John has tutored algebra and SAT Prep and has a B.A. degree with a major in psychology and a minor in mathematics from Christopher Newport University.
Ratios, Rates, and Proportions Topic Overview
''When am I going to ever use algebra in real-life scenarios?'' As a teacher, you have probably heard that question many times over the years.
Well, the good news is ratios, rates, and proportions are indeed applicable to day-to-day life, including:
- Discounts when shopping
- Financial situations involving business
- Tax rate calculations
- Rate: a specific type of ratio in which two completely different units are being compared
- Ratio: a comparison of two terms or items
- Proportion: involves comparing two separate ratios which are equal with one another
- Internet capability
- Writing paper
- Two days to complete this Algebra II assignment
- Ten days to put together a cohesive presentation
For more Algebra II information, visit our Homeschool Curriculum.
Assignment Instructions for Students
20 students in a class prefer Taylor Swift songs, while 15 people in the class prefer Justin Bieber songs. Therefore, the ratio of Taylor Swift to Justin Bieber fans is 20:15.
In gym class, Bobby can do sixty push-ups in four minutes. This yields a rate of 15 push-ups/minute.
A proportion is a bit more complex. For example, Ted has a coupon that saves him $20 for every $100 he spends at the sporting goods store. This means he saves $40 for every $200 he spends. Writing this out in number form gives the proportions 20:100 = 40:200.
Step One - Distance, Rate, and Time
When you rearrange the formula distance = rate * time you get time = distance / rate.
- Susie can ride her bicycle at 25 mph for a 5-mile trip to school.
- Marie can ride her bicycle at 20 mph for a 4-mile trip to school.
- Sally can ride her bicycle at 30 mph for a 6-mile trip to school.
Who arrives first to get the last free breakfast doughnut in the Senior Lounge?
Step Two - Money
Sandy has earned $25.50 from one day of selling at her lemonade stand for the school drama club. She sorts out her coins afterward and finds quarters, dimes, nickels, and pennies. She has 40 more quarters than dimes, 54 fewer nickels than quarters, and six fewer pennies than nickels. How many of each coin does Sandy find?
Step Three - Percentages
A percentage is a way of expressing a ratio from one to one hundred. For example, Sophia answers an impressive 48 out of 50 questions correctly on her application to become an astronaut, so she answered 96%, or 96 out of one hundred, correctly. We will be seeing her fly to the International Space Station soon.
Bill shoots 25 basketball three-pointers and makes five, while Mary shoots 25 three-pointers and makes ten of them. How much higher was Mary's percentage?
Step Four - Taxes and Discounts
Part I - Taxes
Luke is at the mall buying a copy of the latest Star Wars DVD. However, he is scrounging for enough change to pay. The sales tax in his town cost $ 0.84, and the tax rate is six percent. What is Luke's final price?
Part II - Discounts
As Luke is walking out the door, the sales clerk offers him a 10% pre-tax discount for wearing his cool Star Wars knit beanie hat. How much money did the clerk place in Luke's hands?
Step Five - Interest
The compound interest formula is A = P(1 + r/n)nt, where:
- A = final amount
- P = initial principal,
- r = interest rate
- n = number of times interest applied
- t = elapsed time periods.
Mary invests $5000 in her savings account, which earns her 10% interest, compounded quarterly. How much total money will she have accumulated in three years?
- Go online to take a virtual tour of the National Museum of Mathematics.
Remember, a quality setup is often the key to making Algebra II word problems easier.
Since d = r * t, it follows that t = d/r
- t = d/r
- t = 5/25
- t = 1/5 hour = 12 minutes
- t = 4/20
- t = 1/5 hour = 12 minutes
- t = 6/30
- t = 1/5 hour = 12 minutes
Wow! All three ladies arrive at the same time, and they agree to split the tasty doughnut three ways.
Now, this sounds really hard, doesn't it? However, most of your work will be in the setup.
- Let a = number of quarters
- Let a - 40 = number of dimes
- Let a - 54 = number of nickels
- Let a - 60 = number of pennies
- 25a + 10(a - 40) + 5 (a - 54) + 1(a - 60) = 2550
- 25a + 10a - 400 + 5a - 270 + 1a - 60 = 2550
- 41a = 3280
- a = number of quarters = 80
- Sandy has 80 quarters, 40 dimes, 26 nickels, and 20 pennies.
- Bill - 5/25 = 0.20 = 20%
- Mary - 10/25 = 0.40 = 40%
- Since 40 is exactly double 20, Mary's percentage was twice as high.
- Original price * tax rate = sales tax
- OP * 6% = 84 cents
- OP * 0.06 = 0.84
- OP = 0.84 / 0.06
- OP = $14.00
FP = Final Price = $14.00 + 0.84 = $14.84
D = Discount = $14.00 (not $14.84) * 0.10 = $1.40 returned to Luke, who immediately spends it in the food court
Don't give up yet. Yes, this formula does indeed look daunting, but let's break it up into smaller bits for easier mental digesting.
- A = $5000(1 + 0.10/4)3 * 4
- A = $5000(1 + 0.025)12
- A = $5000(1.025)12
- A = $5000(1.3449)
- A = $6,724.50
Now it's time to demonstrate your command of these new Algebra II concepts:
- Apply your skills to a real-world situation, such as starting a bank account and using the compound interest formula to track savings, and then writing a related report or presentation.
- Create a poster explaining your algebra skills by using real world examples with one of these topics.
- ''Be the teacher'' and explain these mathematical concepts by walking through examples in your own YouTube video, Ted Talk, podcast, or making a written lesson plan.
|Define the above three key terms|
|Differentiate between ratio, rate, and proportion|
|Solve a money problem|
|Define percentage and solve a related problem|
|Solve problems involving taxes and discounts|
|Learn the compound interest formula|
|Calculate using the compound interest formula|
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