Math 20 Math 25 Math Tips davidvs.net

# Math Lecture NotesPercent Formulas

## Restaurant Entrée Pricing Methods

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.

### 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.

## Retail 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.

Some stores 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.

Others stores 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 Wholesale Cost Formulas

markup in dollars = wholesale cost × markup rate

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

Markup on Selling Price Formulas

markup in dollars = retail selling price × markup rate

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

Example 1 (Markup on Wholesale Cost)

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?

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

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

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

Example 2 (Markup on Selling Price)

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?

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

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

Notice that the markup was the same 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.

The 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. There are formulas for those mis-matched situations, but understanding them requires more algebra than is involved in this class.

## Retail 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.

Example 3 (Discount Using Two Steps)

A toy originally selling for \$80 is put on sale at 15% off. What is the new cost?

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

y = 0.15 × \$80 = \$12

So the new cost is \$80 − \$12 = \$68

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

Example 4 (Discount Using the "One Minus Trick")

A toy originally selling for \$80 is put on sale at 15% off. What is the new cost?

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

y = (1 − 0.15) × \$80 = 0.85 × \$80 = \$68

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.

Example 5 (Finding the Single Equivalent Discount Rate)

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?

15% off = 85% remaining.

20% off = 80% remaining.

10% off = 90% remaining.

Multiply \$80 × 0.85 × 0.8 × 0.9 = \$48.96

(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.)

What about if we know the single equivalent discount rate, but are missing one of the links?

Example 6 (Starting with the Single Equivalent Discount Rate)

A toy originally selling for \$100 is put on sale at 15% off. A store-wide seasonal sale cuts the price another 10%. The store expects the toy will only sell if the price is lowered to \$60. What additional discount should be applied with a special coupon?

\$100 × 0.85 × 0.9 × r = \$60

\$76.5 × r = \$60

r ≈ 0.78 = 78%

So the missing link has 78% remaining.

The store should apply 100% − 78% = 22% more discount with the coupon.

## Interest

### Simple Interest

Simple interest was a Math 20 topic, discussed here.

The most common real-life examples of simple interest are payday loans, bank certificates of deposits, and bond investments.

Here is an interesting review problem.

Example 7

Scrooge McDuck puts one million dollars into an investment that appreciates 3% its first year. The second year the investment depreciates by 3%. Scrooge is sad, and thinks he is financially back where he started.

When his account statement arrives in the mail he is surprised to see that he has less money than he started with! What happened?

### Compound Interest

Consider an apartment whose monthly rent starts at \$500, and every year it increases by 5%. What is the rent after six years?

What was the overall percent change? Why wasn't the overall percent change equal to 6 years × 5% = 30%?

We can simplify our table using the one plus trick.

The English word "compound" means two different things. It can refer to an item made up of multiple and different parts (for example, a medicine that is a compound of ingredients). It can also refer to an item made greater by a certain type of thing (for example, strong winds compounding the difficulty of putting out a forest fire, or young siblings whose tendency to get into trouble is compounded when their cousins visit).

Compound interest uses the word "compound" in this second sense. The situation for compound interest involves only one principal amount and then watching this change over time. However, the investment grows at a faster rate than with simple interest because the interest earned in one payout will itself earn more interest in future payouts.

Here is a sample monthly compound interest spreadsheet. Why is the annual interest rate of 22% divided by twelve?

Remember that rounding in the middle of a math problem usually creates an incorrect answer. The table above uses one-twelfth of the annual interest rate as a decimal and does not round this number! The calculations would be wrong if it did. (Try it!)

Compound interest tables can transmogrify into a compound interest formula.

The Compound Interest Formula

Final Amount = Principal × (1 + interest rate per payout)number of payouts

Remember that when you are introduced to a new formula you should practice solving for any of its variables. Unfortunately, solving for the interest rate or the number of payouts requires knowing how to undo an exponent. That makes those tasks inappropriate for this class. (It is not actually harder—it merely uses a different calculator key.) As an official part of our curriculum we only can solve for the final amount or the principal.

Also, be alert. (Snakes don't eat lerts.) Most real-life compound interest with bank accounts and credit cards happens monthly. But there are other options used by stocks, bonds, and certificates of deposit. Complete this chart to explore the others.

NameMeaningPayouts per Year
annuallyonce per year
semiannuallytwice per year
quarterlyfour times per year
monthlyonce per month
semimonthlytwice per month
biweeklyevery other week
weeklyonce per week

Most real-life financial fees occur monthly even though the account is open much longer. Examples include bank account interest, credit card interest, mortgage payments, and vehicle loan payments.

However, in real life the formula seldom applies because no one makes a single initial deposit/payment and then passively watches the interest grow. We normally make deposits and withdrawls to bank accounts, at least make minimum payments oncredit cards, and make the required monthly mortgage and vehicle loan payments.

### Annual Effective Interest

A special case of the compound interest formula happens when the principal is \$1, the time for each payout is less than a year, and the number of payouts exactly fits a single year. This situation would measure how much compounding happens each year.

In other words, there really is interest compounding during that year, but we are going to "summarize" the increase as if it were simple interest. Accountants call this the annual effective interest rate.

We can write the normal compound interest formula for this situation.

Final Amount = \$1 × (1 + interest rate per payout)number of payouts

Then we make two changes. First, multiplying by \$1 does nothing, so we can leave that out. Second, we are really looking for the interest, not the final amount, so we subtract the \$1 principal at the end.

The Annual Effective Interest Rate Formula

The number of payouts must fit a single year.

Annual Effective Interest Rate = (1 + interest rate per payout)number of payouts − 1

### Sum of Annuity Due

Similar, but bigger, is the sum of annuity due formula that must be useful for something. As written below it assumes one deposit and payout per year, so the annual interest rate needs no modification using division to find a "per payout" rate.

The Sum of Annuity Due Formula

Final Amount = [ Principal × (1 + rate)(years + 1)    −    Principal × (1 + rate)] ÷ rate

How would you modify the sum of annuity due formula so it considered months instead of years?

## Changes in Likelihood

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?

We could use subtraction to find the absolute change in risk.

Example 8

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.)

Using a percent change to measure "less likely" is a relative change.

Example 9

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!)

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.

Example 10

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.