Example

How many times does your heart beat in a year?

Answer: This question is asking for the rate of heart beats per year. Since a year is a long time to measure heart beats for, if we knew the rate of heart beats per minute, we could scale that quantity up to a year. So the information we need to answer this question is heart beats per minute. This is something you can easily measure by counting your pulse while watching a clock for a minute. Suppose you count 80 beats in a minute. To convert this to beats per year: [latex-display]\displaystyle\frac{80\text{ beats}}{1\text{ minute}}\cdot\frac{60\text{ minutes}}{1\text{ hour}}\cdot\frac{24\text{ hours}}{1\text{ day}}\cdot\frac{365\text{ days}}{1\text{ year}}=42,048,000\text{ beats per year}[/latex-display] This solution is worked out in the video below.

Example

How thick is a single sheet of paper? How much does it weigh?

Answer: While you might have a sheet of paper handy, trying to measure it would be tricky. Instead we might imagine a stack of paper, and then scale the thickness and weight to a single sheet. If you’ve ever bought paper for a printer or copier, you probably bought a ream, which contains 500 sheets. We could estimate that a ream of paper is about 2 inches thick and weighs about 5 pounds. Scaling these down, [latex-display]\displaystyle\frac{2\text{ inches}}{\text{ream}}\cdot\frac{1\text{ ream}}{500\text{ pages}}=0.004\text{ inches per sheet}[/latex-display] [latex-display]\displaystyle\frac{5\text{ pounds}}{\text{ream}}\cdot\frac{1\text{ ream}}{500\text{ pages}}=0.01\text{ pounds per sheet, or }=0.16\text{ ounces per sheet.}[/latex-display] In this problem, we inferred a measurement by using scaling.  If 500 sheets of paper is two inches thick, then we can use proportional reasoning to infer the thickness of one sheet of paper.  Similarly for the weight.  This solution is worked out in the video below.

Example

A recipe for zucchini muffins states that it yields 12 muffins, with 250 calories per muffin. You instead decide to make mini-muffins, and the recipe yields 20 muffins. If you eat 4, how many calories will you consume?

Answer: To answer the question of how many calories 4 mini-muffins will contain, there are several possible solution pathways.  We will explore one.

1. To find how many calories in 4 mini-muffins, it may help to know the number of calories in just 1 mini-muffin.  Then, we would multiply the number of calories in 1 mini-muffin by 4.
2. To find the calories in each mini-muffin, we could first find the total calories for the entire recipe, then divide it by the number of mini-muffins produced.
3. To find the total calories for the recipe, we could multiply the calories per standard muffin by the number of muffins.

Notice that this produces a multi-step solution pathway. It is often easier to solve a problem in small steps, rather than trying to find a way to jump directly from the given information to the solution.  To solve our problem, we follow the pathway backwards: Step 3: [latex]\displaystyle{12}\text{ muffins}\cdot\frac{250\text{ calories}}{\text{muffin}}=3000\text{ calories for the whole recipe}[/latex] Step 2: [latex]\displaystyle\frac{3000\text{ calories}}{20\text{ mini-muffins}}=\text{ gives }150\text{ calories per mini-muffin}[/latex] Step 1: [latex]\displaystyle4\text{ mini-muffins}\cdot\frac{150\text{ calories}}{\text{mini-muffin}}=\text{totals }600\text{ calories consumed.}[/latex] Thus, there are 600 calories consumed by eating 4 mini-muffins. 

View the following video for more about the zucchini muffin problem. https://youtu.be/NVCwFO-w2z4

Example

Is it worth buying a Hyundai Sonata hybrid instead the regular Hyundai Sonata? To make this decision, we must first decide what our basis for comparison will be. What are some factors that would be important to you? to continue.

Answer: For the purposes of this example, we’ll focus on fuel and purchase costs, but there are many other factors that one might consider such as environmental impacts, safety, maintenance costs, resale value, and many more.  (Hence there are many ways to correctly argue this problem!) In order to compare fuel and purchase costs, what information/data would you need to research?

  for one pathway example and the corresponding required data.

Answer: Part 1: Total fuel cost for one year

1. To compare fuel costs, we will find the total cost of gas to run each car for a year.
2. To find the cost of gas to run each car for a year, we will need to know average price of fuel per gallon and the number of gallons needed per year.
3. To find the number of gallons needed per year, we will need to know the the miles per gallon each car gets and the number of miles we expect to drive in a year.

From Hyundai’s website, the 2013 Sonata will get 24 miles per gallon (mpg) in the city, and 35 mpg on the highway. The hybrid will get 35 mpg in the city, and 40 mpg on the highway. Suppose you expect to drive about 12,000 miles a year, about 9,000 city miles a year, and 3,000 highway miles a year.  Finally, we will assume that gas in your area averages about $3.50 per gallon. Part 2: Total purchase price

• The hybrid Sonata costs about $25,850, compared to the base model for the regular Sonata, at $20,895.

Use the given information to calculate the total fuel cost and to decide if the hybrid model is worth buying given this particular set of information.

 

Answer:   To find the total fuel cost, we will follow the pathway laid out in Part 1 above backwards for each vehicle. Step 3: Find the number of gallons of fuel each car needs per year.   Sonata (regular): [latex]\displaystyle{9000}\text{ city miles}\cdot\frac{1\text{ gallon}}{24\text{ city miles}}+3000\text{ highway miles}\cdot\frac{1\text{ gallon}}{35\text{ highway miles}}=460.7\text{ gallons}[/latex] Sonata (hybrid): [latex]\displaystyle{9000}\text{ city miles}\cdot\frac{1\text{ gallon}}{35\text{ city miles}}+3000\text{ highway miles}\cdot\frac{1\text{ gallon}}{40\text{ highway miles}}=332.1\text{ gallons}[/latex]   Step 2: Find the total fuel cost of each vehicle per year. Sonata (regular): [latex]\displaystyle{460.7}\text{ gallons}\cdot\frac{\$3.50}{\text{gallon}}=\$1612.45[/latex] Sonata (hybrid): [latex]\displaystyle{332.1}\text{ gallons}\cdot\frac{\$3.50}{\text{gallon}}=\$1162.35[/latex]   Step 1: Compare the total annual fuel cost of the vehicles. The difference in annual fuel cost is given by: [latex]\$1612.45 - \$1162.35 = \$450.10[/latex].  The hybrid will save $450.10 a year. The gas costs for the hybrid are about [latex]\displaystyle\frac{\$450.10}{\$1612.45}[/latex] = 0.279 = 27.9% lower than the costs for the regular Sonata.   While both the absolute and relative comparisons are useful here, it is still hard to answer the original question, since “is it worth it” implies there is some trade-off for the gas savings. To better answer the “is it worth it” question, we need to determine the difference in purchase price and explore how long it will take the gas savings to make up for the additional initial cost. The difference in purchase price is given by: [latex]\$25,850 - \$20,895 = \$4955[/latex].  [NOTE: In the video, this difference is mistakenly calculated to be $4965].  The hybrid costs $4955 more. With gas savings of $451.10 a year, it will take about 11 years for the gas savings to make up for the higher initial costs. We can conclude that if you expect to own the car for at least 11 years, the hybrid is indeed worth it. If you plan to own the car for less than 11 years, then using only these factors, it would not be worth it.  Of course, as we previously mentioned, there are many other factors that we consider when purchasing a vehicle, so if we included some of these other factors, the hybrid may still be worth it. This is a case where math can help guide your decision, but it can’t make it for you. The video below works through the solution.

https://youtu.be/HXmc-EkOYJE