Percents Part 2 and Error Analysis

Morgan Chase

12 Percents Part 2 and Error Analysis

a sign reading "clearance 399.97 dollars, was 399.99 dollars" — How could anyone pass up that 0.005% discount?

You may use a calculator throughout this module.

Recall: The amount is the answer we get after finding the percent of the original number. The base is the original number, the number we find the percent of. We can call the percent the rate.

When we looked at percents in Module 9, we focused on finding the amount. In this module, we will practice finding the percentage rate and the base.

$\text{Amount}=\text{Rate}\cdot\text{Base}$

$A=R\cdot{B}$

We can translate from words into algebra.

“is” means equals
“of” means multiply
“what” means a variable

Solving Percent Problems: Finding the Rate

Suppose you earned $56$ points on a $60$ -point quiz. To figure out your grade as a percent, you need to answer the question “ $56$ is what percent of $60$ ?” We can translate this sentence into the equation $56=R\cdot60$ .

Exercises

$56$ is what percent of $60$ ?
What percent of $120$ is $45$ ?

Be aware that this method gives us the answer in decimal form and we must move the decimal point to convert the answer to a percent.

Also, if the instructions don’t explicitly tell you how to round your answer, use your best judgment: to the nearest whole percent or nearest tenth of a percent, to two or three significant figures, etc.

Solving Percent Problems: Finding the Base

Suppose you earn $2\%$ cash rewards for the amount you charge on your credit card. If you want to earn $\textdollar50$ in cash rewards, how much do you need to charge on your card? To figure this out, you need to answer the question “ $50$ is $2\%$ of what number?” We can translate this into the equation $50=0.02\cdot{B}$ .

Exercises

$\textdollar50$ is $2\%$ of what dollar value?
$5\%$ of what number is $36$ ?

Solving Percent Problems: Using Proportions

Recall that a percent is a ratio, a fraction out of 100. Instead of translating word for word as we just were, we can set up a proportion with the percentage rate over 100. Because the base is the original amount, it corresponds to 100%.

$\frac{\text{amount}}{\text{base}}=\frac{\text{percent}}{100}$

Let’s try Exercises 1 through 4 again, using proportions.

Exercises

$56$ is what percent of $60$ ?
What percent of $120$ is $45$ ?
$\textdollar50$ is $2\%$ of what dollar value?
$5\%$ of what number is $36$ ?

Now that we have looked at both methods, you are free to use whichever method you prefer: percent equations or proportions.

Exercises

The University of Oregon women’s basketball team made 13 of the 29 three-points shots they attempted during a game against UNC. What percent of their three-point shots did the team make?
A bottle of Dr. Pepper contains 65 grams of added sugars, which is 129% of the recommended daily intake.^[1] What is the recommended daily intake?

Solving Percent Problems: Percent Increase

When a quantity changes, it is often useful to know by what percent it changed. If the price of a candy bar is increased by 50 cents, you might be annoyed because it’s it’s a relatively large percentage of the original price. If the price of a car is increased by 50 cents, though, you wouldn’t care because it’s such a small percentage of the original price.

To find the percent of increase:

Subtract the two numbers to find the amount of increase.
Using this result as the amount and the original number as the base, divide and find the unknown percent.

Notice that we always use the original number for the base, the number that occurred earlier in time. In the case of a percent increase, this is the smaller of the two numbers.

Exercises

The price of a candy bar increased from $\textdollar0.89$ to $\textdollar1.39$ . By what percent did the price increase?
Your author bought an overpriced t-shirt at a Seattle Kraken hockey game with a retail price of $\textdollar40.00$ . Including sales tax, the actual cost was $\textdollar44.04$ . What was the sales tax rate?

Solving Percent Problems: Percent Decrease

Finding the percent decrease in a number is very similar.

To find the percent of decrease:

Subtract the two numbers to find the amount of decrease.
Using this result as the amount and the original number as the base, divide and find the unknown percent.

Again, we always use the original number for the base, the number that occurred earlier in time. For a percent decrease, this is the larger of the two numbers.

Exercises

During a sale, the price of a candy bar was reduced from $\textdollar1.39$ to $\textdollar0.89$ . By what percent did the price decrease?
The estimated population of Portland, Oregon in April 2020 was $652,500$ . The estimated population in July 2022 was $635,100$ . Find the percent of decrease in the population, to the nearest tenth of a percent.^[2]

To summarize, we can determine the percent change using the following formula, which works whether we’re finding a percent increase or a percent decrease.

$\text{percent change}=\frac{|\text{new }-\text{ original}|}{\text{original}}\cdot100$

Relative Error

In Module 5, we said that a measurement will always include some error, no matter how carefully we measure. It can be helpful to consider the size of the error relative to the size of what is being measured. As we saw in the examples above, a difference of $50$ cents is important when we’re pricing candy bars but insignificant when we’re pricing cars. In the same way, an error of an eighth of an inch could be a deal-breaker when you’re trying to fit a screen into a window frame, but an eighth of an inch is insignificant when you’re measuring the length of your garage.

The expected outcome is what the number would be in a perfect world. If a window screen is supposed to be exactly $25$ inches wide, we call this the expected outcome, and we treat it as though it has infinitely many significant digits. In theory, the expected outcome is $25.000000...$

To find the absolute error, we subtract the measurement and the expected outcome. Because we always treat the expected outcome as though it has unlimited significant figures, the absolute error should have the same precision (place value) as the measurement, not the expected outcome.

To find the relative error, we divide the absolute error by the expected outcome. We usually express the relative error as a percent. In fact, the procedure for finding the relative error is identical to the procedures for finding a percent increase or percent decrease!

To find the relative error:

Subtract the two numbers to find the absolute error.
Using the absolute error as the amount and the expected outcome as the base, divide and find the unknown percent.

Exercises

A window screen is measured to be $25\frac{3}{16}$ inches wide instead of the advertised $25$ inches. Determine the relative error, rounded to the nearest tenth of a percent.
The contents of a box of cereal are supposed to weigh $10.8$ ounces, but they are measured at $10.67$ ounces. Determine the relative error, rounded to the nearest tenth of a percent.

The following formula has the same structure as the percent change formula we saw earlier.

$\text{percent error}=\frac{|\text{measured }-\text{ expected}|}{\text{expected}}\cdot100$

Tolerance

The tolerance is the maximum amount that a measurement is allowed to differ from the expected outcome. For example, the U.S. Mint needs its coins to have a consistent size and weight so that they will work in vending machines. A dime (10 cents) weighs $2.268$ grams, with a tolerance of $\pm0.091$ grams.^[3] This tells us that the minimum acceptable weight is $2.268-0.091=2.177$ grams, and the maximum acceptable weight is $2.268+0.091=2.359$ grams. A dime with a weight outside of the range $2.177\leq\text{weight}\leq2.359$ would be unacceptable.

Exercises

A U.S. nickel (5 cents) weighs $5.000$ grams with a tolerance of $\pm0.194$ grams.

Determine the lowest acceptable weight and highest acceptable weight of a nickel.
Determine the relative error of the weight of a nickel with an absolute error of $0.194$ grams.

A U.S. quarter (25 cents) weighs $5.670$ grams with a tolerance of $\pm0.227$ grams.

Determine the lowest acceptable weight and highest acceptable weight of a quarter.
Determine the relative error of the weight of a quarter with an absolute error of $0.227$ grams.

Exercise Answers

Source (PDF file): https://www.fda.gov/media/135299/download ↵
Source: https://www.census.gov/quickfacts/fact/table/portlandcityoregon/PST045222 ↵
Sources: https://www.usmint.gov/learn/coin-and-medal-programs/coin-specifications and https://www.thesprucecrafts.com/how-much-do-coins-weigh-4171330 ↵

License

Icon for the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License