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A priest asked, "What is Fate, Master?"

And he answered, "It is that which gives a beast of burden its reason for existence. It is that which men in former times had to bear upon their backs. It is that which has caused nations to build byways from City to City upon which carts and coaches pass, and alongside which inns have come to be built to stave off Hunger, Thirst and Weariness."

"And that is Fate?" said the priest.

"Fate...I thought you said Freight," responded the Master.

"That's all right," said the priest. "I wanted to know what Freight was too."

- Kehlog Albran, The Profit

Our third application 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 we study this topic, work on making helpful and organized notes, so you have handy the comments, formulas, and example problems you need.

## Margin and Markup

### A Profitable Vocabulary

#### Cost and Price

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.

#### Margin, Two Ways

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 = m

Next 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

#### Markup, Two Ways

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 = m

Next 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.

#### Markup On...

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.

#### Profit

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.

### Ten Exercises

(no ten exercises for this topic)

### 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.

## Pricing

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.

### Markup Pricing

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.

#### Consistency is Not a Virtue

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 = \$12

Then add the five dollars: \$12 + \$5 = \$17

#### Product Maturity

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

#### Bundling

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.

### Margin 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.

#### Consistency is Not a Virtue

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

#### Product Maturity

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.

#### Bundling

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.

### Entrée Pricing

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.

#### Desired Profit Method

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

#### Food Cost Percentage Method

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

### Discount

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

Single Equivalent Discount

#### Missing Link in a Chain Discount

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%!

### 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. Gavin's tool emporium uses a 40% margin rate. Its most popular item has a \$240 selling price. What is the margin amount?

2. Grace works at a store that uses a 40% markup rate. She orders an item for \$90 wholesale cost. What is the markup amount?

3. Georgina's Vitamin Shop uses a 75% margin rate. It needs to stock a certain bottle of vitamins with a selling price of no more than \$3.50. How much can the shop allow a supplier to charge for this bottle of vitamins?

4. Geoffrey works at a store that uses a 30% markup rate. The wholesale cost of an certain item is at minimum \$22. The store's competitors sell an equivalent item for \$30. Will Geoffrey's store undercut the competition if they stock this item?

5. Galina has a clock that cost her \$62.50. She wants to sell it online for \$102.50, for a profit of \$40. What is the margin rate? What is the markup rate?

6. Grafton works at a sporting goods store, and knows that a certain kind of skis will only sell if it is priced \$109.95 or less. The wholesale cost is \$80. What is the markup rate?

7. A fancy new infant car seat has a skim price of \$220 initially, but the sale price eventually settles at the penetration price of \$150. The wholesale cost is \$67. Divide the larger margin by the smaller margin to find the percentage of extra margin from skimming.

8. A restaurant meal that serves six has \$50 food cost, \$70 labor cost, and \$25 other cost. Find the price per plate using to the desired profit method with a 10% desired profit, and then with the food cost percentage method with a 30% scale factor.

9. The manager of a furniture store knows that a certain table will only sell if the sale price \$250 or less. Currently the sale price is \$275. What percent discount is needed?

10. Giselle works at a candy store. She knows from past years' experience that after Valentine's Day she needs to reduce the prices of the special \$30 chocolate boxes down to \$18 to clear out that inventory. She uses a store-wide sale of 10%, hoping that will attract customers. She also distributes a coupon that further discounts the sale price of just those expensive chocolate boxes. What percent discount is needed on the coupon?

### 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

Conquering the Straight Line 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?

## Charge Options

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.

### Four Different Options

#### 1. Layaway Plan

 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.

#### 2. Installment Plan

 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!

#### 3. Zero Down Plan

 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.

#### 4. Credit Card Use

 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.

### Annoying Tables

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?

1. He could use a layaway plan to avoid interest and pay exactly \$80 per month, but he would prefer to avoid the \$50 storage fee, and does not want to wait almost a year to have his dining room set.
2. He could use the furniture store's installment plan that has a \$100 downpayment and then charges \$73 per month to pay off a 15% annual interest rate.
3. He could use the furniture store's zero down plan that costs nothing for six months and then charges \$180 per month to pay off an 18% annual interest rate..
4. He could open a new credit card with a 22% annual rate and pay off exactly the \$80 per month he budgeted.

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.

### 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 to 7. Raynor starts a new credit card that charges 24% annual interest per year to keep his bookkeeping simple when buying a \$1,499 computer. (He will use the card for nothing else.) The credit card charges him one-twelfth of its annual interest rate each month. Raynor pays \$140 per month until the balance is paid off. Fill in the last parts of the table below.

MonthStartingPaymentInterest Due OnInterestEnding
1\$1,499.00\$140\$1,359.00\$27.18\$1,386.18
2\$1,386.18\$140\$1,246.18\$24.92\$1,271.10
3\$1,271.10\$140\$1,131.10\$22.62\$1,153.72
4\$1,153.72\$140\$1,013.72\$20.27\$1,033.99
5\$1,033.99\$140\$893.99\$17.88\$911.87
6\$911.87\$140\$771.87\$15.44\$787.31
7\$787.31\$140\$647.31\$12.95\$660.26
8\$660.26\$140\$520.26\$10.41\$530.67
9\$530.67\$140\$390.67\$7.81#1
10#1\$140#2#3#4
11#4\$140#5#6#7
12#7#7\$0nonepaid off!

8. Continuing the previous problem, find his total interest in dollars.

9. Continuing the previous problem, find what percentage of his first month's payment was interest.

10. Continuing the previous problem, find what percentage of his tenth month's payment was interest.

### 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.