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 Health Decisions 

In college I learned that biology is really chemistry, chemistry is really physics, physics is really math, and math is really cool.
    - Anonymous

topic logo

Our first big Math 25 topic is Health Decisions. This is a great place to start because the issues are interesting and simple. There are a lot of subtopics. Each subtopic has its own formula, and it will be easy to know when to use each formula.

Work on making helpful and organized notes, so you have handy the comments, formulas, and example problems you need.

Real life is more complicated than any math formula. We will also discuss the context and limitations of each formula. How accurate is it? For whom? Why is it used even though it is imperfect? Does it cause any issues in real life?

This topic is both important and completely phony. For example, if you study nursing you will see the metabolism formulas a lot. But while at work you will probably never actually do the computation by hand with a calulator, paper, and pencil. A computer will do the computation automatically.

So we learn the metabolism formulas for two reasons. First, so you really understand what the computer is doing for you. Second, because while at work you will have to do some calculations, and those probably will not be any easier or harder than those formulas. So the calculations you do in class also serve as a phony stand-in for the unpredictable calculations you will actually do by hand some day at work. The goal is for your future employer to trust that you understand what the important formula is about when the computer solves it for you, and also that you have the ability to do calculations when needed.

Before looking at more formulas, we should start by taking a step back to look at what a formula really is.

Patterns

What is math?

My scale tells me I weigh 150 pounds.

That's just a number. Numbers are part of math.

Here is one of the first health formulas we will use in class, one of the Mifflin-St.Jeor BMR Formulas:

A Man's BMR = (weight × 4.55) + (height × 15.88) − (age × 5) − 161 

This formula expects us to plug in a man's weight, height, and age, and then it provides a basal metabolic rate (BMR). Formulas are part of math.

We could actually use the formula to calculate an answer. I weigh 150 pounds, am 66 inchess tall, and am 47 years old. So the formula claims that my basal metabolic rate is (150 × 4.55) + (66 × 15.88) − (47 × 5) − 161 ≈ 1,335 calories per day. Calculations are part of math.

Here is a quick version of the famous barometer story.

The Barometer Story

A physics test asked the question, "How could you measure the height of a tall building, using a barometer?"

One student gave the expected answer: "I could measure the barometric pressures at the top and the bottom of the building, and use the formula about pressure to determine the building’s height," but instead of stopping there provided many other answers.

That story talks about numbers and formulas and computations. But it does not actually include any. Thinking about numbers and formulas and computations is also math.

This math class focuses on all four parts of math.

We use lots of numbers! No big deal. Numbers themselves are rarely exciting.

We learn a bunch of famous formulas! We will plug in numbers and use these formulas. But equally important is to learn how to use the formulas. For example, that BMR formula is used by nearly every doctor's office and insurance company. It it being used appropriately? How accurate is it? Does it apply equally well to all people?

We do a lot of computation! This is what the website welcome page called the "small problems with right answers". Doing computations correctly helps you feel good about your math ability.

The computations we do are not themselves hard. They can be hard when they are new, just because anything can be hard when it is new! But once they stop being new, they also stop being hard.

Finally, we do a lot of thinking about numbers and formulas and computations.

Most of our thinking about numbers and formulas and computations is planning.

cover image from Aha Insight book by Martin Gardener

How do you plan a diet to manage your weight? How do you plan a meal to feed a hundred people? How do you plan a household budget, or plan what size house to buy to fit a budget? How do you plan saving for retirement? How do you plan the pricing strategy for a store or restaurant?

These plans are what the website welcome page called the "big problems without right answers".

But before we can start discussing big plans, we need to look a little more at what math really is.

videos

Domain of Science

The Map of Mathematics

Making Patterns from Tables

Linear Patterns

Where do formulas come from? How are they invented? Why do they work? How do they make sense?

All those questions have the same answer: patterns.

The easiest way to watch patterns become formulas is to make a table. In each row put which place in the pattern we are at, what the pattern looks like, and the number-value for that place. We can then look at the table to analyze the pattern and create a formula.

1. This pattern is an increasing number of X shapes, each made by four toothpicks. Can you explain the formula that describes how many toothpicks it takes to make a row of X shapes?

a toothpick boxes pattern

1. Each row adds four toothpicks. The pattern is y = n × 4.

By the way, we use n as the formula's input letter instead of x because of tradition. Using n shows that only "normal" counting numbers are inputs—never fractions, decimals, or negative numbers. A formula with x would allow those.

Here are more patterns to analyze and turn into formulas. To help you meet your classmates, work on each pattern with a different partner, and introduce yourselves as you work quietly together.

2. Can you explain the formula that describes how many toothpicks it takes to make a row of boxes?

a toothpick x pattern

2. Each row adds three toothpicks in a C shape, and there is always one extra toothpick at the far right edge to close the right-most box. The pattern is y = (3 × n) + 1.

Notice that we do not need the parenthesis. The order of operations for arithmetic already has us multiply before adding. But the parenthesis do help communicate where the pattern came from.

3. This pattern looks like a row of houses that gets longer and longer.

a toothpick house pattern

3. Each row adds five toothpicks in the shape of a house with no right wall, and there is always one extra toothpick at the far right edge to close the right-most house. The pattern is y = (5 × n) + 1.

As before, we do not need the parenthesis because the order of operations already has us multiply before adding. But the parenthesis do help communicate where the pattern came from.

videos

Khan Academy

Patterns and Mathematical Reasoning - Tables

Patterns and Mathematical Reasoning - Townhouses

Area Patterns

Time for some area patterns.

4. How does the number of tiles in a square increase? Do we know this formula's name?

square area pattern

4. The pattern is y = n × n, which can also be written y = n2.

This pattern is "tautological" because the formula does what its name says. The reason we call an exponent of two squaring a number because it makes a square whose side length is the number.

5. The area of a rectangle is equal to its length times its width. We cannot rediscover that whole formula with only one table. But we can find a specific version of it. What is the pattern for these rectangular areas?

rectangle area pattern

5. In this pattern the rectangles have sides of length n and (n + 1). The area formula multiplies these sides.

Our answer is y = n × (n + 1).

The formula A = l × w is also tautological. In our illustration is the second rectangle three rows of two tiles, or two columns of three tiles? Either way, making copies of an amount is simply what multiplcation does by definition.

6. How about a triangle of tiles? This pattern seems harder than the previous two! But there is a trick that makes it easy. You can find the trick by comparing this pattern to the rectangle pattern.

triangle area pattern

6. Notice that each triangle in this pattern is half the size of the corresponding rectangle in the previous pattern. Since the previous pattern was y = n × (n + 1), we want half of that. We need to divide by two at the end.

Our answer is y = n × (n + 1) ÷ 2

The formula we just found is called the Triangle Formula. Outside of a math classroom it is not as famous as the Square Formula or the Rectangle Area Formula. But it does deserve its own name because it is very useful.

videos

Mark Willis

Finding the nth term of the triangular numbers

In-Class Activity

We have all seen how dust floats in the air. Each dust particle has weight and is pulled down by gravity—but the upward force of air resistance can be equally strong. Just like water strider bugs can walk on water because they do not weigh enough to sink through the surface tension of water they stand on, a dust particle can "stand" on the air below it.

7. Get into groups of three or four students. Use the table below to explain why rain falls but clouds float.

growing diamond pattern

The Triangle Formula

We just found the Triangle Formula.

The Triangle Formula

The triangle pattern goes 1, 3, 6, 10,... with each step increasing additively by one more than the previous step.

Its formula is y = n × (n + 1) ÷ 2

The Triangle Formula appears surprisingly often in real-life applications. Here are a three pattern problems that seem tricky until you realize how the answer is made by tweaking the Triangle Formula.

8. This pattern involves how many toothpicks are in a triangle that grows downward. Each step in the pattern adds another row to the bottom of the previous triangle.

toothpick triangles pattern

8. Each step in the pattern is three times as big as the Triangle Pattern. So we need to multiply by three at the end.

The pattern is y = n × (n + 1) ÷ 2 × 3.

9. Now toothpicks make squares of increasing size. What is the pattern for how many toothpicks are in each square? (We are not looking at the area of the squares.)

toothpick squares pattern

9. Each step in the pattern is four times as big as the Triangle Pattern. So we need to multiply by four at the end.

If we divide by two and then multiply by four, the overall result is simply multiplying by two. Instead of ÷ 2 × 4 we can simply do × 2.

The pattern is y = n × (n + 1) × 2.

10. What is the most number of pieces you can make with straight cuts on a pizza? You will have to cut messy and not have every cut go through the center! One cut must make 2 pieces. Two cuts cannot make more than four pieces. The picture below shows a way three cuts can make seven pieces. Four cuts can make eleven pieces! And so on.

pizza pieces pattern

10. Each step in the pattern is one more than the Triangle Pattern. So we need to add one at the end.

The pattern is y = n × (n + 1) ÷ 2 + 1.

Ten Exercises

Try these ten exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. After you are very happy with your answers, you can use this form to ask me to check your work. Can you get at least 8 out of 10 correct?

1. 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

2. 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

3. 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

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

growing wide plus shapes pattern

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

hollow diamond shapes pattern

6. 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

7. 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.

8. 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?

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

10. 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."

Random Exercises

Try these exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. Check your work when you are done.

Interlude from Factfulness

Factfulness book cover Conquering the Single Perspective Instinct

The eighth chapter of Factfulness includes a conversation that Hans Rosling had in 2004 with Pascoal Mocumbi, then the prime minister of Mozambique. Hans had asked how he knew that Mozambique was making great economic progress.

"I do look at those figures," he said. "but they are not so accurate. So I have also made it a habit to watch the marches on May first every year. They are a popular tradition in our country. And I look at people's feet, and what kind of shoes they have. I know that people do their best to look good on that day. I know that they cannot borrow their friend's shoes, because their friend will be out marching too. So I look. And I can see if they walk barefoot, or if they have bad shoes, or if they have good shoes. And I can compare what I see with what I saw last year."

The point is that wise perspectives come from patterns.

Looking at a single perspective often leads to poor decisions. Looking at a pattern (whether or not it is patterns of numbers) often leads to a better decision. Looking at several related patterns often leads to the best decision.

So making good decisions requires some good habits:

Look for changes over time. There is very little value in knowing how many calories you eat today. Knowing the pattern of how that number number changes over time is hugely more important.

Dissect the generalizations. Actually, there is very little value in knowing the number of calories you eat each day. What kinds of calories are these? How much of each kind do you eat?

Get feedback about your conclusions. You finally have good data: useful categories of calories, and how each is changing over time. Do not immediately trust your own ideas about applying this knowledge. First bring your conclusions to other people, in this case people who know about nutrition or dieting or developing new habits. What do they consider the strengths and weaknesses of your ideas?

chapter 8 logo

Calories

The reason is that you eat too many foods that are high in "calories," which are little units that measure how good a particular food tastes. Fudge, for example, has a great many calories, whereas celery, which is not really a food at all but a member of the plywood family, provided by Mother Nature so that mankind would have a way to get onion dip into his mouth at parties, has none.
    - Dave Barry, Dave Barry's Guide to Life

Calories and Exercise

Calories are a measure of energy.

Our body uses energy even when resting for breathing, blood circulation, maintenance of body temperature and growing and repairing cells. When we exercise we need even more energy.

There is a pattern that shows how many calories are burned when exercising!

Before we read about it, first realize that this pattern might be very complicated. It surely depends upon the type of exercise, how intensely it is done, and for how long the person exercises. It could also sensibly depend upon the person's age, weight, height, sex, and other factors.

Also, a pattern about real-life data will never be perfectly accurate. Even if the same person repeated the same exercise for the same length of time, tiny changes in how they moved or what their metabolism was doing would result in a slightly different number of calories burned.

The Exercise Formula

In this case, there is great news. A simple pattern has enough accuracy for every day use.

Burning calories only depends upon the weight of the person exercising, how long they exercise, and what they are doing. We could describe this pattern using tables (like above). But let's skip right to the formula.

The Exercise Formula

Calories Burned = weight × minutes × scale factor 

Activity
 
Calories Per Pound
Per Minute
ballroom dancing0.023
grocery shopping0.028
walking0.037
weight lifting0.039
bicycling0.045
aerobic dancing0.061
basketball0.063
swimming0.070
running0.090

The chart on the right that estimates, for different activities, how many calories are burned per minute per pound of the person exercising. These numbers are the "scale factors" in the formula.

This formula uses a person's weight measured in pounds. If we wanted to use kilograms, we would need a different list of scale factors.

If we wanted more accuracy, we would need a better list of scale factors. Bicycling slowly along the Amazon Creek Trail is not as intense exercise as bicycling at racing speed over the Donald Street Pass! But this is a just math class, so this will be our official list of scale factors to use on homework and tests.

If you are curious about other activities, more charts such as this one are easy to find in books and on the internet. But their information should always be taken with some skepticism. The formula is always only an estimate.

11. Arthur weighs 150 pounds. He runs for an hour and then does a half hour of weight lifting. How many calories does he burn?

11. For Arthur's running we multiply:
155 pounds × 60 minutes × 0.09 calories per pound per minute ≈ 810 calories.

For Arthur's weight lifting we multiply:
150 pounds × 30 minutes × 0.039 calories per pound per minute ≈ 176 calories.

So Arthur's total is about 810 + 181 = 986 calories.

12. Odette weighs 110 pounds. She bicycles to the gym in 10 minutes, plays basketball for two hours, and then bicycles home in 20 minutes. How many calories does she burn?

12. For Odette's bicycling (30 minutes total) we multiply:
110 pounds × 30 minutes × 00.04509 calories per pound per minute ≈ 149 calories.

For Odette's basketball we multiply:
110 pounds × 120 minutes × 0.063 calories per pound per minute ≈ 832 calories.

So Odette's total is about 149 + 832 = 981 calories.

13. Odette weighs twice as much as her daughter. If they both do the same exercise, does Odette always burn twice as many calories?

13. In theory, yes, doubling the weight will double the answer. But in real life it seldom happens that an adult and child will exercise in exactly the same way for a long length of time. Perhaps Brianna's daughter is unusual, and when they go on walks will stay by her mother's side instead of frequently running ahead, or falling behind and then running to catch up.

Fat vs. Carbohydrates Burned

A person wanting to lose weight desires to burn as many fat calories as possible, not only carbohydrate calories.

The practical lesson is simply "never mind and exercise a lot." But this is a math class, so we will make the issue complicated by noticing that two effects work against each other.

When you do a more intense exercise then your metabolism burns a smaller percentage of fat calories. This happens because busy muscles need carbohydrate calories for quick energy and health.

But a more intense exercise burns many more calories total! The total number of fat calories burned still comes out ahead.

14. When walking, about three-quarters of the calories burned will be fat calories and one-quarter will be carbohydrate calories. When running, about half of the calories burned will be fat calories and half carbohydrate calories. Consider two people who both weigh 150 pounds who both exercise for 30 minutes: one person walks and the other person runs. How many fat and carbohydrate calories do each burn?

14. For the walker's total calories we multiply:
150 pounds × 30 minutes × 0.037 calories per pound per minute ≈ 167 total calories burned.

75% of those 167 calories are fat:
0.75 × 167 calories ≈ 125 fat calories.

25% of those 167 calories are carbohydrate:
0.25 × 167 calories ≈ 42 carbohydrate calories.


For the runner's total calories we multiply:
150 pounds × 30 minutes × 0.09 calories per pound per minute ≈ 405 total calories burned.

50% of those 405 calories are fat, and 50% are carbohydrate:
0.5 × 405 calories ≈ 203 each of fat and carbohydrate calories.

The runner burned many more fat calories (203 versus 125) even though a smaller percentage of calories burned were fat calories (50% versus 75%).

Testing the Formula

Breathing or Heart Rate

Is there any way that we check the Exercise Formula?

We cannot directly measure how many calories we burn. But we can estimate indirectly.

When we exercise our muscles need more oxygen. For aerobic exercise, the amount of extra oxygen needed is proportional to the amount of energy used exercising. To get that extra oxygen to our muscles, our breathing and heart rate increase.

Maybe our breating rate, or our heart rate, can help use have confidence that the Exercise Formula is reasonable.

If we do the same exercise twice as long, or three times as long, does either rate double or triple? That would be like the minutes in the Exercise Formula changing by ×2 or ×3.

Here is a table made by a group of Math 25 students in Spring Term 2019. One group member ran for longer periods of time.

running for a longer time

If we do the same exercise twice as quickly, or three times as quickly, does either rate double or triple? That would be like the scale factor in the Exercise Formula changing by ×2 or ×3.

Here is a table made by a group of Math 25 students in Winter Term 2019. One group member walked down a long hallway. Then he tried to jog that same hallway in half the walking time. Then he tried to run that same hallway in one-third the walking time.

walk jog run

Here is another table made by another group of Math 25 students in Winter Term 2019. One group member did jumping jacks for 30 seconds. First she did them very slowly. Then she did twice as many as at first. Then she did three times as many as at first.

jumping jacks

In all thre tables, any patterns did create a formula where was multiplied by a number. That is what we hoped to see! That means doubling or tripling the minutes or the scale factor in the Exercise Formula does result in doubling or tripling the answer.

(If the data made pattern whose forumla did not have any n, or if it made a formula that needed n2, that result would be very different from the Exercise Formula.)

In-Class Activity

15. Your turn! Get into groups of three or four students. Pick an exercise whose time or intensity you can double and triple. Make your own chart of breathing and heart rates. Does your data make any patterns?

Calories and Food

Calories in a Gram

Not all calories are created equal! One gram of fat has more than twice the calories of a gram of carbohydrate or protein.

carbohydrate protein fat

Calories per Gram

      • Carbohydrates have 4 calories per gram

      • Proteins have 4 calories per gram

      • Fats have 9 calories per gram

Those are measurement conversion rates. They let us switch between grams and calories.

bagel nutrition information

16. Arthur is in a rush this morning and eats a plain 4" bagel in the car for his breakfast. How many of the bagel's calories are from carbohydrates? from proteins? from sugar?

16. The bagel has 47 × 4 = 188 calories from carbohydrates.

It has 9 × 4 = 36 calories from protein.

Sugar is a kind of carbohydrate, so it also has 5 × 4 = 20 calories from sugar.

17. The bagel package says the bagel has 1 gram of fat, which should be 9 calories. But the package also says the bagel has 13 calories from fat. What is the most likely explanation for this difference?

17. The bagel probably does have 13 calories from fat. This would be 13 ÷ 9 ≈ 1.4 grams of fat. The manufacturer rounded down to "1 gram" for the label.

17b. 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. Change to calories the amounts of fat, carbohydrate, sugar, and protein.

17b. The milk has 5 × 9 = 45 calories from fat.

Its has 12 × 4 = 48 calories from carbohydrates.

(All of those 48 calories from carbohydrates are sugar.)

It has 9 × 4 = 36 calories from protein.

(By the way, milk tastes good because it contains a lot of natural sugar!)

18. One serving of Lite Chocolate Frosted Sugar Bombs has 1.3 grams of fat, 22 grams of carbohydrates (including 9 from sugar), and 2.5 grams of protein. Change to calories these amounts of fat, carbohydrate, sugar, and protein.

18. The cereal has 1.3 × 9 ≈ 12 calories from fat.

Its has 22 × 4 = 88 calories from carbohydrates.

As part of those 88 carbohydrate carbohydrates, there are 9 × 4 = 36 sugar calories.

It has 2.5 × 4 = 10 calories from protein.

(By the way, the nutrition information for this fake cereal is actually from a popular "healthy" cereal!)

Calories in a Day

The USDA has recommendations for what someone on a 2,000 calorie per day diet should be eating. (Labels on food packages refer to this information.)

2,000 Calories per day Diet

• 300 grams of carbohydrate per day (1,200 calories)
• at least 53 grams of protein per day (212 calories or more)
• less than 65 grams of fat per day (585 calories or less)

Of these amounts, at least 25 of the carbohydrate grams should be dietary fiber, no more than 50 of the carbohydrate grams should be added sugars, and no more than 20 of the fat grams should be saturated fat.

Those recommendations naturally make people wonder how much of a calorie category they have already eaten that day.

Recall that we find a percentage by dividing a partial amount by the whole amount.

Percentage of Whole Formula

Percentage = part ÷ whole

That formula creates an answer in decimal format. Recall we can move back and forth between decimal format and percent format by scooting the decimal point twice. The acronym RIP LOP summarizes moving between decimal format and percent format.

The RIP LOP Acronym

Right Into Percent, Left Out of Percent.

Notice that RIP LOP does not tell us "two places of decimal scoots". But we can remember that we always scoot twice. (Because percents are always about 100, and 100 always has two zeroes.)

19. Odette is on a 2,000 calorie per day diet. She eats two eggs for and two slices of toast for breakfast. That food has 13 grams of fat. What percentage of her recommended daily fat has she eaten?

19. To express the size of a part as a percentage use the formula:

Percentage = part ÷ whole.

The part is the 13 grams she has already eaten. Then whole is the 65 grams recommended per day.

13 ÷ 65 = 0.2.

Then use RIP LOP by scooting the decimal point twice to the right, to move into percent format. Also remember to label the answer of word problem.

0.2 = 20% of her recommended daily fat

What about a person who is more active or larger and eats more? The recommendations for a 2,500 calorie per day diet have slightly higher amounts.

2,500 Calories per day Diet

• 375 grams of carbohydrate per day (1,500 calories)
• at least 70 grams of protein per day (280 calories or more)
• less than 80 grams of fat per day (720 calories or less)

Of these amounts, at least 30 of the carbohydrate grams should be dietary fiber, no more than 63 of the carbohydrate grams should be added sugars, and no more than 25 of the fat grams should be saturated fat.

Most people do not fit either set of guidelines exactly.

20. Arthur should be eating 2,250 calories per day. What are his USDA recommendations for grams per day of carbohydrates, protein, and fat?

20. Arthur is halway between the 2,000 Calories Per Day guidelines and the 2,500 Calories Per Day guideliness. So we should average those.

For carbohydrates: (300 + 375) ÷ 2 ≈ 338 grams per day.

For protein: (53 + 70) ÷ 2 ≈ 62 grams per day.

For carbohydrates: (65 + 80) ÷ 2 ≈ 73 grams per day.

In-Class Activity

21. Your turn! Get into groups of three or four students. Brainstorm a list of what someone could (or really did) eat during a normal day. Sort those foods into the four types of calories described above.

Ten Exercises

Try these ten exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. After you are very happy with your answers, you can use this form to ask me to check your work. Can you get at least 8 out of 10 correct?

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

2. 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 3. 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?

4. 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?

5. 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?

6. 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?

7.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?

8. 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?

9. 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?

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

Random Exercises

Try these exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. Check your work when you are done.

Interlude from Factfulness

Factfulness book cover Conquering the Gap Instinct

In the first chapter of Factfulness, Hans Rosling writes:

"One reason the old labels are so popular is that they are so simple. But they are wrong! So, to replace them, I will now suggest an equally simple but more relevant and useful way of dividing up the world. Instead of dividing the world into two groups I will divide it into four..."

At that point in the book he was talking about global income levels.

But his words are equally true for calories!

Most diets (and thus most people) divide the world of food into "good calories" versus "bad calories". But this is too simple to be accurate or useful. So many foods are somewhere in the middle. As with most things in life, bad decisions are made when we focus our attention on the extremes.

Instead, divide food into four categories:

Of course the best foods are both nutritious and filling, and the worse foods are neither. But most people eat a lot of food that is only nutritious or filling, not both.

The best advice is usually to focus on foods that are filling, and aim for as much nutritional value as possible. The foods that are nutritious but not filling (like juice or dried fruit) can trick us into either eating too much as we try to get full, or into considering that our meal is done but we are still hungry when having dessert.

chapter 1 logo

Metabolism

Why is it an insult to say someone eats like a horse? A thousand pound horse eats about twenty pounds daily. Compare that to a hummingbird who eats its body weight daily, and doubles its weight when gorging before migrating!
    - Anonymous

Next we look at formulas that describe how our bodies work.

Heart Rate Formulas

Your maximum safe heart rate decreases with age. Since the 1970's, the guideline has been "220 minus your age". That is a simple calculation, but it is not accurate.

A better rule is "211 minus 64% of your age".

When doing aerobic exercise you want your heart rate between 50% and 85% of this maximum.

Heart Rate Formulas

maximum safe heart rate = 211 − (0.64 × age)

lower limit for aerobic exercise = maximum safe heart rate × 0.5 

upper limit for aerobic exercise = maximum safe heart rate × 0.85 

These formulas are accurate to about 6%. An actual value can be measured at a doctor's office by doing different exercises while monitored by an ECG machine.

22. Arthur is 35 years old. What is his maximum safe heart rate, minimum aerobic exercise heart rate, and maximum aerobic exercise heart rate?

22. maximum safe heart rate = 211 − (0.64 × age) = 211 − (0.64 × 35) = 211 − 19.2 ≈ 192 beats per minute.

lower limit for aerobic exercise = 192 × 0.5 = 96 beats per minute.

upper limit for aerobic exercise = 192 × 0.85 ≈ 163 beats per minute.

Metabolism Formulas

Basal Metabolic Rate (BMR)

sedentary sitting

Earlier we noted that our body uses energy even when resting for breathing, blood circulation, maintenance of body temperature and growing and repairing cells. Digesting food also uses up energy.

That amount of energy is called the "basal" (similar to the word "baseline") metabolic rate. The abbreviation BMR avoids having to say or write that long name.

A person who was lying in a hospital bed all day, not moving, would burn as many calories as his or her BMR. If that person ate more calories than that, he or she would gain weight. If that person ate fewer calories, he or she would lose weight.

We cannot measure BMR in class. But we can estimate it.

The name "basal metabolic rate" was made in 1918 by two scientists named J. Arthur Harris and Francis G. Benedict. They discovered that it could be predicted quite accurately by measuring the surface area of the person's skin. But measuring the surface area of a person's skin is difficult, so this did not help much.

A year later those two scientists found an estimate for the BMR based on weight, height, age, and sex. This was much more useful! The "Harris-Benedict BMR Formulas" were used for more than sixty years. They still appear in many medical books. But we will not use them in our math class.

In 1990, the formulas were updated by Mark D. Mifflin and Sachiko T. St. Jeor. For most people these new formulas are the most accurate. Their answer is usually within 10% of a person's actual BMR.

The Mifflin-St.Jeor BMR Formulas

Women's BMR = (weight × 4.55) + (height × 15.88) − (age × 5) + 5 

Men's BMR = (weight × 4.55) + (height × 15.88) − (age × 5) − 161 

These formulas use weight measured in pounds and height measured in inches. The numbers would be different if we used weight measured in kilograms and height measured in centimeters!

23. Arthur is a 35-year-old man who is moderately active, weighs 150 pounds, and is 5' 8" tall. What is his BMR?

23. Use the Mifflin-St.Jeor BMR Formula for men:
(150 lbs × 4.55) + (68 inches × 15.88) − (35 years × 5) − 161
      ≈ 1,426 calories per day

24. Odette is a 30-year-old very active woman who weighs 110 pounds and is 5' 1" tall. What is her BMR?

24. Use the Mifflin-St.Jeor BMR Formula for women:
(110 lbs × 4.55) + (61 inches × 15.88) − (30 years × 5) + 5
      ≈ 1,324 calories per day

Here is a list of these and other BMR formulas translated into SI units, with notes about their original research populations. For example, the The Mifflin-St.Jeor BMR Formulas do tend to overestimate the BMR for young Hispanic women and for Asian women.

Daily Calorie Intake (DCI)

portions of DCI

Fortunately, we are not completely lazy and do move about during the day.

The Daily Calorie Intake, or DCI, is how many calories a person really burns each day.

A person's DCI is always bigger than their BMR because they move about during the day and use more energy. So we need to increase the BMR to get the DCI.

DCI Formulas

DCI = BMR × scale factor

What is this scale factor?

An accurate scale factor would include information about all your movement and exercise. So in our math class we must estiamte.

The World Health Organization has discovered that you can ask people to rate their typical activity into only three categories. That provides a "close enough" estimate of DCI.

DCI Scale Factors

minimal
(couch potato)
moderate
(normal)
very active
(exercise routines)
Women1.561.641.82
Men1.551.782.10

Thus an athletic woman each day burns about her BMR × 1.82 calories.

We can also think about the DCI as the number of calories a person would eat each day to maintain their current weight. If the person eats less than their DCI they lose weight. If they eat more than their DCI they gain weight.

So an athletic woman could eat BMR × 1.82 calories each day and maintain her current weight.

(For the sake of completeness, know that none of these BMR or DCI formulas work for professional atheletes. Doing that much exercise changes how metabolisms function on a deeper level. The "athletic" category is for people who exercise quite often but are not professional atheletes.)

25. What is Arthur's DCI?

25. Arthur's BMR was 1,426 calories per day. For a moderately active man the scale factor is 1.78.
BMR × 1.78 = 1,426 × 1.78 ≈ 2,539 calories per day

26. What is Odette's DCI?

26. Odette's BMR was 1,324 calories per day. For a very active woman the scale factor is 1.82.
BMR × 1.82 = 1,324 × 1.82 ≈ 2,410 calories per day

To finish discussing DCI we need to debunk two common myths.

First, exercising more seldom leads to weight loss. For most people, doing additional exercise causes an increase in appetite that nearly cancels out the extra calories burned in the new exercise program. An effective weight loss program needs to change both exercise and diet.

Second, for most people strenuous exercise doesn't elevate your metabolism for a while. But there is good news for older people (over 50) who do both resistance exercise and aerobic exercise. For them, the strenuous exercise does noticeably raise their Basal Metabolic Rate—by as much as 30%!

Body Mass Index

Both BMR and DCI are personal measurements. The formulas were invented to measure how your own body behaves. The formulas are not completely accurate, but at least we are using them as intended.

Unfortunately, that is not true for another measurement that you have probably seen at the doctor's office or on health insurance paperwork.

Body Mass Index

BMI = (weight ÷ 2.2) ÷ (height ÷ 39.37)2 

This formula uses weight measured in pounds and height measured in inches. Remove the 2.2 and 39.37 if using weight measured in kilograms and height measured in centimetes.

The developers of BMI never intended it to be used to assess individuals. It was designed as a tool to measure statistical differences in groups of people. It should be a group measurement.

BMI ignores how much body weight is from muscle versus fat, and whether people are "apple shaped" or "pear shaped".

Even the Federal government admits that "BMI can be used to screen for weight categories that may lead to health problems but it is not diagnostic of the body fatness or health of an individual."

Nevertheless, some doctors and health insurance providers consider a "healthy" BMI to be between 18.5 and 25.

As a general rule, either ignore your BMI, or estimate it annually to check if there is a change that does not come from increasing exercise. That kind of change might be a wake-up call that your habits for diet or sleep have changed in surprising ways.

We will not use the BMI formula in this class. Using it properly to measure statistical differences in groups involves overly complicated math.

Percent Body Fat

The simplest measure for if someone's weight is healthy is their percent body fat. The American Council on Exercise has categorized how much body fat people should have to be healthy, based upon the person's usual activity levels.

Not Enough Fat
crazy abs
"scary"
Atheletes
well-defined abs
"hard"
Fit
slight abs
"lean"
Average
hidden abs
"love handles"
Obese
flab not abs
"squishy"
Women 13% or less 14% to 20% 21% to 24% 25% to 31% 32% or more
Men 5% or less 6% to 13% 14% to 17% 18% to 24% 25% or more

For every 20 years older than 18 years old, a person should expect to have 2% more body fat for their activity level than shown in the table.

Unfortunately, percent body fat is very difficult to measure accurately. A measurement with even ±4% accuracy requires a trip to the doctor and all sorts of caliper measurements. A more accurate measurement requires a fancy room for underwater weighing or air displacement measurements.

Please do not search online for a formula to estimate percent body fat. You will find some, but they are all garbage.

Perhaps more practical than a math formula is the saying, "Nothing tastes as good as feeling fit." If that saying is genuinely true for you, and your waist feels more lean than love-handlish, you almost certainly have a very healthy percent body fat.

Ten Exercises

Try these ten exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. After you are very happy with your answers, you can use this form to ask me to check your work. Can you get at least 8 out of 10 correct?

1. Caroline is an 80 year old woman, not physically active, who weighs 120 pounds and is 5' 3" tall. What is Caroline's maximum safe heart rate, minimum aerobic exercise heart rate, and maximum aerobic exercise heart rate?

2. Zachary is a 35 year old man, moderately active, who weighs 150 pounds and is 5' 7" tall. What is Zachary's maximum safe heart rate, minimum aerobic exercise heart rate, and maximum aerobic exercise heart rate?

3.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?

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

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

6. What is Caroline's BMR and DCI?

7. What is Zachary's BMR and DCI?

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

9. 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.

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

Random Exercises

Try these exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. Check your work when you are done.


Food Preparation

It's hard to know exactly what category these recipes fall in 'specially since some of them barely count as food, but I've done my best to put them in order as First Courses, Main Dishes, Misc. Savouries (and some are very misc.), Pudding and Misc. Sweets. Dwarf cookery deserves a place of its own, probably as a boat anchor.
    - Nanny Ogg's Cookbook

Imagine that a friend with a garden gives me a dozen wonderful carrots. I decide to be adventurous and make a family dinner using a recipe that I have not used before. I look through one of my favorite recipe books to find something that uses a dozen carrots.

That kind of household cooking is following a recipe on easy mode. We can estimate a lot. We cook the food already at home instead of going shoping for ingredients that match the recipe. We do not care exactly how big the carrots are. We do not worry if the recipe makes six servings or ten.

In a professional kitchen everything is more complicated. The business tracks how much a recipe is made. The recipe is scaled up to a huge size to create it's share of the shopping list. That bigger amount must often be expressed in different units of measurement. The produce on the shopping list is modified again because extra must be purchased to account for the portion lost when trimming. The price of the purchased amount is put into the restaurant's accounting.

Let's look at those steps one at a time.

Recipe Scaling

When the numbers are friendly it is easy to scale a recipe. If we want a triple batch we multiply every amount by three. If we want a half batch we multiply every amount by two.

Usually the numbers are not so convenient. We have one relationship (perhaps 1.5 cups of sugar for 12 servings) that does not compare nicely to another relationship (perhaps we want to shop for 200 servings, and we notice that 12 is not a factor of 200).

Proportions in General

Recall that those relationships are called ratios. In real life most ratio will have labels, so we also call it them rates.

To avoid careless mistakes we will scale recipes using a proportion.

Definition

A proportion is an equation of the format "ratio equals ratio" (or "rate equals rate").

But not all things that look like proportions actually are. What makes a real proportion different from a mistake?

Below are four situations, each involving a pair of events. For each situation four possible proportions are listed. In groups, decide proportions are correct. When most groups are done we will discuss what everyone thinks..

which proportions are valid

The pattern your group should have found was that the two events needs to be "kept together" symmetrically, either vertically or horizontally.

which proportions are valid

Most students remember this with the rule The labels on the right must match the labels on the left—do not flip them!

(It also works to have the labels on top match the labels on the bottom. But for most people this looks less natural and intuitive.)

If the two events are spread out upside down compared to each other then the proportion will not be correct.

which proportions are valid

If the two events are spread out diagonally then the proportion will not be correct.

which proportions are valid

Proportions for Recipe Scaling

Now we are ready to do some recipe scaling.

Here are two examples. For the moment please ignore how the answers are unrealistic.

25. A recipe uses 1.5 cups of sugar for 12 servings. Your restaurant wants to shop for 200 servings. How many cups of sugar do you add to the shopping list?

25. The situation can be written as the proprtion:

proportion

We can solve by cross-multiplying. The answer is 1.5 × 200 ÷ 12 = 25 cups of sugar

26. A recipe uses 30 ounces of canned pumpkin for 8 servings. Your restaurant wants to shop for 350 servings. How many ounces of canned pumpkin do you add to the shopping list?

26. The situation can be written as the proprtion:

proportion

We can solve by cross-multiplying. The answer is 30 × 350 ÷ 8 ≈ 1,313 ounces of pumpkin

Most students who take Math 25 are used to solving proportions by cross-multiplying. If you need more review for this process than those two examples, please say so!

Changing Units of Measurement

The previous two example problems had unrealistic answers. But they were unrealistic in different ways.

The answer "25 cups of sugar" is unrealistic because a professional kitchen would measure sugar by weight (not volume). The entire problem should really be about pounds of sugar.

The answer "1,313 ounces of pumpkin" is unrealistic because a weight over 16 ounces would be converted to pounds (or a mix of pounds and ounces). A restaurant kitchen might have a recipe that used a weight measured in ounces, but the shopping list would instead use pounds.

Let's look at several measurement unit conversion rates.

Ounces and Pounds

The Roman Empire used weights from which we get the words "pounds" and "ounces". After the Roman Empire, an ounce saw various definitions across Europe, usually 450 or 480 grains of barleycorn because that grain is very uniform.

The Roman pound was 12 ounces. But soon after the Roman Empire ended, most European merchants switched to a sixteen ounce pound. Their balance scales used a set metal weights for 1, 2, 4, 8, and 16 ounces, and they preferred a pound that was 16 ounces, which could be represented by a single metal weight.

27. Convert 1,313 ounces to pounds.

27. If we have done this kind of problem a lot, we would instinctively know to divide:

1,313 ounces ÷ 16 ≈ 82 pounds

But what if we were unsure whether to multiply or divide? Almost everyone has had a "brain fart" when taking a test and become suddenly uncertain about math calculations.

As a foolproof method, write the situation as a proprtion:

proportion

They we can solve by cross-multiplying, which will tell us when to multiply and when to divide.

Parts of a Gallon

Volume was not used much until the eighteenth century. A standard size bag was much more difficult to build (or compare to verify accuracy) than a standard rod/cord for length, or a metal weight.

For most of history, volume measurements were not used. When they were used, they were merchants' jargon for the volume taken up by a certain weight of a certain trade good. For example, a "gallon" was originally the volume of eight pounds of wheat.

28. Convert 30 quarts into cups.

28. If we have done this kind of problem a lot, we would instinctively know to multiply:

30 quarts × 2 × 2 = 120 cups

But what if we were unsure whether to multiply or divide? As before, a foolproof method is to use proprtions. First we convert quarts into pints:

proportion

Then we convert pints into cups:

proportion

As before, when we solve by cross-multiplying it tells us when to multiply and when to divide. Using proportions might require more writing, but it is nice an reliable (especially on tests!).

Is there an easy way to remember the parts of a gallon? You betcha.

Here is our gallon.

parts of a gallon

If we chop it into fourths we get quarts. The name quarts is like "quarters".

parts of a gallon

Each quart is two pints.

parts of a gallon

Each pint is two cups. Think about two short one-cup milk cartons stacked on top of each other to become the size of a one-pint carton of whipping cream.

parts of a gallon

Some students prefer to combine all of those diagrams into a single, more complicated picture.

parts of a gallon

Mind Your P's and Q's

Did you ever wonder why people say "Mind your p's and q's"?

Pubs in England used to give credit to regular customers. People would only get paid once per month, and in the meanwhile the bartender would mark p's and q's (for pints and quarts) below their name on the wall behind the bar. At the end of the month this record would show what they owed to bar. So "Mind your p's and q's" meant "Don't drink too much before payday!"

Which is a bit silly, because nowadays we usually say that phrase to young children.

Be aware that there is a different unit of volume called a "dry quart" which is not quite the same amount as liquid quarts. We will not use dry quarts.

The dry quart is almost the same amount as a liquid quart, so for grocery shopping no one notices the difference.

But if you are part of the Willamette Valley grass seed business and shop for bushels of grain the dry quart is important!

Measuring Spoons: Parts of a Cup

measuring spoons

In our math class we will seldom do problems that include measuring spoon amounts. Our focus is not on properly spicing recipes, but on making huge shopping lists for scaled-up recipes.

But it is important to know these measurement unit conversion rates too.

For the sake of clarity, this website abbrevates teaspoons as tsp, and tablespoons as Tbsp.

(Some cookbooks use tbsp or Tsp for tablespoon. Using both the capital T and the b avoids needless confusion.)

By the way, remember how volume measurements are a surprisingly recent invention? The first standardized set of measuring spoons were invented by Fannie Farmer in 1896. She was the director of the Boston Cooking School, and introduced her invention and philosophy in The Boston Cooking School Cook Book. Not everyone approved of Ms. Farmer adding more math to the art of cooking.

Tangentially, if we divide the last rate by four we get a shortcut that might also be worth memorizing:

29. A restaurant recipe that makes 30 servings requires 2 cups of butter. You want to scale the recipe down to make 6 servings. How many tablespoons of butter will it have?

29. The original 2 cups of butter is equal to 2 cups × 16 = 32 Tbsp

The recipe will be scaled down by a factor of 30 ÷ 6 = 5.

So the scaled down recipe will have 32 ÷ 5 ≈ 6 Tbsp of butter.

As a final detail about measurement unit conversions, sometimes the trickiest problems involve a decimal less than one.

30. Convert 0.2 cups into tablespoons.

30. 0.2 cups × 16 ≈ 3 Tbsp.

Those "less than one" problems happen when you are writing a cookbook. Imagine that you are used to using a favorite large restaurant recipe. You scale down the recipe to family size for the cookbook you are writing. When the calculations are done, a certain ingredient has 3.2 cups.

31. Express 3.2 cups as 3 cups and some tablespoons.

31. The previous problem told us that 0.2 cups equals 3 Tbsp. The 3 cups is a bigger amount that just hangs out in front. So our final answer is 3 cups and 3 Tbsp.

Trimming Produce

Gift Certificate Trickiness

Before we continue talking about recipe scaling and shopping, here is a problem that seems simple but does not fit our natural intuition. Its numbers rush into the wrong places in our heads!

gift certificate

32. As a birthday present, Janice receives a $50 gift certificate to a fancy favorite restaurant. She invites a friend to join her for dinner there. She expects great service and plans to pay an extra 25% for the tip and tax. What is the most she can pay for the food if she wants the gift certificate to also cover the tip and tax?

32. The incorrect but very natural answer that almost everyone gets first involves multiplying first and then subtracting. $50 × 0.25 = $12.50, and then $50 − $12.50 = $37.50.

The correct answer is $40. Once we are told that correct answer it makes a lot of sense! We can see that $40 × 0.25 = $10, and then $40 + $10 = $50.

How do we get to an answer of $40 instead of $37.50? That is the important question, which this answer key will not yet spoil. Ponder it yourself!

The point of the previous problem is that 25% of $50 is a different amount than 25% of $40.

When surprising, this distinction is not surprising. Of course the same percentage of a bigger number is a bigger amount. 25% of a millionaire's income is a lot more than 25% of my income.

But when the numbers are smaller, as in the previous problem, our brains naturally forget that distinction and rush to an incorrect answer.

A Fateful Decision

avacado trimming

Did you know that after removing the pit, cubing the avacado flesh makes it pop out of the skin easily?

Anyway, imagine that somone buys 100 pounds of avocados. The peel and pit them, and find out that they have 79 pounds of usable stuff for cooking.

What do we say about the 100 and the 79?

We could start from the point of view of the shopping. We buy the "original" amount (after all, that is what we had on the counter before we got a knife from the cutting board) and trimming gives us 79% that is usable.

Or we could start from the point of view of the usable ingredient. The trimmed amount is the "original" amount (after all, that is what the recipe cares about) and when shopping we need to buy 127% to have enough after trimming.

Both choices make sense in their own ways.

Which should be the baseline for our comparison, the purchased amount or the amount in the pot?

Yield Percent

ProduceYield Percent
avocados79%
carrots81%
cauliflower60%
celery69%
chives80%
onions89%
potatoes78%
scallions83%
spinach65%
turnips81%

The culinary industry has chosen the first option and named it Yield Percent. We need to remember that it measure how much of the purchased ingredient actually makes it into the pot.

The table to the right lists the yield percent for a few common types of produce. This will be the official table we use in class, so that our answers agree when solving problems.

Other tables of Yield Percents are available online (two examples). Note that these values are estimates. There are small variances because no two chefs or cauliflowers behave identically.

There are also Yield Percents for meats. But these have even more variation, because ways to trim meat also depend on how the meat will be cooked. As one example, see pages 27-45 of the USDA 2012 guidelines.

When using yield percent values, remember to continue our habit of using RIP LOP changing each percent into decimal format before doing other arithmetic with it.

It is straightforward to use yield percent to scale down from the amount actually purchased.

33. A restaurant buys 100 pounds of potatoes. How many pounds of trimmed potatoes will this make?

33. The yield percent for potatoes is 78%. Use the decimal 0.78.

100 pounds purchased × 0.78 = 78 pounds of trimmed potatoes

But much more common in real life is to instead answer the question, "If we want a certain amount of this produce item in the recipe, how much do we actually shop for?" Then we must work backwards (scale up instead of scale down) and divide by the yield percent.

34. A restaurant needs 100 pounds of trimmed potatoes. How many pounds of potatoes should they buy?

34. The yield percent for potatoes is 78%. Use the decimal 0.78.

100 pounds trimmed potatoes ÷ 0.78 ≈ 128 pounds purchased

Scale Factors Between 0 and 1

arrows up and down

How we use yield percents is an example of an important general rule.

When a scale factor is between zero and one, multiplying makes the number smaller.

When a scale factor is between zero and one, dividing makes the number bigger.

Remember that. We will use it again!

If the rule seems strange, that is because our real-life experience mostly uses numbers bigger than one.

We are used to doubling and tripling. When a scale factor is bigger than one, such as ×2 or ×3, then multiplying does make the number bigger.

We are used to halves and thirds. When a scale factor is bigger than one, such as ÷2 or ÷3, then dividing does make the number bigger.

Putting The Steps Together

We began our discussion of food preparation by listing five steps. The first step was for the business to track how much a recipe is made. Our problems start at the time when that is done. The business already knows how many servings of the recipe to account for in the upcoming shopping.

The other four steps were:

We can finally put those four steps together.

35. A recipe that makes 28 servings requires 5 pounds of carrots. How many pounds of carrots should you purchase if you are scaling up the recipe to make 105 servings? If carrots cost $0.75 per pound, what will that ingedient cost?

35. First we find how much we are scaling up the recipe.

105 desired servings ÷ 28 recipe servings =3.75

Then we scale up the ingredient amounts.

5 pounds recipe carrots × 3.75 scale factor = 18.75 pounds of trimmed carrots needed

But we are not done yet. The yield percent for carrots is 81%. Use the decimal 0.81.

18.75 pounds trimmed carrots ÷ 0.81 ≈ 23 pounds purchased carrots

Finally, we multiply to find the price.

23 pounds purchased carrots × $0.75 per pound = $17.25

36. A recipe that makes 16 servings requires 3 pounds of celery. How many pounds of celery should you purchase if you are scaling up the recipe to make 240 servings? If celery cost $1.05 per pound, what will that ingedient cost?

36. First we find how much we are scaling up the recipe.

240 desired servings ÷ 16 recipe servings = 15

Then we scale up the ingredient amounts.

3 pounds recipe celery × 15 scale factor = 45 pounds of trimmed celery needed

But we are not done yet. The yield percent for celery is 69%. Use the decimal 0.69.

45 pounds trimmed celery ÷ 0.69 ≈ 65 pounds purchased celery

Finally, we multiply to find the price.

65 pounds purchased celery × $1.05 per pound = $68.25

By the way, a general rule when doing a math problem that involves multiple steps is to not round until the very end. But we also rounded to whole numbers at the third step during those previous two problems. We rounded to be realistic. Most shopping orders involve a whole number of pounds, and the yield percent is only an estimate anyway. But you can round less if you want. Either style of rounding is okay.

In-Class Activity

vegetable trimming

Your turn! Get into groups of three or four students. Weigh your group's fruit or vegetable on the electric kitchen scale. Then trim it appropriately, and weigh it again.

37. Write both weights as a decimal number of pounds.

38. Write both weights as pounds and ounces.

39. What is the yield percent for your trimming?

40. If you needed 10 pounds of your item for a recipe, how many pounds would you need to shop for?

Ten Exercises

Try these ten exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. After you are very happy with your answers, you can use this form to ask me to check your work. Can you get at least 8 out of 10 correct?

1. 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?

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

3. 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?

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

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

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

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

8.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

9. 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

10. 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?

Random Exercises

Try these exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. Check your work when you are done.

Interlude from Factfulness

Factfulness book cover Conquering the Size Instinct

The fifth chapter of Factfulness includes an analysis of what medical staff can best do to save lives in very poor countries. The kind urge is to prioritize more doctors and hospital beds to aid people who are clearly sick. But numerous studies show that it is vastly better to spend time, money, and training on preventive measures. (Including literacy: mothers able to read and write is responsible for half the increases in child survival rates.)

Hans Rosling advises:

Getting things out of proportion, or misjudging the size of things, is something that we humans do naturally. It is instinctive to look at a lonely number and misjudge its importance. It is also instinctive to misjudge the importance of a single instance or an identifiable victim.

The most important thing you can do to avoid misjudging something's importance is to avoid lonely numbers. Never, ever leave a number all by itself. Never believe that one number on its own can be meaningful.

When I see a lonely number in a news report, it always triggers an alarm: What should this lonely number be compared to? What was that number a year go? Ten years ago? What is it in a comparable country or region? And what should it be divided by? What is the total of which this is a part? What would this be per person? I compare the rates, and only then do I decide whether it is really an important number.

Just because a number is big does not make it important or scary.

One of the many examples in that chapter is global infant mortality rates. In 2016, 4.2 million infants died. This lonely number seems immense and a cause for despair.

But in 1950 the number was 14.4 million. Now the first number, although still tragic and needing improvement, suddenly seems a cause for hope. Making a relevant comparison shows an improvement by over 10 million.

Moreover, in 1950 the infant mortality rate was 15% (there were 97 million children born) and in 2016 it was 3% (there were 141 million children born). Now we can feel great hope. Looking at rates shows a five-fold improvement.

Finally, the global mortality rate for all ages is about 1%. If the current rate of improvement continues, within forty years it will no longer be especially dangerous to be an infant.

So even if you have never bought 65 pounds of celery (as in Problem #36) you can now see where that big shopping amount came from. And hopefully you will develop the habit of always being curious about where lonely big numbers come from, and using tools like comparisons and rates to see the core of the issue.

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