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Math 25
 Homework 

About the Homework

Homework is the foundation of civilization.
    - Tom Foster

Before You Start

What to Do

Before you try any homework problems, go back to the math topics and actively read the online notes a second time.

While you do this, review your notes. Where are they missing an example problem? Where can you improve an explanation? Fix up any weak spots in your notes. Make flashcards if those help you.

If possible, do homework with others. Students who participate in study groups almost always pass the class. You can ask your instructor to help students find study groups.

Why to Do That

There are Three Levels of Math Understanding. (Thanks go to Cathy Miner for this insight.)

It is completely normal to have different levels of math understanding for different math topics and subtopics. And it is completely normal to not know what level you are at for a topic until you try "open book" and "closed book" problems.

So part of the purpose of math homework is to determine your Level of Understanding for each new subtopic. This is the purpose of the above recommendation to actively read the online notes a second time. They are prompt homework problems that should be done as soon after class as possible.

Prompt Homework Problems

After class attempt a few problems "open book". If you get stuck on any problem, skip it while making a note that you are at the first level of understanding for that problem. If you can answer it, try a similar problem "closed book". Then you will know whether you are at the second or third level of understanding.

When you are done, plan how much time you will need for studying, and how to pace yourself.

Later you can do what we call enough homework problems. This is not as urgent. Waiting a day or two after the lecture can be okay. But it is usually best to structure every day to include some homework time. Really doing the homework problems might take more time than you can afford to dedicate to a single study session.

Enough Homework Problems

Do all the assigned homework problems, and even more problems if you need extra practice. Now you are working to master each topic and reach the third level of understanding.

Use all your resources for help. Join a study group if you can. Work forward from the concepts, not backwards from the answer key.

Please do not worry if a topic is difficult. It is normal to have different levels of understanding. It is normal to have trouble with some topics for a few days. A day or two of struggle is not a danger sign. But if a math topic remains difficult after a three days then make sure to get help promptly, so you do not fall behind. Visit office hours, the MRC, or get help from a friend.

Three Types of Homework

This page has three types of homework.

When you have learned all that homework can teach you, start doing the appropriate problems on practice finals. Pretend these are real tests: find a place to work without distractions, use your own notes but not the website, and save hard problems for last.


Random Problems for More Practice

Imperial Civil Service Examination in Song Dynasty

Each time you load the page these problems change! The answer keys also change.

Generate as much practice as you need. When you are confident, move on to the next type of homework which has no answer key.

Health Decisions

Patterns

Calories

Metabolism

Food Preparation

Personal Finance Decisions

Typicality

Mortgages

Saving for Retirement

Business Decisions

Pricing

Markup and Discount

Charge Options

Likelihood


Small Problems to Turn In

Homework

Small homework problems without an answer key are your chance to prove that you can mimic sample problems to get correct answers. You demonstrate that you have learned the skills needed to follow procedures and use formulas.

Completing these assignments requires getting 80% or more of the problems correct. If you do worse, you might be asked to fix your work and resubmit it. Or you might be asked to try a different set of similar problems.

Health Decisions

Patterns

S1. Triangular tables are placed in a row to seat more people. One table has 3 seats. Two tables have 4 seats. Create a formula where we put in the number of tables (as n) and get out the number of seats (as y).

row of triangular tables problem

S2. Now we switch to square tables. We still make a row of tables to seat more people. One table has 4 seats. Two tables have 6 seats. Create a formula where we put in the number of tables (as n) and get out the number of seats (as y).

row of triangular tables problem

S3. This shape is sort of like a V or W that gets wider with more wiggles. What is the pattern for how many squares are in each row? (As an optional, extra challenge you can also find the pattern for how many toothpicks are in each row!)

growing W shapes pattern

S4. How about this extra-wide plus shape? What is the pattern for how many squares are in each row?

growing wide plus shapes pattern

S4b. How about this hollow diamond shape? What is the pattern for how many squares are in each row?

hollow diamond shapes pattern

S5. This shape looks somewhat like the stand that holds up a road construction sign. With each step it gets longer in each direction. What is the pattern for how many cubes are in each row?

growing in five directions

S6. There are many versions of an old story about the inventor of the game of chess. One version appears below. On which day will the total grains of rice exceed 3 million?

The Chessboard Story

chess story

King Radha of India was bored of backgammon, and desired a new game. Sessa, his minister invented chess. King Radha was pleased and asked Sessa what he desired in payment.

Sessa asked that a single grain of rice be placed on the first square of the chessboard, two grains on the second square, four grains on the third, and so on, doubling each time.

King Radha saw that this would require far more rice than his kingdom would ever produce, and had Sessa executed for impudence.

S6b. Two trains are approaching each other on parallel tracks. Both are traveling at 30 miles per hour. What is the overall speed at which they approach?

S6c. Continuing the previous problem, the two trains start 9 miles apart. How many minutes does it take for them to pass each other?

S6d. Continuing the previous problem, a fly zooms back and forth from the headlight of one train to headlight of the other. It starts when the trains are 9 miles apart. By the time it arrives at the other train, the two trains have gotten closer. It instantly reverses direction and heads back to the first train. And so on. The fly moves at 20 miles per hour. How far does it travel before the trains pass?

The Trains and Fly Story

One day at Los Alamos, Richard Feynman noticed something interesting. When he asked a physicist to solve the Trains and Fly problem they all used the shortcut (as above) and got the answer immediately. When he asked a mathematician, they all calculated the fly's trip bit by bit and finding the answer took several minutes.

Eventually Feynman brought the Trains and Fly problem to the most astounding mathematician of the group, John von Neumann, who immediately answered.

"That's not right!" protested Feynman. "You're a mathematician. You're supposed to sum the series, not use the shortcut!"

"What shortcut?" asked von Neumann. "I did sum the series."

Calories

S7. Liam weighs 165 pounds. If he swims for 45 minutes, how many calories does he burn?

S8. One serving of oatmeal has 3 grams of fat, 31 grams of carbohydrates (including 1 gram from sugar), and 6 grams of protein. Change to calories these amounts of fat, carbohydrate, sugar, and protein.

An old woman from Game of Thrones S9. Caroline is an 80 year old woman, not physically active, who weighs 120 pounds and is 5' 3" tall. She walks for 15 minutes to the grocery store, shops for 20 minutes, and walks 15 minutes home. How many calories does she burn?

S10. When Caroline walks, 60% of the calories she burns are fat calories (because of her age and other factors). How many fat calories did she burn from her 30 minutes of walking to and from the grocery store?

S11. One 1-cup serving of 2% lowfat milk has 5 grams of fat, 12 grams of carbohydrates (all 12 from sugar), and 9 grams of protein. Throughout the day Caroline drinks three cups of 2% milk. How many calories of fat is this?

S12. Caroline's DCI is 90% of the USDA expectation of 2,000 calories per day. So her recommended daily fat intake is 65 grams for a 2,000 calorie diet × 0.9 ≈ 59 grams of fat per day. What percentage of her recommended daily fat intake are those three cups of lowfat milk?

S13.A man wearing scale armor Zachary is a 35 year old man, moderately active, who weighs 150 pounds and is 5' 7" tall. He bikes for 15 minutes to the gym, lifts weights for 30 minutes, and then bikes 15 minutes home. How many calories does he burn?

S14. One 38-gram serving of dark chocolate has 15 grams of fat, 19 grams of carbohydrates (10 from sugar), and 3 grams of protein. Throughout the day Zachary eats four servings of dark chocolate. How many calories of sugar is this?

S15. Zachary's DCI is very close to the USDA variation of 2,500 calories per day. He can simply use its recommendation of no more than 63 grams sugar per day. What percentage of his recommended daily sugar intake are those four servings of dark chocolate?

S16. Write a formula that outputs the total calories in a food, if you plug in the grams of fat, carbohydrate, and protein.

Metabolism

S17. What is Caroline's maximum safe heart rate, minimum aerobic exercise heart rate, and maximum aerobic exercise heart rate?

S18. What is Zachary's maximum safe heart rate, minimum aerobic exercise heart rate, and maximum aerobic exercise heart rate?

S19.Renaissance Fair archery Caroline and Zachary meet at the local Renaissance Fair, as they do every year, for their archery competition. They are nearly silent, and few other fair participants notice the intensity of their rivalry. They shoot for twenty minutes. Archery burns 0.03 calories per pound per minute. How many calories total are burned during this epic struggle?

S20. For that archery competition, write Caroline's calories burned as a percentage of Zachary's calories burned.

S21. For that archery competition, write Zachary's calories burned as a percentage of Caroline's calories burned.

S22. What is Caroline's BMR and DCI?

S23. What is Zachary's BMR and DCI?

S24. Write Caroline's DCI as a percentage of Zachary's DCI.

S25. Caroline has a best friend named Greta, who is also an 80 year old woman, also not physically active, also 5' 3" tall, and who weighs 105 pounds. Find Greta's DCI.

S26. Write Greta's DCI as a percentage of Caroline's DCI.

Food Preparation

S27. A recipe that makes 8 servings requires 1.5 pounds of trimmed scallions. How many pounds of trimmed scallions will you need if you are scaling up the recipe to make 30 servings?

S28. One gallon of whole milk weighs 8.6 pounds. What is the weight of 6.5 cups of whole milk?

S29. One cup of uncooked long-grain brown rice weighs 7.2 ounces. What is the weight (in pounds) of one gallon of uncooked long-grain rice?

S30. Express 2.88 cups as 2 cups and some tablespoons.

S31. Express 6.15 cups as 6 cups and some teaspoons.

S32. After scaling up a recipe you get 19 teaspoons of cinnamon. Express this amount in tablespoons and teaspoons.

S33. You purchase 20 pounds of celery. How much trimmed celery can you expect this to make?

S34.photograph of spinach and rice casserole photograph of spinach and rice casserole Imagine that you are preparing a large meal for a non-profit fundraiser. Your task is to make the provided meal for 60 people. Find how much of each ingredient you need to purchase, keeping in mind the yield percent for produce. When using eggs you will need to round fractional amounts to the nearest whole number. (Note: spices and directions are omitted for the sake of simplicity.)

three recipe meal

S35. Use this list of prices to find the total price for the recipes after scaling all three recipes to 60 servings. (Water is considered free.)

  • brown rice: $0.76 per pound
  • carrots: $0.55 per pound
  • cauliflower: $1.01 per pound
  • heavy cream: $2.87 per quart
  • dry milk: $4.30 per pound
  • honey: $3.31 per pound
  • large eggs: $2.75 per dozen
  • milk: $2.65 per gallon
  • onions: $0.22 per pound
  • spinach: $2.35 per pound
  • vegetable broth: $2.28 per quart
  • whole wheat flour: $0.57 per pound

S36. An elementary school fundraiser includes serving ice cream. The school has a very good idea how much ice cream is needed because it has hosted this event for many years There will be 300 guests attending. 60% of the guests will want a half-cup serving, and the remaining 40% will want one cup of ice cream. As for flavors, 50% of the servings will be vanilla, 30% will be chocolate, and 20% will be strawberry. How many gallons of each flavor ice cream will need to be bought?

Personal Finance Decisions

Typicality

S41. Find the mean of these six numbers: 135, 95, 11, 5, 33, 15.

S42. Continuing the previous probem, find the median of those six numbers.

S43. The histogram below (original source) shows the number of books read by different children. What is the mean number of books read?

the number of books read by twelve children

S44. Continuing the previous problem, what is the median number of books read?

S45. The home values on a certain street, in thousands of dollars, are: 384, 364, 342, 346, 360, 356, 265, 417, and 530. What is the mean of these home values?

S46. Continuing the previous problem, what is the median of those home values? Why does this type of average better communicate the "typical" value of a home on that street?

S47. A shipping company needs to transport seven freight containers. Their weights are 10, 16, 16, 18, 20, 60, and 77 tons. What is the mean and median weight of these freight containers?

S48. Two company clerks receive a report that only contains the mean and median weights, and number of containers, from the previous problem. The first clerk tries to find the total weight by multiplying the mean by the number of containers. The second clerk tries to find the total weight by multiplying the median by the number of containers. Which clerk is correct? Why? How much error does the other clerk have?

S49. Some news articles make a big deal when many countries have an average temperate increase well above global average (Alaska, Canada, Russia, Norway, Finland, Switzerland, China, Singapore, Australia, South Africa, etc.) How does a better understanding of averages explain that having many items above average is neither surprising nor sensationalism?

S50. During the 2007 strike of the Writer's Guild of America, two different news reports painted very different pictures of these screen and television writers.

• According to CNBC, there were 4,434 guild writers who worked full-time in 2006, and their average salary was $204,000. (CNBC headline, October 11, 2007)

• According to the Los Angeles Times, the median income of the writers from their guild-covered employment is $5,000 a year. (Howard A. Rodman, October 17, 2007)

Were Hollywood's writers very wealthy and going to strike even though they earned much more than most Americans? Or were they poor and going on strike to defend the few thousand dollars they could earn from their writing? The headlines leave out two important facts. First, almost half of the guild's writers don't write anything in a given year (their salary that year is $0). Second, a very few writers earn millions of dollars. How does a better understanding of averages explain the situation more clearly?

Bell Curves

The most interesting part of bell curves is how they are used to sort people and shape society. That is not a great source of small math problems! So please pardon a brief tangent into our local contributions towards climate change.

In Oregon, a household's energy usage per month forms a histogram that resembles a bell curve reasonably well for real-life data.

The histogram below shows the monthly electrical useage for a household that has electric heat.

electric usage histogram

S51. In which month does this household use the most electricity?

S52. Which month has the least deviation from year to year?

S53. Someone asks, "What is this household's typical monthly electrical usage?" What would make a numeric answer to this question meaningful?

S54. When did this family replace their old electric furnace with a modern and more efficient heat pump?

S55. The total electrical usage for 2017 was 14,780 kilowatt hours. The total electrical usage for 2019 was 11,513 kilowatt hours. How much less electricity was used in 2019?

S56. Electricity costs an average of 11.3 cents per kilowatt hour. How much less money was spent on electricity in 2019 than in 2017?

S57. What was the percentage decrease of this household's total annual electricity usage when comparing 2017 to 2019?

S58. In this city 80% percent of the electric power is from carbon-free hydroelectric energy, making an overall CO2 emission of 16.2 grams per kilowatt hour. How many kilograms of CO2 did this household's electric use create in 2019?

S59. A typical gasoline automobile's emission is 8.8 kilograms of CO2 per gallon. This household's car gets 30 miles per gallon. How many miles would they need to drive to the equal CO2 emissions of their annual electricity use?

S60. That household actually drives 10,000 miles per year, which is equivalent to about 333 gallons of gasoline. Find this household's total kg of CO2 emissions for house and car, and then divide by 1,000 to convert kilograms to metric tons. Purchasing a carbon offset costs about $14 per metric ton of CO2. What is the value of the carbon offset cost for this household's house and car?

For most households the energy used to grow and transport food is a larger carbon footprint than home heating or vehicle driving. You can use a website such as CarbonFootprint to estimate your own numbers.

The Central Lane Metropolitan Planning Organization has found that in the Eugene-Springfield area the mean household carbon footprint is 31.9 metric tons of CO2, and the mean carbon footprint per person is 13.8 metric tons of CO2.

Mortgages

S61. Brenda can afford to spend $900 per month on mortgage payments, with a 30-year loan. Currently mortgage rates are 5% per year. What price home can she afford?

S62. Brenda gets a loan with a 25% downpayment. What will be the size of Brenda's actual mortgage and monthly payment?

S63. Brenda's loan has a 2% mortgage fee, first month pre-paid, and $1,000 other up-front costs. What is the total of her downpayment and these other up-front costs?

S64. Zane has an annual income of $65,000. He wants to spend 30% of his income on a mortgage, with a 15-year loan. The interest rate is 5%. What price home can he afford?

S65. Zane gets a loan with a 15% downpayment, 4% mortgage fee, first month pre-paid, and $1,300 other up-front costs. What is the total of these up-front costs?

S66. What will be the size of Zane's actual mortgage and monthly payment?

Remember the Wahl family? Their numbers were:

S67. Does doubling the monthly payment double the size of the other numbers? Consider the Moneybag family that has initially budgets $2,200 per month on a mortgage, with everything else the same as the Wahl family. Do the other numbers (except for the fixed "other up-front costs") double?

S68. Does halving the length of time halve the size of the other numbers? Consider the Short family that uses a 15 year loan for their mortgage, with everything else the same as the Wahl family. Do the other numbers (except for the fixed "other up-front costs") halve?

S69. Does halving the interest rate halve the size of the other numbers? Consider the Timing family that uses a 3% interest for their mortgage, with everything else the same as the Wahl family. (Their Amortization Table value is $4.22.) Do the other numbers (except for the fixed "other up-front costs") halve?

S70. Does doubling the interest rate double the size of the other numbers? Consider the Calamity family that uses a 12% interest for their mortgage, with everything else the same as the Wahl family. (Their Amortization Table value is $10.29.) Do the other numbers (except for the fixed "other up-front costs") double?

Saving for Retirement

S71. Tom buys a painting for $100. His friend, Eric, buys a painting for $200. Both paintings increasse in value by $30. What is the percent increase for each? How can the two items have different percent appreciation if they both increased by $30?

S72. A business buys a copy machine for $2,500 by borrowing that $2,500 with a loan of 15% simple interest for three years and three months. What is the total cost (the copy machine plus the loan's interest)?

S73.payday advance loan neon sign Oregon Senate Bill 1105 limits payday loan interest rates to 36% or less. Before that bill was passed, payday loans in Oregon often had interest rates of 120%. Nationally, payday loans can have interest rates as high as 7,000%. How much was borrowed at 120% annual simple interest for two weeks if the interest was $16.15?

(Note: Here is an article describing how check-cashing businesses can be helpful for their community. Nationally, the average payday loan is a two-week advance on $350.)

S74.Huey, Dewey, and Louie When Huey, Dewey, and Louie entered kindergarten their uncled started a college account for them. Each account had $5,000. That one deposit grew at 9% annual interest for 13 years. For Huey the interest was compounded weekly. For Dewey the interest was compounded monthly. For Louie the interest was compounded annually. How much was each account worth at the end of the 13 years?

S75.Einstein Mr. Largo gives his 20-year-old sibling a $100 wedding present. The sibling invests it at 11% annual compound interest for 30 years. Then the sibling adds an extra $10,000, and moves the combined funds in a different investment that earns 4% annual compound interest for 10 years. What is the final value, when the sibling is 60 years old?

S76. A credit card has an 18% annual interest rate. Payouts happen monthly. The loan uses compound interest. What is the annual effective interest rate?

S77. If $3,865 is invested with 9% annual compound interest for 40 years, how much will it grow to be worth? How much less than $176,696 is your answer? These questions are interesting because research by Vanguard shows that the average 25-year-old American has saved $3,865 for retirement, and the average 65-year-old American has saved $176,696. We are examining how much more the average American saves beyond the nest egg they have at age 25.

average savings chart

S78. Fiddle around using the sum of annuity due formula to find what annual deposit (principal) would grow to $176,696 over 40 years at 9% annual interest. (Hint: an annual deposit of $400 is too small, but $550 is too big.)

The two previous problems show us that it is easy to save as much as an average American. Setting aside less than $500 per year is enough to be average. Do not be embarassed if you start small with retirement savings. About half of Americans have a below average nest egg at age 25. You can become above average by saving more later in life as your career progresses.

child care photo from the Michigan Health Lab

S79. How does the long-term financial cost of infant care compare to saving for kids' college expenses? The "big three" child care centers in Eugene cost roughly $13,000 per year. (Vivian Olum at UO, Early Learning Childrens Community at LCC, and Child Development Center at EWEB. Oregon is an unusually expensive state for child care.) Use the sum of annuity due formula for two years (when the infant is age 1 and 2), and then the compound interest formula for sixteen years (until the child is 18). If the family instead saved the infant care money and earned a 5% annual interest rate, how much would be saved towards the child's college tuition?

Archer Queen from Clash of Clans

S80. In a certain computer game the archer queen gets 10% stronger every time she gains a level. Two brothers play the game. The younger brother has a level 20 archer queen. The older brother has a level 27 archer queen. Roughly how much stronger is the older brother's archer queen?

Business Decisions

Markup and Discount


Big Issues

Icing on a Gingerbread House

You demonstrate your ability to use and communicate deductive reasoning when you find answers for big issues.

Can you write about how a word problem applies to a broader real-life situation? Can you solve a word problem that requires using more than one skill in a way that does not mimic an example problem?

These problems have no due date. They do not have one correct answer. But sloppy work will be noticed, and you will be asked to improve and resubmit it.

Health Decisions

B1. Apply some of our health decision formulas to yourself, or to a friend or family member.

B2. Research the history and current use of the BMI and Body Fat formulas. How reliable are they? For what issues do they provide help? For what issues do they cause problems?

B3. Research how health monitoring devices use metabolism formulas to supplement their measured data for heart rate (for watches, chest monitors, and scales) and perhaps weight (only scales). Some health monitoring devices also measure blood oxygen level: what does that number mean, what formulas use it, and for whom would it be useful to monitor?

B4. Create a realistic large meal plan usable by a restaurant or catering service. The menu should include at least four dishes, and consider yield percent when making a shopping list. Price the shopping list and analyze the meal's cost per plate. Then compare your results with a similar meal at an actual restaurant or catering service.

Personal Finance Decisions

vacation I can afford meme B5. An average American vacation that costs about $1,100 per person. Plan two versions of a vacation. First describe a vacation with that average cost, split among categories of expenses. Then refine your plans by cutting costs to make a more affordable but still satisfactory version.

B6. An average American wedding that costs about $26,000. Plan two versions of a wedding. First describe a wedding with that average cost, split among categories of expenses. Then refine your plans by cutting costs to make a more affordable but still satisfactory version. (Weddings, unlike vacations, have well-established categories of expenses. These include per-guest costs (pre-reception food and drinks, the reception meal, reception drinks, tables and chairs at the reception) as well as fixed costs that do not depend upon how many guests attend (invitations, music, photography, flowers, dress, rings, bridesmaid expenses, rehearsal dinner, the ceremony room, and honeymoon).

B7. Compare the costs of renting a house versus buying a house. How do the categories of expenses differ? How do the costs differ? Overall, how much does monthly rent differ from the combination of monthly mortage payment and average monthly home maintenance cost? And how many hours each month on average does the owner of a house that size spend on home maintenance?

B8. Plan and complete a construction project involving a scale diagram. In past terms students have planned gardens, built trailers, and refurbished a deck. Describe the initial layout you have to work with, as well as the purpose and goals of the construction. Detail the preparations that need to be made to prepare the materials and/or land. Draw a scale drawing of the layout you wish to create, with all the features labeled. Create a shopping list to budget your expenses. Finally, do the work and include before and after photographs of the area.

B9. Analyze a month's spending on groceries. Keep track of which groceries you use. Then price your grocery list at three or more different grocery stores. First find which store would be least expensive overall. Then break down your costs at each store by category (dairy, produce, meat, beverages, frozen foods, dry goods, prepared meals, bulk food, bathroom supplies, cleaning supplies, baby items, etc.). Does your analysis suggest any shopping strategy?

B10. Some financial advisors use the 50-30-20 Rule for a household budgeting its after-tax income: budget 50% for necessities (including housing), 30% for luxuries, and 20% for paying of debt or saving for the future. Start by practicing with the Speck family's record of their income and expenses on the spreadsheet below (also a Google Sheet).

Most of their expenses are the same from month to month. Garbage pickup is billed quarterly. Their home has about $3,000 of property tax due each November, which they budget for by saving throughout the year. In March they receive some anniversary money from family. In April they need to fly to a wedding, and income tax is due.

Find how closely the Speck family expenses fit the 50-30-20 Rule. Then track your own household expenses for a month or more, and find how closely your own expenses fit the 50-30-20 Rule. How could you improve budget?

Speck Family Budget screenshot of Google Sheets link

B11. Pick one type of charitable activity, such as microfinance, disaster relief, medical aid, environmental work, etc. Use the website Charity Navigator to compare four or more non-profit organizations that do that type of activity. Which is the best to financially support to ensure your dollars help the world as much as possible? What information did you use to reach that conclusion?

B12. The website Portfolio Charts is a fun way to look back in time and observe how successful various investment strategies would have been. Pretend that you are someone nearing retirement age. What types of investment strategies are understandable? Which would have helped you be safely prepared for retirement?

B13. Gina's mother keeps pestering her to "shop for a husband while you are young and pretty." Gina's friends assure her that this advice is outdated, and Gina can focus on her career at least until the age of 30 if not 35. Although there is no correct answer, you can help them make peace by using this graph of census data to argue who is correct. Notice that a woman who is 35 years old in 2014 appears in four different places on this graph!

2014 Never Married Census Data graphed by Dalrock

Business Decisions

B14. The book Priced to Influence, Sell & Satisfy notes a few examples of interesting pricing issues. Find your own examples of these three pricing phenomenon during your own shopping or by contacting businesses.