I will also need to add another 24% to figure out the final projected cost of a gallon of gas. Since 25% of $4.17 is about $1.04, the projected cost of a gallon of gas should be around $9.40.”

$4.17 + 4.17 + (0.24 ∙ 4.17) = 2.24 x 4.17

  • A sweater is marked down 33%. Its original price was $37.50. What is the price of the sweater before sales tax?

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The discount is 33% times 37.50. The sale price of the sweater is the original price minus the discount or 67% of the original price of the sweater, or Sale Price = 0.67 x Original Price.

  • A shirt is on sale for 40% off. The sale price is $12. What was the original price? What was the amount of the discount?

                       

The sale price is 60% of the original price. This reasoning can be expressed as 12 = 0.60p. Dividing both sides by 0.60 gives an original price of $20.

  • At a certain store, 48 television sets were sold in April. The manager at the store wants to encourage the sales team to sell more TVs and is going to give all the sales team members a bonus if the number of TVs sold increases by 30% in May. How many TVs must the sales team sell in May to receive the bonus?  Justify your solution.

Solution: The sales team members need to sell the 48 and an additional 30% of 48. 14.4 is exactly 30% so the team would need to sell 15 more TVs than in April or 63 total (48 + 15).

  • A salesperson set a goal to earn $2,000 in May. He receives a base salary of $500 as well as a 10% commission for all sales. How much merchandise will he have to sell to meet his goal?

Solution: $2,000 - $500 = $1,500 or the amount needed to be earned as commission. 10% of what amount will equal $1,500. Because 100% is 10 times 10%, then the commission amount would be 10 times 1,500 or 15,000.

  • After eating at a restaurant, your bill before tax is $52.60 The sales tax rate is 8%. You decide to leave a 20% tip for the waiter based on the pre-tax amount. How much is the tip you leave for the waiter? How much will the total bill be, including tax and tip? Express your solution as a multiple of the bill. The amount paid = 0.20 x $52.50 + 0.08 x $52.50 = 0.28 x $52.50

Solution: The amount paid = 0.20 x $52.50 + 0.08 x $52.50 = 0.28 x $52.50 or $14.70 for the tip and tax. The total bill would be $67.20,

  • Example: Stephanie paid $9.18 for a pair of earrings. This amount includes a tax of 8%. What was the cost of the item before tax?

Solution: One possible solution path follows: $9.18 represents 100% of the cost of the earrings + 8% of the cost of the earrings. This representation can be expressed as 1.08c = 9.18, where c represents the cost of the earrings. Solving for c gives $8.50 for the cost of the earrings.

  • Skateboard Problem 1. After a 20% discount, the price of a SuperSick skateboard is $140. What was the price before the discount?

  • Skateboard problem 2. A SuperSick skateboard costs $140 now, but its price will go up by 20%. What will the new price be after the increase? The solutions to these two problems are different because the 20% refers to different wholes or 100% amounts.

Notice that the distributive property is implicitly involved in working with percent decrease and increase. For example, in the first problem, if x s the original price of the skateboard (in dollars), then after the 20% discount, the new price is x −20%•x . The distributive property shows that the new price is 80%•x : x − 20%•x = 100%•x − 20%•x = (100% − 20%)x = 80%•x