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Math Lecture Notes
Pricing

In-Class Activity link to here link to index

Charge Options

We have already discussed what interest is. We learned how to use the formulas for simple interest, compound interest, and sum of annuity due.

Unfortunately, the situations that these formulas describe are unrealistic. Most of the time loans do not begin with an initial amount (principal) and then nothing else happens as interest grows. In real life people have to make monthly payments for morgages and credit cards. Saving for retirement seldom involves repeatedly contributing exactly the same amount. And so on.

In other words the formulas are normally tools we use to in parts of a more complex situation. They are not usually answers all by themselves.

Remember the table we made about six years of increasing rent? Some real life situations require tables. There is no place for the simple and compound interest formulas to help.

1. 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.

computer pay for credit card problem chart

1. 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.

2. Express Geoffrey's total interest as a percentage of the computer's cost.

2. The percent change was $193.80 ÷ $3,000 ≈ 0.65 = 6.5%

Geoffrey's problem was a bit long and unpleasant.

Unfortunately, real life can be even more complicated! Here is a practical situation that involves interest in subtle ways.

3. 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. Which option is best for him?

list of charge options

3. 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.

Saving up to pay cash up front is of course the least expensive. It has $0 interest! But that might not be what makes Fendrick happiest.

If Fendrick can get together the $100 downpayment then the installment plan is the best way to get the dining room set today. The total interest is only $54.33

If Fendrick cannot get together the $100 downpayment then he pays significantly less total interest for the credit card ($81.75) than the zero down plan ($118.83).

Facts link to here link to index

Markup and Discount

Markup

examples of markup

Fortunately, retail markup is a straightforward application of percent sentences. We can even save a step with the one plus trick.

Unfortunately, retail markup is also confusing. This is because stores disagree about which value (wholesale cost versus retail selling price) is the "original" amount.

The information provided must match, unless we get involved with more algebra than is used in this class. A store that bases markup on wholesale cost must also know the item's wholesale cost. A store that bases markup on retail selling price must also know the item's retail selling price.

First consider the stores that use markup on wholesale cost. In their mindset, they purchase their goods at an "original" wholesale cost, and then work to use enough markup to earn an acceptable profit.

Markup on Wholesale Cost Formulas

markup in dollars = wholesale cost × markup rate

retail selling price = wholesale cost × (1 + markup rate)

4. A store uses a 40% markup on wholesale cost. What is the markup in dollars for an item with a wholesale cost of $100? What is the retail selling price of that item?

4. In this case, markup answers the question "What is 40% of the wholesale cost?"

markup in dollars = wholesale cost × markup rate = $100 × 0.4 = $40

retail selling price = wholesale cost × (1 + markup rate) = $100 + (1 + 0.4) = $140

Next consider the stores that use markup on selling price. In their mindset, they look how to price goods competitively and then from that "original" competitive retail price work to find an affordable wholesale supplier.

Markup on Selling Price Formulas

markup in dollars = retail selling price × markup rate

wholesale cost = retail selling price × (1 − markup rate)

5. Another store uses a 40% markup on selling price. What is the markup in dollars for an item with a retail selling price of $100? What is the wholesale cost of that item?

5. In this case, markup answers the question "What is 40% of the retail price?"

markup in dollars = retail selling price × markup rate = $100 × 0.4 = $40

retail selling price = retail selling price × (1 − markup rate) = $100 × (1 − 0.4) = $60

Notice that the markup was the same dollar amount in both examples. In one case we added the markup to the wholesale cost. In the other example we subtracted the markup from the retail price. We were using two different kinds of markup appropriately. First we were told the wholesale cost and that the markup is based on that wholesale cost. Second we were told the retail selling price and that the markup is based on that retail selling price.

Remember that our math does not work if we try to use markup inappropriately. Consider the problem "Sam works at a jewelry store that uses a 40% markup on selling price. His store ordered a necklace for $1,250. For how much should he price the item?" None of our formulas apply, since we know the wholesale cost but are asked to use a markup on selling price.

Discount

The use of retail discount is easier because there is no confusion about what value is the "original" amount. The original price is the older, before the sale, more expense one.

Discount is again a straightforward application of percent sentences, and we can save a step with the one plus trick.

6. A toy originally selling for $80 is put on sale at 15% off. What is the new cost? Find the discount, then subtract it from the original cost.

6. We ask, "What is 15% of $80?"

y = 0.15 × $80 original = $12 discount

So the new cost is $80 original − $12 discount = $68 new cost

Because we are subtracting to find a smaller number, we change the one plus trick into a "one minus trick".

7. A toy originally selling for $80 is put on sale at 15% off. What is the new cost? Find the new cost in one step using the "one minus trick".

7. We ask, "What is (100 % − 15%) of $80?"

y = (1 − 0.15) × $80 original = 0.85 × $80 original = $68 new cost

A more complicated situation is a chain discount, when more than one discount applies.

We use the "one minus trick" 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.

8. A toy originally selling for $80 is put on sale at 15% off. Then that selling price is reduced another 20%. Then a coupon cuts the price another 10%. What is the final cost?

8. We change each of the three discounts into a percentage remaining after that discount.

15% off = 85% remaining

20% off = 80% remaining

10% off = 90% remaining

Multiply the original cost by all three of these:

$80 original × 0.85 × 0.8 × 0.9 = $48.96 new cost

(In the previous example problem, it does seem more natural to try somehow combining the original $80 with 0.15, 0.2, and 0.1. Go ahead and try to do the work that way; you will see for yourself why it does not work unless you break the problem into an annoying number of small steps.)

Formulas link to here link to index

Entrée Pricing Methods

restaurant inventory software

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.

Food Cost Percentage Method

The food cost percentage method estimates that the food costs are 25% to 30% of the total expenses. This percentage is used as a scale factor to determine the cost per plate.

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 (for example, "of the total cost 25% was the food"). But we want to use it "backwards". So, as with the produce yield percent, we divide instead of multiply.

9. 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.

9. Use the formula for the food cost percentage method.
cost per plate
= food cost ÷ scale factor ÷ servings
= $32 ÷ 0.3 ÷ 4
= $26.67

Desired Profit Method

The desired profit method estimates the other costs as dollar amounts, sums these costs, and then uses a scale factor to increase that total to make a profit. Traditionally the profit is 10% to 15%. Because we want the total cost per plate (not merely the profit per plate) we use the "one plus trick".

Desired Profit Method (for for Cost Per Plate)

Use a scale factor traditionally between 1.10 and 1.15

cost per plate = (food cost + labor cost + other costs) × scale factor ÷ servings

In this case we are using a scale factor originally intended to scale up, so we multiply by it as usual. If we did not increase the scale factor using the one plus trick the formula would only tell us the profit per plate.

10. Using those same costs, find the price per plate according to the desired profit method with a 10% desired profit?

10. 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

Homework link to here link to index

More Practice

pricing

22. 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 food cost percentage method with a 30% scale factor, and then with the desired profit method with a 10% desired profit.

22. For the food cost percentage method, the cost per plate = food cost ÷ scale factor ÷ servings = $50 ÷ 0.3 ÷ 6 = $27.78 For the desired profit method, the cost per plate = (food cost + labor cost + other costs) × scale factor ÷ servings = ($50 + $70 + $25) × 1.1 ÷ 6 = $26.58

23. Grace works at a store that uses a 40% markup on wholesale cost. She orders an item for $90. What will the markup be in dollars?

24. Grafton works at a store that uses a 30% markup on wholesale cost. He orders an item for $200. For how much should he price the item?

25. Gavin sees that an item whose selling price is $240 has a 40% markup on selling price. What is the dollar amount of this markup?

26. Georgina works at a vitamin store that uses a 75% markup on selling price. She needs to stock a certain bottle of vitamins for no more than $3.50. How much can she allow a supplier to charge her store for this bottle of vitamins?

27. 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 markup rate if measured as a markup on wholesale cost? What is the markup rate if measured as a markup on selling price?

28. Ginger 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. Currently the price is $120. What percent discount is needed?

29. Grant buys a jacket. What a great deal! It normally cost $275, but there was a store-wide 20% off sale and he was also able to use coupon to reduce that sale price another 15%. How much did he pay for the jacket?

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

23. For a markup on wholesale cost, markup in dollars = wholesale cost × markup rate = $90 × 0.4 = $36.

24. For a markup on wholesale cost, retail selling price = wholesale cost × (1 + markup rate) = $200 × 1.3 = $260.

25. For a markup on selling price, markup in dollars = retail selling price × markup rate = $240 × 0.4 = $96.

26. For a markup on selling price, wholesale cost = retail selling price × (1 − markup rate) = $3.50 × (1 − 0.75) = $3.50 × 0.25 = $0.87.

27. The markup rate if measured as a increase from the cost is $40 ÷ $62.50 = 0.64 = 64%. The markup rate if measured as a decrease from the selling price is $40 ÷ $102.50 = 0.39 = 39%.

28. The change is $120 − $109.95 = $10.05. So the discount rate is $10.05 ÷ $120 = 0.08375 ≈ 8%.

29. The final price is $275 × (1 − 0.2) × (1 &miuns; 0.15) = $187.

30. The discount chain equation is $30 × (1 − 0.1) × r = $18. Combine the first two numbers to get $27 × r = $18. Divide both sides by 27 to get r = 67%. Since that is what should remains after the coupon decreases the price, the discount should be 100% − 67% = 33%.

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Homework

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