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Math Lecture Notes
Percent Facts

Mortgages link to here link to index

amortization table

Amortization is the paying off of debt with repeating, schedules repayments.

Imagine that you had a home or car loan, and needed to make monthly payments over many years. What would be the proper monthly payment amount? This problem is tricky, because the debt shrinks with payments and grows with interest at the same time. But accountants have solved the problem many times, and saved the answer as an amortization table.

In this class we will use the amortization table to the right. In real life, the amortization table is huge. It has more rows, for interest rates with decimals (usually to the hundredths place). It has more columns, for loans with other durations of months and/or years.

The table values show us the correct monthly payment for a $1,000 loan with that interest rate and duration. Multiply this number to scale it up by how many thousands are in the actual loan amount. (For example, if the mortgage is $130,000 then multiply the table value by 130.)

Yield Percent link to here link to index

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

Most recipe scaling is pretty obvious. Use scale factors to adjust the ingredient amounts.

For example, if you are tripling the recipe, use three times as much of each ingredient. Or to halve the recipe, use half as much of each ingredient.

But one aspect of making a very large recipe is often overlooked by people without culinary experience. For large recipes the amount of produce needed to purchase is greater than the amount mentioned in the recipe. This scale factor is called Yield Percent.

For example, if you buy 10 pounds of potatoes, after you trim and peel them there is only (on average) 7.8 pounds of usable potato for a recipe that requires peeled potato. As another example, 10 pounds of cherries, after pitting, results in about 8.2 pounds of usable cherries.

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

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

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

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

Notice that yield percent is a scale factor that scales down from the amount actually purchased. It answers the question, "How much of this produce item makes it into the meal?" To answer that question we would use the scae factor normally and multiply. But much more common in real life is to instead answer the question, "If we want a certain amount of this produce item in the recipe, how much do we actually shop for?" Then we must work backwards (scale up instead of scale down) and divide by the yield percent.

On normal days, most people cooking at home ignore the yield percent step. They begin their meal planning by considering what is in the refrigerator rather than making a shopping list. However, if you were planning a big recipe (perhaps something for a Thanksgiving dinner for 20 people, or for a thank-you meal at a non-profit for 50 people) then you would shop specifically for that meal and not want to be surprised that you bought too few potatoes!

Three Kinds of Probability link to here link to index

Probability

The probability of a situation happening is the ratio of desirable outcomes to total outcomes. (This ratio is often changed into percent format.)

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.

table of possibilities for three gumballs of three colors

Example of Probability

How likely is it to get at least one blue gumball?

Nineteen of the twenty-seven possibilities have at least one blue gumball. So the probability is 19/27 or about 70%.

Odds

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.)

Example of Odds

What are the odds of getting at least one blue gumball?

The odds of getting at least one blue gumball are "19 to 8", which can be reduced to "2 to 1".

Expected Value

The expected value of a group of situations measures the "average result" of that group.

To find an expected value, 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.

Example of Expected Value

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?

Here is a sample spreadsheet that shows Billy spends an average of about 2.1¢ each trip to the grocery store.

table to find expected value for gumballs

For many students the most commonly used expected value table is finding their overall grade in a class, often again with a spreadsheet.

table to find expected value for a grade