Math OER Zoom Room Textbook YouTube Distance Learning Tips |
Our third big Math 25 topic is Business Decisions. This topic is easier, and less tricky, than personal finance decisions.
As with health decisions, none of the subtopics look too much alike. Yay!
Unlike personal finance decisions, the subtopics do not get mixed up and it is clear which to use for each problem. Yay!
What makes the business decisions topic different is that we switch our focus from formulas to processes.
There certainly are formulas. But these formulas summarize a process, and you might need to modify the process so it stays useful even in real-life situations that are more complicated than the formulas allow.
As always, work on making helpful and organized notes, so you have handy the comments, formulas, and example problems you need.
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The word "price" can be ambiguous. Are we talking about the price a business pays to its supplier, or the price a customer pays to the business?
Our discussion will always say wholesale cost or selling price to be clear.
The difference between an item's selling price and wholesale cost is called the margin amount.
This gives us two formulas.
Margin Amount
margin amount = selling price − wholesale cost
or
wholesale cost + margin amount = selling price
1. Dora's Dress Shop can get an item for $60 wholesale cost, and sell it for a $80 selling price. What is the margin amount for that item?
1. $80 selling price − $60 wholesale cost = $20 margin amount
Many businesses prefer to express margin as percent change. Recall that formula:
Percent Change Formula
percent change = change ÷ original
After the division, use RIP LOP to change the decimal into percent format.
People thinking about the margin rate start by considering the selling price and then "look down" to the wholesale cost. In this mindset the change is the margin amount and the original is the selling price.
Margin Rate
margin rate = margin amount ÷ selling price
After the division, use RIP LOP to change the decimal into percent format.
2. Dora's Dress Shop can get an item for $60 wholesale cost, and sell it for a $80 selling price. What is the margin rate for that item?
2. margin rate = $20 margin amount ÷ $80 selling price = 0.25 = 25% margin rate
3. Another business uses a 40% margin rate. What is the margin amount for an item with a selling price of $300? What is the wholesale cost of that item?
3. First find the margin amount.
We know that margin rate = margin amount ÷ selling price.
So we plug in 0.4 = m ÷ $300
To get m by itself the opposite of ÷ $300 would be × $300.
So we do × $300 to both sides of the equation.
$120 margin amount = 0.4 × $300 = mNext we find the wholesale cost.
We know that wholesale cost + margin amount = selling price.
So we plug in w + $120 = $300
To get w by itself the opposite of + $120 would be − $120.
So we do − $120 to both sides of the equation.
That gives us a w = $300 − $120 = $180 wholesale cost
The difference between an item's selling price and wholesale cost is also called the markup amount.
Markup Amount
This is only a name change compared to the margin amount!
markup amount = selling price − wholesale cost
or
wholesale cost + markup amount = selling price
4. Dora's Dress Shop can get an item for $60 wholesale cost, and sell it for a $80 selling price. What is the markup amount for that item?
This is only a name change compared to Problem #1!
4. $80 selling price − $60 wholesale cost = $20 markup amount
The difference is that people thinking about the markup rate start by considering the wholesale cost and then "look up" to the selling price. In this mindset the change is the markup amount and the original is the wholesale cost.
Markup Rate
markup rate = markup amount ÷ wholesale cost
After the division, use RIP LOP to change the decimal into percent format.
5. Dora's Dress Shop can get an item for $60 wholesale cost, and sell it for a $80 selling price. What is the markup rate for that item?
5. markup rate = $20 margin amount ÷ $60 wholesale cost ≈ 0.33 = 33% markup rate
6. A third business uses a 40% markup rate. What is the markup amount for an item with a wholesale cost of $180? What is the selling price of that item?
6. First find the markup amount.
We know that markup rate = markup amount ÷ wholesale cost.
So we plug in 0.4 = m ÷ $180
To get m by itself the opposite of ÷ $180 would be × $180.
So we do × $180 to both sides of the equation.
$72 markup amount = 0.4 × $180 = mNext we find the selling price.
We know that wholesale cost + markup amount = selling price.
So we plug in $180 wholesale cost + $72 markup amount = $252 selling price
Notice that both Problem #3 and Problem #6 had a wholesale cost of $180. But the problems were very different. In Problem #3 we looked at 40% of the $300 selling price. In Problem #6 we looked at 40% of the $180 wholesale cost. Naturally taking 40% of a bigger number yields a bigger result for the margin/markup amount.
So we have four terms:
margin amount | markup amount |
margin rate | markup rate |
Some books and websites try to be simpler and only use two terms.
They call margin rate the more confusing term markup on selling price. We will not do this.
They call markup rate the more confusing term markup on wholesale cost. We will not do this either.
You can understand the intention. Those texts wish to be simple by skipping the word "margin" entirely and only use its synonym "markup".
But careful language only uses markup to mean an increase from wholesale cost. Trying to use the word markup to mean a decrease (uhg!) from selling price (uhg!) is twice sloppy.
Also, those "markup on..." phrases are vague about when they are talking about a dollar amount and when they are talking about a percent rate.
Let's avoid that confusing terminology. Four terms are better than two for clarity. We are clear about whether we are looking up or down. We are clear about whether we are talking about a dollar amount or rate.
Business have many other expenses besides their wholesale costs: payroll, insurance, rent, utilities, advertising, taxes, etc.
The profit a business makes will be the gains from its total margin amounts reduced by these other expenses.
We can imagine a five-year-old who wants to sell lemonade from his front yard on a hot summer day. His parents help him set up his table, and freely provide all the cups and ice. They charge him a nickel for each cup of lemonade, and he sells each cup for a quarter. Since he is cute, and the day is hot, he sells a dozen glasses and is happy with his first "business".
That child got to keep his entire total margin amount. He earned 20¢ per cup of lemonade. But a real business does not get to keep their total margin amount. Profit is always smaller.
We start this topic by looking at the three fundamental pricing methods that businesses use.
These methods are less exciting than the pricing tactics people usually think about when someone mentions retail pricing. Retail prices can be carefully set to attract attention, communicate value, influence opinion, build brand comfort and loyalty, and direct customers to buy specific inventory items. However, there is much more psychology than math in these tactics. For more about that aspect of pricing, use our classroom library.
Many businesses base their decisions upon their wholesale suppliers. Their primary concern is to build relationships with these suppliers, build brand loyalties, and to acquire goods inexpensively.
The owners of these businesses are certainly aware of the other businesses they compete with. Too keep competitive, they might make short-term modifications to their selling prices or inventory, or they might have habits of selling certain items only seasonally.
These businesses normally set prices by using a markup rate.
Markup Pricing
selling price = wholesale cost × (1 + markup rate)
7. Granny's Gardening Supplies prices items with a 40% markup rate. If the wholesale cost of an item is $50, what selling price would the store use?
7. selling price = wholesale cost × (1 + markup rate) = $50 × 1.4 = $70
However, markup pricing has its issues.
Not all goods should be given the same percent increase. Goods with a higher volume of sales can be assigned a lower percent increase (to attract customers and build brand loyalty) while still remaining profitable. Trendy goods can temporarily be given a higher percent increase to increase profit while demand is high.
Many business expenses are a general overhead cost not tied to wholesale cost of any particular good. These could be totalled and averaged to spread them out evenly among all selling prices. But that is seldom the most strategic option.
8. Granny's Gardening Supplies prices its best-selling tools by first using a 50% markup rate and then adding $5. If the wholesale cost of its best-selling rake is $8, what selling price would the store use?
8. First apply the rate.
selling price = wholesale cost × (1 + markup rate) = $8 × 1.5 = $12Then add the five dollars: $12 + $5 = $17
Newly invented goods are usually at first expensive to produce. Then, as their technology matures, their wholesale cost decreases. The history of a good can also reflect its penetration into society.
Businesses must adapt. The appropriate amount to increase selling price above wholesale costs will change over time as the technology matures. (You might enjoy a well-written article about microwaves.)
9. Hector's Home Goods sells electric pressure canners. There is more demand for newer models with more features, but some customers are looking for an older and less expensive model. The store prices its pressure canners by increasing its wholesale costs by 60% for a new model, 30% for last year's model, and 20% for even older models. If the wholesale cost of its newest model is $90, what selling price would the store use?
9. selling price = wholesale cost × (1 + markup rate) = $90 × 1.6 = $144
A business can bundle items to smooth out the effect of wholesale costs over several goods.
Consider a store selling vegetables. Conventional lettuce and carrots are sold at high volume with negligible margin. Organic spinach and kale are sold at lower volume but with greater margin. The store could try using attractive display of mixed greens to achieve a specific balance of volume sold and margin per item.
10. Pamela's produce sells conventional Romaine lettuce with a 10% markup on its $0.70 per pound wholesale cost, and organic baby spinach with a 70% markup on its $3.20 per pound wholesale cost. How many pounds of the lettuce must be sold to earn as much markup as one pound of the spinach?
10. The markup on one pound of the spinach is $3.20 × 0.7 = $2.24.
The markup on one pound of the lettuce is $0.70 × 0.1= $0.07.
That seven cents must be earned $2.24 ÷ $0.07 = 32 times to make $2.24.
A strategy similar to bundling is to offer a portfolio of slightly different products, so customers willing to pay a bit more for extra features have the opportunity to do so. This is often done with three options, and called good-better-best pricing.
Other businesses must base their decisions based upon their competition. Their primary concern is to monitor how their own selling prices compare to the selling prices of similar goods. They know that their inventory has weak brand loyalty, and customers will shop elsewhere if they see a better deal. They keep their selling prices as high as a tough market allows with an acceptable margin.
The owners of these businesses are certainly aware of the importance of building relationships with wholesale suppliers, and of increasing brand loyalties. But they sometimes cannot afford these luxuries. If customers will only pay a certain amount for an item, they must drop that item from their inventory if they cannot find a supplier who can supply it inexpensively enough.
These businesses normally set prices by using a margin rate.
Margin Pricing
wholesale cost = selling price × (1 − margin rate)
11. Guinevere's Gardening Supplies uses a 40% margin rate. If the selling price of an item is $50, what is the largest wholesale cost the store can afford pay a supplier?
11. wholesale cost = selling price × (1 − margin rate) = $50 × (1 − 0.6) = $30
In other words, Guinevere's Gardening Supplies can only stock goods for which they have found a supplier who can provide the items for 60% or less of the selling price.
There is no benefit in pricing all goods to equally undercut the competition. To the contrary, a successful business will set some selling prices above those of the competition.
Businesses that use margin pricing frequently use loss leaders to attract customers with great deals on items with an unprofitable selling price, and then compensate with profit from their other inventory. We have all seen the mailers and large storefront displays that advertise a grocery's store's current deals.
A very few customers are willing to travel among similar stores to buy each item at its least expensive price. Businesses that use margin pricing learn to ignore those extremely price-conscious customers. Not only are they a small minority, but their shopping does not provide much profit for any of the businesses they use. When thinking about when and how much to undercut the competition, focus on the more typical customer.
12. Businesses that use markup pricing can also use loss leaders. A famous example is Costco's rotisserie chickens. Costco sells about 60 million chickens each year. Each chicken is sold at about a $0.57 loss. How much of a total loss are these sales?
12.$0.57 × 60 million ≈ $34 million lost
Businesses that use margin pricing must balance skim pricing (charging more for trendy goods) with penetration pricing (the price eventually reached with a long-term supply and demand balance).
Many customers are happy—even excited—to pay more for new and innovative products. To some extent skim pricing is a natural part of the cycle of product innovation and development. But excessive or inappropriate skimming will create a public relations backlash.
A large business fighting for market share can also minimize skimming, attempting to increase market share by rushing closer to the penetration price. Sometimes losing short-term profit is the best long-term strategy.
13. In November 2006 the new Sony PlayStation 3 was priced at $500. During the next three years the selling price was lowered in steps, and eventually settled at $180. If the wholesale cost is $120, divide the larger margin by the smaller margin to find the percentage of extra margin from skimming.
13.The larger margin was $500 − $120 = $380
The smaller margin is $180 − $120 = $60
The percentage is $380 larger margin ÷ $60 smaller margin ≈ 633% extra margin on the initial skim price compared to the eventual penetration price.
A business can bundle items to smooth out or disguise the effect of skimming over several goods.
A new model of electronics can be priced high if bundled with plugs and cords that are advertised as a loss leader. Or the other way around: the new model can be priced to undercut the competition and bundled with plugs and cords whose higher than typical price is not noticed by the excited customer.
14. Phineas's Phones sells a popular cell phone for $200. It also offers a $320 bundle that includes the phone and a two-year pre-paid data plan. The wholesale cost of the phone is $160, and the wholesale cost of the data plan is $40 per year. Which has the higher percentage margin, the phone by itself or the bundle?
14. For the phone alone margin rate = margin amount ÷ selling price = $40 change ÷ $200 original = 20%.
The bundle's wholsale cost is $160 + $40 + $40 = $240.
The bundle's margin amount is $320 − $240 = $80.
For the bundle margin rate = margin amount ÷ selling price = $80 change ÷ $320 original = 25%.
The bundle has a higher margin rate. Perhaps the store's location attracts customers looking for phones more than customers looking for pre-paid plans? That would be one explanation for why the store tries to use bundling to encourage more sales of plans.
The modern world is full of data. Businesses can now immediately measure the changing demand for goods and services, and automatically adjust pricing to match that demand. This is called adaptive pricing.
Adaptive pricing is why the selling price of the plane trip you are pondering will increase if you repeatedly window shop from the same computer, why the selling prices at Amazon.com are in constant flux, and why parking meters in San Francisco charge more as parking spaces fill up.
Sometimes adapting prices by constantly monitoring customer behavior is needlessly complex or expensive. A simpler yet similar strategy is to automatically adjust prices based on predicted customer behavior. This is called dynamic pricing.
For example, many online retailers adjust their prices during the times of day that most customers shop.
The second chapter of Algorithms to Live By in our classroom library has more about adaptive and dynamic pricing.
(The terms adaptive and dynamic pricing are still new enough that many older books and articles use them interchangeably.)
15. Dirk Farwood wants to self-publish a cookbook. He guessing it would sell for $10 to $20, but is not sure what selling price to use to maximize his total margin. He decides to experiment by using to equally popular online retailers. On one site he offers a standard version of the cookbook for $10. On the other site he offers a version with glossy photos for $18. (He doubts any customers care about glossy photos, but can use the legitimate difference to avoid claims of unfairness when testing two prices simultaneously.) His wholesale cost for printing on demand is $8 for either version. He sells 400 copies at the lower price and 100 copies at the higher price. Which version made more total margin?
15. The lower price version earned $2 margin amount each × 400 copies = $800 total margin.
The higher price version earned $10 margin amount each × 100 copies = $1,000 total margin.
So the higher price version earned him a greater total margin amount, even though it sold only a quarter as many copies.
Selling multiple versions of an item in different places to test which sells best, as in the previous problem, is called A/B Testing.
There are two ways restaurants define "cost per plate".
Both take into consideration that feeding a large group of people includes many expenses other than what the food costs. In fact, these other expenses (labor for cooking, serving and cleaning, material costs for cleaning before and after the meal, cost of the room, etc.) usually make up more of the meal's cost than the food.
The desired profit method is a version of markup based on wholesale costs.
This method first estimates the other costs as dollar amounts, and sums these costs. Then is uses a scale factor (and the the One Plus Trick) to increase that total to make a profit. As a final step it divides this per-dish selling price by the number of servings.
For most restaurants, the scale factor is 10% to 15%.
Desired Profit Method (for for Cost Per Plate)
Use a scale factor traditionally between 0.10 and 0.15
cost per plate = (food cost + labor cost + other costs) × (1 + scale factor) ÷ servings
If we did not use the one plus trick, the formula would only tell us the profit per plate. But we want the cost per plate that includes both wholesale costs and profit.
16. A restaurant meal that serves four has $32 food cost, $60 labor cost, and $15 other cost. Find the price per plate according to the desired profit method with a 10% desired profit?
16. Use the formula for the desired profit method method.
cost per plate
= (food cost + labor cost + other costs) × scale factor ÷ servings
= ($32 + $60 + $15) × 1.1 ÷ 4
= $29.43
The food cost percentage method is unlike any of the above pricing strategies.
This method starts by estimating that the food costs are 25% to 30% of the total expenses. This percentage is used "backwards" as a scale factor to determine the per-dish selling price. As before, finish by dividing by the number of servings.
Food Cost Percentage Method (for Cost Per Plate)
Use a scale factor traditionally between 0.25 and 0.30
cost per plate = food cost ÷ scale factor ÷ servings
Notice that the scale factor was stated as an amount to scale down the bigger final amount of per-dish selling price (for example, "25% of the per-dish selling price was the food"). But we want to go from the food cost "backwards" to the per-dish selling price. This should remind you of how we used produce yield percent. As before, we divide instead of multiply when using a scale factor "backwards".
17. A restaurant meal that serves four has $32 food cost, $60 labor cost, and $15 other cost. Find the price per plate using to the food cost percentage method with a 30% scale factor.
17. Use the formula for the food cost percentage method.
cost per plate
= food cost ÷ scale factor ÷ servings
= $32 ÷ 0.3 ÷ 4
= $26.67
When an item's selling price is discounted the math looks exactly like margin pricing. The selling price by a certain percentage. To find the remaining amount in one step we use the "One Minus Trick".
Discount
discounted price = old selling price × (1 − discount rate)
Philosophically there is a big distinction. Margin pricing moves down from the selling price to find the maximum affordable wholesale cost. Discount moves down from the selling price to put an item on sale.
But the formulas are only different because of the words we use.
18. A toy originally selling for $80 is put on sale at 15% off. What is the discounted price?
18. discounted price = old selling price × (1 − discount rate) = $80 × (1 − 0.15) = $68
A more complicated situation is a chain discount, when more than one discount applies.
We use the formula for each link in the chain, to consider what remains after each price reduction.
The end result of a chain discount is called the single equivalent discount rate.
19. A toy originally selling for $100 is put on sale at 15% off. A store-wide seasonal sale reduces the selling price by another 20%. Then a coupon cuts the price another 10%. What is the final discounted price?
19. For the first discount discounted price = old selling price × (1 − discount rate) = $100 × (1 − 0.15) = $85.
For the second discount discounted price = old selling price × (1 − discount rate) = $85 × (1 − 0.2) = $68
For the third discount discounted price = old selling price × (1 − discount rate) = $68 × (1 − 0.1) = $61.20
We could solve a discount problem in two steps, by first finding the discount amount and then by subtracting. However, chain discount problems already involve many steps. It feels smoother to use the formula with its "One Minus Trick" to only do half as many steps.
So in the previous problem, it does seem more natural to try somehow combining the original $100 with 0.15, 0.2, and 0.1. Go ahead and try to do the work that way. You will see for yourself why six steps are needed to do the work that way.
Gregg Learning
What about if we know the single equivalent discount rate, but are missing one of the links?
Let's revisit the previous problem but imagine we are missing the coupon's discount rate.
20. A toy originally selling for $100 is put on sale at 15% off. A store-wide seasonal sale reduces the selling price by another 20%. The store expects the toy will only sell if the price is lowered to $61.20. What additional discount rate should be applied with a special coupon?
20. The first two steps are the same as in the previous problem.
For the first discount discounted price = old selling price × (1 − discount rate) = $100 × (1 − 0.15) = $85.
For the second discount discounted price = old selling price × (1 − discount rate) = $85 × (1 − 0.2) = $68
For the third discount we need to work carefully.
discounted price = old selling price × (1 − discount rate)
$61.20 = $68 × (1 − r)
Divide both sides by $68.
0.9 = (1 − r)
So r must be 0.1, which is 10%!
Conquering the Negativity Instinct
The third chapter of Factfulness discusses the straight line instinct. Most trends appear in the short term to be straight lines, but in the long term are not.
The product maturity graph we saw above is an excellent example. During the 1910s and 1920s it must have seemed that everyone would get an automobile, but in fact automobile penetration into society was about to decrease during the 1930s. The penetration of washers and driers was also not at all a straight line. Did these curves suprise the manufacturers of those goods?
After learning about mortgages, we briefly talked about bank safety. Banks give people mortgages because that choice minimizes the risk the loan will default (the bank loses that principal), even though the reward is dramatically smaller than how much that principal could typically earn if invested.
That was just one example of how a problem all businesses face. They need to balance risk and reward.
Furniture stores are famous for having different kinds of payment plans. These plans give the business several options about how much risk they accept.
Layaway Plan No Risk: immediate income while safely storing furniture No Reward: tiny or zero interest rate, downpayment only covers storage costs |
The simplest option is a layaway plan.
After the purchase, the store puts the item in storage. The customer pays a storage fee that covers the cost of this storage. This storage fee is usually an additional cost, not a downpayment on the item.
The customer then makes monthly payments with little or zero interest. Eventually the customer has paid for the item, and finally receives his or her furniture.
This plan has no risk for the store. They are keeping the item safe, and can re-sell the furniture if something goes wrong. Their storage costs are covered.
Installment Plan Small Risk: immediate income and can probably reclaim furniture Small Reward: lowest interest rate, downpayment avoids interest |
The next option is an installment plan.
After the purchase, the customer takes the item home. The customer pays a downpayment.
The customer then makes monthly payments with a small interest rate. Eventually the customer has paid for the item.
This plan has a small risk for the store. The furniture has left the store, and the store might not be able to reclaim and re-sell it if something goes wrong. The downpayment means the customer is probably financially reliable. The downpayment is conveniently immediate income, but also an amount on which the store does not earn interest.
Some stores have a rent to own plan that works in a similar way, with the added benefit that customers might pay more than the value of the furniture!
Zero Down Plan Medium Risk: delay of income, but can probably reclaim furniture Medium Reward: medium interest rate |
The final store option is a zero down plan.
After the purchase, the customer takes the item home. The customer pays no downpayment. In fact, the customer pays nothing for several months.
After those "zero down" months are complete, the customer begins monthly payments with a medium interest rate. Eventually the customer has paid for the item.
This plan has a medium risk for the store. The furniture has left the store, and the store might not be able to reclaim and re-sell it if something goes wrong. The store is out of touch with the customer for several months, which increases risk. The interest rate is the highest the store can get away with charging, and is how the store tries to offset the risk.
Credit Card Use Highest Risk: cannot reclaim furniture Highest Reward: highest interest rate |
The highest risk case happens when a customer pays for the item with a credit card.
This risk is not assumed by the furniture store (which receives its sale price immediately) but by the credit card company.
The credit card company cannot reclaim and re-sell the furniture if something goes wrong. It would instead need to work with a debt collecting agency, which is not cheap. That is why credit cards have higher interest rates than any of the store plans.
Some real life situations require tables. Unfortunately, there is no place for the simple and compound interest formulas to help.
21. Geoffrey Crayon buys a $3,000 computer. To keep his bookkeeping simple, he starts a new credit card that charges 22% annual interest per year (compounded monthly) and will use the card for nothing else. Geoffrey pays $400 per month until the balance is paid off. Finish the table below to find his total interest in dollars.
21. Lots of numbers! Work is done on a Google spreadsheet. The total interest is $193.80.
You can also save your own copy of that spreadsheet and try fiddling with the starting balance, monthly interest rates, or other values to instantly see how the total interest changes.
22. Express Geoffrey's total interest as a percentage of the computer's cost.
22. The percent change was $193.80 part ÷ $3,000 whole ≈ 0.065 = 6.5%
Geoffrey's problem was a bit long and unpleasant.
Unfortunately, charge option problems work the same way.
23. Fendrick wants to buy a new dining room set for $900. He is considering four methods of payment. After looking at his budget as well as his actual expenses for the past few months, he thinks he can save $80 per month towards this purchase. He has four options, described in detail on the table below. How much does each option cost?
23. Lots of numbers! Work is done on a Google spreadsheet.
You can also save your own copy of that spreadsheet and try fiddling with the starting balance, monthly interest rates, or other values to instantly see how the total interest changes.
The layaway plan costs only the $50 storage fee.
The installment plan costs $54.33 interest, and the "cost" of having to somehow get together a $100 downpayment
The zero down plan costs $118.83 interest.
The credit card plan costs $81.75 interest.
24. Which option is best for him?
24. Frederick should use the installment plan if he can get together the $100 downpayment.
The layaway plan is only $4.33 less expensive, definitely not worth waiting so long.
If Fendrick cannot get together the $100 downpayment, he might have to use the layaway plan or resign himself to paying the credit card interest.
The zero down plan has the most interest and should be avoided.
We get to play with more toys. Yay!
Each group of students shares one plastic spinner with a clear base.
We can put a spinner onto a circle like the one below to make a game.
The first circle is a very boring game. Half the time a player wins $1. Half the time a player loses $1.
What we want to focus on is that the game is fair. This means that if someone played it a whole lot—long enough for any rare streaks of good or bad luck to cancel out—then overall they would not gain or lose money.
25. How can we draw lines to make a spinner game fair if winning has twice the gain of losing?
25. Pretend the circle is a pie, and we are cutting the pie into slices.
You could think that the +2 needs to happen half as often as the −1.
Or you could think that the −1 needs to happen twice as often as the +2.
Either way, we should give the −1 two pie slices, but only give the +2 one pie slice.
That is a total of 2 + 1 = 3 pie slices. So we cut the pie into thirds. The result looks like this:
26. How can we draw lines to make a spinner game fair if winning has three times the gain of losing?
26. Pretend the circle is a pie, and we are cutting the pie into slices.
You could think that the +3 needs to happen one-third as often as the −1.
Or you could think that the −1 needs to happen three times as often as the +3.
Either way, we should give the −1 three pie slices, but only give the +3 one pie slice.
That is a total of 3 + 1 = 4 pie slices. So we cut the pie into quarters. The result looks like this:
27. How about this spinner game with three possible outcomes? Can it be made fair?
27. Pretend the circle is a pie, and we are cutting the pie into slices.
If we give the +1 and +2 each one slice of pie, the slice for the +2 counts double. We effectively have three slices of pie for winning +1.
This means we need three slices of pie for the −1.
The two winning numbers each get one slice. The −1 needs three slices.
That is a total of 2 + 3 = 5 pie slices. So we cut the pie into fifths. The result looks like this:
The probability of a situation happening is the ratio of desirable outcomes to total outcomes. (This ratio is often changed into percent format.)
Problems that involve probability almost always involve a bunch of counting. Usually there are no convenient formulas to help us. We need to make lists or tables to count the outcomes.
A classic example of probability is rolling two dice and adding their values.
28. When rolling two dice, what is the probability of the sum being seven?
28. Looking at the green boxes on the chart, we see that six out of thirty-six possibilities have a sum of seven. So the probability is ^{6}⁄_{36}, which we should reduce to ^{1}⁄_{6} or change to about 16.7%.
29. When rolling two dice, what is the probability of the sum being ten?
29. Looking at the pink boxes on the chart, we see that three out of thirty-six possibilities have a sum of seven. So the probability is ^{3}⁄_{36}, which we should reduce to ^{1}⁄_{12} or change to about 8.3%.
Imagine there is a gumball machine with equal amounts of three colors of gumballs: red, green, and blue. The table below shows all twenty-seven possibilities for getting three gumballs.
30. If you get three gumballs, how likely is it to get at least one blue gumball?
30. Nineteen of the twenty-seven possibilities have at least one blue gumball. So the probability is ^{19}/_{27} or about 70%.
The odds of a situation happening is the ratio of desirable outcomes to undesirable outcomes. (This ratio is often reduced, but never changed into percent format.)
31. When rolling two dice, what are the odds of the sum being seven?
31. Looking at the green boxes on the chart, we see that six out of thirty-six possibilities have a sum of seven, and thirty do not. So the odds are 6 to 30, which could be reduced to 1 to 5.
32. When rolling two dice, what are the odds of the sum being ten?
32. Looking at the pink boxes on the chart, we see that three out of thirty-six possibilities have a sum of ten, and thirty-three do not. So the odds are 3 to 33, which could be reduced to 1 to 11.
33. If you get three gumballs, what are the odds of getting at least one blue gumball?
33. The odds of getting at least one blue gumball are 19 to 8, which can be reduced to 2 to 1.
In this class we will always write ratios using the word "to". For example, 1 to 5. Other math books, websites, and real-life contexts might use a colon instead, and write the same ratio 1 : 5.
Math Antics
Unfortunately, there is vagueness about how to measure the probability of a change.
Imagine a medicine that can reduce one of your family member's cancer risk from 44 cases among 10,000 people down to 11 cancer cases among 10,000 people. The medicine has some bad side effects. Is the reduction in cancer risk worth suffering these side effects?
34. We could use subtraction to find the absolute change in risk.
Absolute Change Example
The old risk is 0.44%. The new risk is 0.11%.
0.44% − 0.11% = 0.33%
We could say that with the medication the risk is reduced by 0.33%. (Only a third of one percent? That does not sound like much.)
We could also use a percent change to measure how much less likely is the new risk. Like all percent changes, this is a ratio comparing change to original.
(The funky part of this problem is how the change and original amount are both percentages.)
35. Using a percent change to measure "less likely" is a relative change.
Relative Change (Less Likely)
The old risk is 0.44%. The new risk is 0.11%. Subtracting tells us the change is 0.33%. Then we do change ÷ original.
0.33% ÷ 0.44% = 0.75 = 75%
We could say that with the medication the occurrence of cancer is 75% less likely than before. (That sounds impressive!)
36. We could use a ratio to talk about whether the new risk is as likely as the old risk. Since this is another ratio, it is also called a relative change.
Relative Change (As Likely)
The new risk is 0.11%. The old risk is 0.44%.
0.11% ÷ 0.44% = 0.25 = 25%
We could say that with the medication the occurrence of cancer is only 25% as likely than before. (That still sounds impressive!)
The moral of the story is to pay attention (especially when dealing with small numbers) to whether a speaker is using an absolute change or a relative change. The former made the medicine sound like it is probably not worth the risk of its side effects. The latter made the medicine sound amazing.
The weighted average of a group of situations measures the "average result" of that group.
To find an weighted average, use a table. Each possible outcome is a row. Work across with multiplication: the value for that outcome times its percent probability. Then add those products.
37. When rolling two dice, what is the weighted average?
37. This answer requires making a table, as below. The answer is not surprising. Most people already know that the "average value" when rolling two dice is seven. The expected value table confirms that common knowledge is precise instead of rounded: the expected value is indeed seven exactly, not slightly more or less. (The original Google spreadsheet is here.)
38. Every time little Billy is taken to the grocery store he takes three pennies for this gumball machine. He wants a blue gumball and will spend up to three cents trying to get one. What is the average number of pennies he spends?
38. Here is a sample spreadsheet that shows Billy spends an average of about 2.1¢ each trip to the grocery store.
For many students the most commonly used weighted average table is finding their overall grade in a class.
39. A student has earned the grades below. What is the student's overall grade in the class?
39. Here is a sample spreadsheet that shows the overall grade is 81.9 in the class.
The weighted average is sometimes called the expected value. It does make sense to say "the expected value of the sum of two dice is 7". It almost makes sense to say "the expected value of one of Billy's trips to buy a gumball is 2.1¢." But it does not make sense to call the overall class grade an "expected value" because that situation does not involve condensing mutually exclusive outcomes into an average outcome.
Conquering the Negativity Instinct
The second chapter of Factfulness discusses ways to avoid negativity. Hans Rosling advises us that good situations, especially gradual improvements, are seldom reported. So most news is bad news—and when we hear bad news we should ask both "What good situation was not reported?" and "Is this bad situation, although bad, getting better?"
These online math notes will conclude with two additional attitudes (not from the book) that counteract negativity.
The first attitude is confidence. This word has a special meaning in the setting of personal growth, including any college class.
If we knew a situation would have success, we would have certainty. If we were less sure but had a lot of hope for success, we would have optimism.
If we instead embraced the uncertainty, recognized that life is more complex than success or failure, and acted with the expectation that the situation will be worthwhile for teaching us something about life or ourselves—that is confidence.
Be confident! One lesson of weighted average math problems is that worthwhile situations have many possibilities, and we can consider them all without focusing on success or failure.
In the words of Mark Manson:
Happiness comes from solving problems. The keyword here is "solving." If you're avoiding your problems or feel like you don't have any problems, then you're going to make yourself miserable. If you feel like you have problems that you can't solve, you will likewise make yourself miserable. The secret sauce is in the solving of the problems, not in not having problems in the first place.
When the standard of success becomes merely acting—when any result is regarded and progress and important, when inspiration is seen as a reward rather than a prerequisite—we propel ourselves ahead.
Because here's something that's weird but true: we don't actually know what a positive or negative experience is. Some of the most difficult and stressful moments of our lives also end up being the most formative and motivating. Some of the best and most gratifying experiences of our lives are also the most distracting and demotivating. Don't trust your conception of positive/negative experiences. All that we know for certain is what hurts in the moment and what doesn't. And that's not worth much.
The second attitude is holistic philosophy.
Our earliest understanding of self is based on self-observation and information from authority figures. A girl's parents tell her "You love to dance!" when she was three years old. She had not really bothered to think about it, but they were right.
Cartesian philosophy breaks wholes into parts for understanding. This can allow a different understanding. Dancing involves moving lots of bones and muscles. They joy a dancer feels is part of endorphins and other aspects of brain chemistry.
(But it would be dreadful to assume the Cartesian thought must somehow oppose or debunk the earlier type of thought. No one would say, "Little girl, your dancing is just bones and muscles moving. Your joy is just brain chemistry.")
Systems Theory philosophy views parts in networks. Once the girl who loves dance is a little older she starts to understand how she is part of a family, and a school, and a community, and a dance class, etc.—and what those connections offer her and what she offers to others.
Quantum Mechanical philosophy teaches us to see things as clouds of possibilities. That girl as a young teen wants to grow up to be a dance teacher. She might! But maybe she will also become a writer. That would also be nice. And so on. All those possibilities are a part of her. But not everything is a possibility. She is not going to grow up to be an umbrella, or a shepherd in Alaska.
Modern philosophy notices that Systems Theory philosophy always looks from the outside at a network. If we asked the young woman who loves dancing what she herself thinks about her network, what would she say? What does her network think of her? Then words like "justice", "inspiration" and "integrity" appear that do not have a place in Systems Theory.
All these philosophies can exist together in a holistic and complimentary way. For not only does our fictional young woman see herself in all these ways, but she wants others to see her in all those ways too. She has a certain height and appearance. Her years of ballet have done some harm to her ankles and toes, which affects her today. She is a mother, teacher, and friend. She still has dreams and possibilities. She helps inspire people to better understand themselves and their community.
It is often a challenge to see other people holistically. But doing so—considering their appearance, parts, networks, possibilities, and internal thoughts and points of view—is a key part of treating other people as we treat ourselves.
Be holistic! One lesson of weighted average math problems is that people are a mix many possibilities, and it helps to consider a person as a cloud of possible future versions of themselves.
Watch this video of Ryan Hayashi's coin magic. He is uncertain! He is nervous! His hands shake like crazy! But he is utterly convinced that the situation is worthwhile and meaningful, and has moved beyond thinking about success or failure. And in response, for the first and only time, Penn and Teller tell a magician that they got so drawn into his act, and wrapped up in his confident energy, that they actually forgot to keep analyzing his routine.
Watch Mindwalk for a relxing overview of the evolution of philosophy. Note that the film is from 1990 and thus stops with Systems Theory philosophy.