Mastering Percents: A Quantitative Literacy Exploration

Sick Days at Work: Funny Business -- or Ordinary Math?
"One of the classic jokes in the workplace is the fact that 40% of sick days are taken on Mondays and Fridays. The comic strip Dilbert used this to launch a series in which the boss sets out to solve this problem. Of course, calmer heads will also notice that 40% of the work gets done on Mondays and Fridays, since those two days account for 40% of the work week.
"This is one example of innumeracy, or sleight of number. By stating that "40% of sick days are being taken on Mondays and Fridays" we create the impression that this is somehow abnormal, and it is easy to avoid checking the facts before reacting. Thinking about numbers before reacting. Not falling prey to scams based on numbers....that's what [quantitative literacy] is all about.." [esmerel.com/circle/numeracy/sleight.html]

I. Percents: Definition, Meaning, and Conversion
The word percent comes from Latin meaning "for every hundred". Thus, "25 percent" would read "25 per 100" ...which can be written as a single fraction 25÷100 ...and which can then also be converted into decimal form: 0.25 Examples: Convert 1/5, 1.4, and 0.002 into percents. 1/5 = (1) %divide; (5) = (1·20)÷(5·20) = (20)÷(100) which is "20 out of 100" = 20% but also equal to 0.2 1.4 = 140÷100 = 140% 0.002 = (0.02)÷(10) = (0.2)÷(100) = 0.2% Exercise: Convert 5/8 into percent.

I. Percents: Definition, Meaning, and Conversion

The word percent comes from Latin meaning "for every hundred".

Thus, "25 percent" would read "25 per 100"

...which can be written as a single fraction 25÷100

...and which can then also be converted into decimal form: 0.25

Examples: Convert 1/5, 1.4, and 0.002 into percents.

1/5 = (1) %divide; (5) = (1·20)÷(5·20) = (20)÷(100) which is "20 out of 100" = 20% but also equal to 0.2

1.4 = 140÷100 = 140%

0.002 = (0.02)÷(10) = (0.2)÷(100) = 0.2%

Exercise: Convert 5/8 into percent.

II. "Percent of" Situations

To ask what "15 percent of" some given number equals, is to ask what portion of that given number has the same ratio (the portion to the given number) as 15 has to 100.

For instance, to ask "what is 20% of 240?" is to ask what number is (20/100)th of 240, which is the same as asking what number is (1/5)th of 240, which we can find by multiplying (1/5)· 240 = 48.

Notice that in a "percent of" statement there are three numbers: the part the whole, and the percent. For example above we can say that "20% of 240 is 48" or equivalently, that "48 is 20% of 240" and in either case, 48 is the part, 240 is the whole, and 20% is of course the percent.

Below we will review the three main types of "percent of" problems following which you will be guided (or reminded) on how to solve each of them both mechanically (which may be most familiar) and conceptually (which may be less familiar to you).

"Mechanically" finding "percent of": This includes methods with which you are most familiar: "just show me the formula to use to punch in on my calculator". Such mechanical methods are sometimes the fastest route to the "final answer," but they have two serious drawbacks:

First, mechanical methods are error prone: mistakes in either memorizing the procedure, mistakes in knowing which of several procedures to use, and mistakes about where in a formula a number "goes" (or slips of the finger on a calculator) are surprisingly common.
Second, the workplace of the 21st century requires workers who can not only "get the answer" but understand and be able to communicate what they are doing, how and why it works, and supporting their claims to a diverse audience. A related problem with relying exclusively on mechanical methods is that people in such situations often have real difficulties solving problems which are quite similar but whose "set up" is different.

Keep this in mind if the conceptual methods below seem, at first, "the longer way" -- they are actually fun to play with mentally -- once you get the hang of them. And there isn't one single "conceptual method" but rather many conceptual methods. However they are all, by definition, meaning-based: we look for one or more ways of understanding the meaning behind the numbers and behind the question posed by the problem or situation, seeking a short-cut or a way of breaking up the problem into several easier-to-perform and, crucially, easier-to-mentally-check (verity) steps, so that we are less likely to have an error in the final answer.

To contrast these two methods let's take a very simple example: "Find 20% of 40"

The mechanical approach is to punch in (20/100)·40 into a calculator and to get (0.2)·(40) with a final answer of 8.

A conceptual approach might be based on your noticing that 20% is twice as much as 10%, and that 10% of 40 is 1/10th of 40 (which would be 4) and thus the answer must be two times 4 or 8. You would, in fact, keep in mind the more general fact that 10%of any number is 1/10th that number and thus that 20% of any number is not only "one-fifth" but easily computed mentally by "take 1/10th and then double." For example we can write:

(20% of 40) = 2 · (10% of 40) = 2 · (4) = 8

Other advantages of conceptual approaches are that anyone trying to follow your work can (if you show your steps) follow, understand, and thus also check and verify your results. And it's easier for you to catch if your answer is reasonable or unreasonable.

Exercise: Use a conceptual method of your choosing (and no calculator) to find 60% of 90, by writing a multi-step equation similar to the above example, e.g. using the fact that 50% of any number is half that number.

(60% of 90) = (50% of 90) + (10% of 90) =__________________________________

Exercise: Use similar conceptual methods to find 60% of 84 (Recall that to find 1/10th of a number, one just moves the decimal one position tot he left; for example, 1/10th of 73 would be 7.3). Show several equal signs and steps:

(60% of 84) =_____________________________________________

Exercise Using conceptual methods to find 98% of 120 (Suggestion: use that 98% = 100% - 1% - 1%)

(98% of 120) =_____________________________________________

Exercise Without finding out the answer, explain which two different methods (both conceptual) might be used for finding 80% of 130. (Can you think of more ways than two?)

_____________________________________________________________________________________

III. The other two types of "percent of" situations

You have just seen and used multiple types of conceptual (as well as the basic mechanical) ways for finding the part when we are given the whole and the percent. The other two types of "percent of" question are:

→ B. Finding the percent when given the part and the whole.

→C. Finding the whole when given the part and the percent.

To perform B mechanically, one needs to first divide "Part÷Whole" and to then also convert the (decimal) answer into a percent. However students and in fact most citizens regularly make mistakes with this method (or mistype a number into a calculator) and sadly then have no idea the answer is wrong (even if "way off" due to lacking any conceptual mental compass as to roughly what the answer might be. As seen above, conceptual approaches let one use mental "stepping stones," allowing one to obtain the answer in several steps, each one of which is fairly easy/convenient, so that throughout the process as well as at the end, one can check that one's work and final answer make good sense.

Example: 12 is what percent of 48? (Equivalently, "what percent of 48 is 12?)

Conceptual Solution: We notice that if 12 is P% of 48, then 6 is P% (that is, the same percent) of 24 -- why?, and likewise 3 is P% of 12. But 3 is one-quarter or 12, so 3 is 25% of 12. Check that 12 is 25% of 48.

Similarly, to solve "12 is P% of 72" is same as solving "6 is P% of 36" by dividing by 2 since the proportions are unchanged. Similarly by dividing by three we see that this problems in turn has the same solution as "2 is P% of 12" And since 2 is 1/6 of 12, P must be 1/6 or 0.1666... which is 16.666... ("16 point repeating 6") percent, or 16 and 2/3 of one percent.

A different but related conceptual method lets us solve "12 is P% of 40" by noticing that our P is three times bigger than the answer to "4 is Q% of 40" (that is P=3·Q so we'll know P once we find Q), since 12 is three times bigger than 4. However "4 is Q% of 40" is easy to solve; clearly Q% is 10%. Thus P% is 30%. (Check this!)

Figuring "percent of" problems conceptually: Some practice problems. Show all steps and explain your reasoning carefully for solving:

12 is ____% of 50.
____ is 60% of 20
12 is 15% of __80__
________________________________________________________________(continues on next page)
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________

Bonus Question: is 20% of 80 equal to 80% of 20?

What about if we use other numbers? Investigate and then explain fully and clearly what's going on. Show an example where it does not work, or else prove it always works

IV. Percents over 100

The key thing to keep in mind is the basic fact stated at the beginning: namely, that "per cent" means "per 100"

Thus, "15 is P% of 5" can be found conceptually as 15 = 3·5 = (300/100)·5 = ("300 per 100" of 5) = 300% of 5. Or break up 15 = 5 + 5 + 5 = (100% of 5) + (100% of 5) + (100% of 5). Similarly, in "15 is P% of 10" we can find P conceptually as follows:

15 = 10 + 5 = (100% of 10) + (50% of 10) = 150% of 10.

Solve:

"45 is P% of 20"
"35 is P% of 20" (Hint: 35 = 20 + 10 + 5)
33 is Q% of 18 (hint: write 33 = 18 + ...)

V. Percentage Increase/Decrease Problems

To increase a number A by P percent we just add P percent of A to A, and similarly to decrease by a certain percent, we subtract the "percent of" from the original number, so:

(The number obtained by increasing A by P%) = A + (P% of A)
&
(The number obtained by decreasing A by P%) = A - (P% of A)

Thus any "increase by P%" or "decrease by P%" situation contains a "percent of" problem within it, which we just add to (or subtract from) the original number to find the final answer. There are three types of problems:

Type 1: Given an initial amount and a percent it increases (or decreases), to find the final amount. Example: If a $25 item is marked 20% off, what would the sale price be?
Type 2: Given the initial and final amounts to find the percent the initial amount increased (or decreased) by to arrive at the final amount. Example: If the price changed from $2 per gallon to $3.50 per gallon, what percent increase would that be?
Type 3: Given a final amount and a percent-increase (or percent-decrease) that an initial amount underwent, find the initial amount. Example: Given the sale price is 90% and that everything is 10% off, what was the original amount?

Exercises:

If we increase 20 by 25% what would we get?
If we increase 25 by 20% what would we get?
If we decrease 25 by 20% what would we get?
If we decrease 25 by 40% what would we get?
If we increase 24 by 20% what would we get?
If we decrease 24 by 20% what would we get?
$A increased by 33 and 1/3 percent is 16. What is A?
The old price of $2 became $5.50 what was the percentage increase?
In pairs/teams, each person creates several problems of Type 1 and at least one problem of Type 2 and Type 3, respectively -- then have a teammate solve it. Or, on your own create, and then solve at least one problem of each type, and doing several of Type 1. Can you solve them conceptually?
Mathematically analyze and explain the following cartoon: Explanation:_______________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

A common mistake:

One article noted that world population increased from about one billion near the beginning of the industrial revolution, to 6 billion people in 2000 and (mistakenly) referred to this as "an increase of 600%" Let's use smaller numbers like 1 and 6 to see why this is a mistake.

In fact, 6 is "600% of 1" since 6÷1 = 6 = 600/100 = 600%

However we only need to increase 1 by 500% to get:

(1, after being increased by 500%) = 1 + (500% of 1) = 1 + 5 = 6.

Question: To make a number 3 times bigger we must increase it by ___%

To make a number 7 times bigger we must increase it by ___%

VI. Percents Versus Percentage Points

Consider the following scenario:

Suppose Maryland has a population of 6,000,000 and that two years ago, 20% of Maryland residents went on a trip to the beach in Ocean City during May, and that last year, instead, 30% of Maryland residents went to the beach in Ocean City during May. Quantify the increase in beach attendance during May in terms of percents.

It may be tempting to think that there was a 10% increase in beach attendance, however this is not the case, as the following calculation shows:

Two Years Ago Last Year

Percent who went to beach in May 20% 30%

Number of people who
went to beach in May 1,200,000 1,800,000

	Two Years Ago	Last Year
Percent who went to beach in May	20%	30%
Number of people who went to beach in May	1,200,000	1,800,000

The percent increase from 1,200,000 to 1,800,000 can be calculated directly:

1,800,000 = 1,200,000 + 600,000 = (100% of 1,200,000) + (50% of 1,200,00)

Thus we find that (although 1,800,000 is 150% of 1,200,000 in relative size) we had an increase by (an additional) 50% [Alternatively, (1,800,000)÷(1,200,000) = 1.5 = 1 + 0.5 = 100% + 50% so a 50% increase over the original amount].

Thus, the number of people going to the beach in May increased by 50%. What about the "10 percent" difference between the original "20%" and the new "30%" though? This is called a difference of 10 percentage points -- that-- is, when the increase was from 20% attendance to 30% attendance.

Notice that each of these two concepts does have it's own separate real-world meaning. How many additional people went to the beach in May last year, versus two years ago? An extra 600,000 people went. This 600,000 represents 10% of Maryland's population. Thus when late-spring beach-going became more popular, an additional 10% of Marylanders went to the beach; this is the "10 percentage points" by which beach-going went up. But that's not the percentage increase! After all the new figure (1,800,000) is the same as the old figure (1,200,000) plus an additional HALF of the old figure, so there was, indeed, a 50% increase in the number of people who attended the beach in May.

Exercise: Suppose sales tax used to be 5% and is now 7%. Describe the increase in sales tax, in full sentences and also showing all your steps, in both "percent increase" and "percentage points" increase. What is the real-world significance of each?

Challenge: Look for similar problems in various real-world situations. For example, http://quickfacts.census.gov/qfd/states/24000.html notes that in 2000 about 80% of Americans over 25 had graduated from high school, and that about 24% of Americans over 25 had a Bachelor's degree or higher. The figures were about 84% and 31%, respectively, for Maryland. Could you compare what percentage of those who graduated from high school would make it to graduating from college, in Maryland versus in the US? What assumptions might you be making? Do these numbers tell the whole story? This is just one of many possible open-ended questions you now have the tools to investigate.

Challenge (open-ended exploration): If we take your salary, increase it by 10%, and then decrease the resulting modified salary by 10%, will be get back to the original salary? What if we decrease by 10% first, and then increase by 10%? What if we use other percents (other than 10%)? Investigate this, and use algebra to explain some of the "why" [Reminder: to increase a number N by P% we can multiply N by (1 + (P÷100)) and to decrease a number N by P% we can multiply N by (1 - (P÷100))].

Summary Of Key Points

Don't confuse "percentage of" situations with percent increase/decrease situations
Avoid pitfalls associated with percents greater than 100
When the numbers you are starting with are, themselves, percents, don't confuse "percentage of" nor percentage increase/decrease by, with how many percentage points this increase/decrease represented.
Always go back to the meaning of the underlying situation in mind and keep also in mind that conceptual methods can help you sort out the above pitfalls, and to avoid mistakes and "lies told by my calculator"

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Created 2000-2006 by Dr. Harel Barzilai. Some rights reserved. Reproduction and use with attribution for noncommercial educational purposes is permitted and encouraged. Details at http://creativecommons.org/licenses/by-nc-sa/2.0/

Acknowledgement: Part of this work was made possible with kind support (Fall 2005-Spring 2006) from the Professional Development Mentors Program (PDMP) at Salisbury University, a collaboration of the Faculty Development and the Learning Technology committees.

	Created 2000-2006 by Dr. Harel Barzilai. Some rights reserved. Reproduction and use with attribution for noncommercial educational purposes is permitted and encouraged. Details at `http://creativecommons.org/licenses/by-nc-sa/2.0/` Acknowledgement: Part of this work was made possible with kind support (Fall 2005-Spring 2006) from the Professional Development Mentors Program (PDMP) at Salisbury University, a collaboration of the Faculty Development and the Learning Technology committees.