4.3.2: Percentage Contexts

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Illustrative Mathematics
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Lesson

Let's learn about more situations that involve percentages.

Exercise $\PageIndex{1}$: Leaving a Tip

Which of these expressions represent a 15% tip on a $20 meal? Which represent the total bill?

$15\cdot 20$

$20+0.15\cdot 20$

$1.15\cdot 20$

$\frac{15}{100}\cdot 20$

Exercise $\PageIndex{2}$: A Car Dealership

A car dealership pays a wholesale price of $12,000 to purchase a vehicle.

1. The car dealership wants to make a 32% profit.

By how much will they mark up the price of the vehicle?
After the markup, what is the retail price of the vehicle?

Figure $\PageIndex{1}$: Cars , by Pexels. Public Domain. Pixabay. Source.

2. During a special sales event, the dealership offers a 10% discount off of the retail price. After the discount, how much will a customer pay for this vehicle?

Are you ready for more?

This car dealership pays the salesperson a bonus for selling the car equal to 6.5% of the sale price. How much commission did the salesperson lose when they decided to offer a 10% discount on the price of the car?

Exercise $\PageIndex{3}$: Commission at a Gym

For each gym membership sold, the gym keeps $42 and the employee who sold it gets $8. What is the commission the employee earned as a percentage of the total cost of the gym membership?
If an employee sells a family pass for $135, what is the amount of the commission they get to keep?

Exercise $\PageIndex{4}$: Card Sort: Percentage Situations

Your teacher will give you a set of cards. Take turns with your partner matching a situation with a descriptor. For each match, explain your reasoning to your partner. If you disagree, work to reach an agreement.

Summary

There are many everyday situations where a percentage of an amount of money is added to or subtracted from that amount, in order to be paid to some other person or organization:

Table $\PageIndex{1}$
	goes to	how it works
sales tax	the government	added to the price of the item
gratuity (tip)	the server	added to the cost of the meal
interest	the lender (or account holder)	added to the balance of the loan, credit card, or bank account
markup	the seller	added to the price of an item so the seller can make a profit
markdown (discount)	the customer	subtracted from the price of an item to encourage the customer to buy it
commission	the salesperson	subtracted from the payment that is collected

For example,

If a restaurant bill is $34 and the customer pays $40, they left $6 dollars as a tip for the server. That is 18% of $34, so they left an 18% tip. From the customer's perspective, we can think of this as an 18% increase of the restaurant bill.
If a realtor helps a family sell their home for $200,000 and earns a 3% commission, then the realtor makes $6,000, because $(0.03)\cdot 200,000=6,000$, and the family gets $194,000, because $200,000-6,000=194,000$. From the family's perspective, we can think of this as a 3% decrease on the sale price of the home.

Practice

Exercise $\PageIndex{5}$

A car dealership pays $8,350 for a car. They mark up the price by 17.4% to get the retail price. What is the retail price of the car at this dealership?

Exercise $\PageIndex{6}$

A store has a 20% off sale on pants. With this discount, the price of one pair of pants before tax is $15.20. What was the original price of the pants?

$$3.04$
$$12.16$
$$18.24$
$$19.00$

Exercise $\PageIndex{7}$

Lin is shopping for a couch with her dad and hears him ask the salesperson, “How much is your commission?” The salesperson says that her commission is $5\frac{1}{2}\%$ of the selling price.

How much commission will the salesperson earn by selling a couch for $495?
How much money will the store get from the sale of the couch?

Exercise $\PageIndex{8}$

A college student takes out a $7,500 loan from a bank. What will the balance of the loan be after one year (assuming the student has not made any payments yet):

if the bank charges 3.8% interest each year?
if the bank charges 5.3% interest each year?

(From Unit 4.2.4)

Exercise $\PageIndex{9}$

Match the situations with the equations.

Mai slept for $x$ hours, and Kiran slept for $\frac{1}{10}$ less than that.
Kiran practiced the piano for $x$ hours, and Mai practiced for $\frac{2}{5}$ less than that.
Mai drank $x$ oz of juice and Kiran drank $\frac{4}{3}$ more than that.
Kiran spent $x$ dollars and Mai spent $\frac{1}{4}$ less than that.
Mai ate $x$ grams of almonds and Kiran ate $1.5$ times more than that.
Kiran collected $x$ pounds of recycling and Mai collected $\frac{3}{10}$ less than that.
Mai walked $x$ kilometers and Kiran walked $\frac{3}{8}$ more than that.
Kiran completed $x$ puzzles and Mai completed $\frac{3}{5}$ more than that.

$y=2.33x$

$y=1.375x$

$y=0.6x$

$y=0.9x$

$y=0.75x$

$y=1.6x$

$y=0.7x$

$y=2.5x$

(From Unit 4.1.5)

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