Mastering Percents: A Quantitative Literacy Exploration (with Solutions)

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) %divides; (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.

    First solution: 5/8 = 5÷8 = (5/4)÷2 = (1.25)÷2 = 0.625 = (62.5)÷100 = 62.5%
    Alternatively, one can notice and mentally calculate that 5/8 = 4/8 + 1/8 = (1/2) + (1/8) = 50% + (half of 1/4) = 50% + (half of 25%) = 50% + 12.5% = 62.5%


    • 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:

    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?)

    _____________________________________________________________________________________

    _____________________________________________________________________________________

    _____________________________________________________________________________________

    One solution: 80% of 130 is 50% plus 20% plus 10% or 1/2 plus 1/5 plus 1/10.

    Or 80% of a number is 50% plus 10% + 10% + 10% so add half of that number to triple of what one-tenth of that number is..

    A third way is to use that 80% = 100% - 10% - 10% (or use 80% = 100% - 20%).

    In addition to these five methods and similar ones which "break down" the 80% into the sum or difference of simpler, easier parts, we can also break the 130 apart:

    (80% of 130) = (80% of 100) + (80% of 10) + (80% of 10) + (80% of 10) which is 80 + 8 + 8 + 8 or 80+24 = 104
    [Students check to verify]


    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:

    1. 12 is ____% of 50.

    2. ____ is 60% of 20

    3. 12 is 15% of __80__

      _____________________________________________________________________________________

      _____________________________________________________________________________________

      _____________________________________________________________________________________

      _____________________________________________________________________________________

      _____________________________________________________________________________________

      _____________________________________________________________________________________

      _____________________________________________________________________________________

    Solutions Section

    (a) 12 is ____ of 50

    Mechanical solution: 12/50 = 0.24 = 24%

    One conceptual solution: Break 12 up into pieces that are convenient for this problem: 12 = 10 + 2.

    These pieces are convenient since 10 is an easy to identify fraction (1/5 or 20%) of 50 while 2 is 2/50 (or 4/100, thus 4%) of 50. Thus:

    12 = 10 + 2 = (1/5 of 50) + (4/100 of 50) = (20% of 50) + (4% of 50) = 24% of 50.

    Another conceptual solution is possible, by breaking up the number 50 into pieces that are convenient with respect to what we're trying to "measure with", namely with respect to the 12. Namely, convenient sized pieces are 5 and 1, that is: 5 is 10% of 50, and 1 is 2% of 50 (since 1/50 = 2/100). So, mentally it is not difficult to add up, and on paper it's easy to do so:

    12 = 5 + 5 + 1 + 1 = (10% of 50) + (10% of 50) + (2% of 50) + (2% of 50) = 24% of 50

    (b) ____ is 60% of 20

    One possible conceptual solution is to use that 60% = 50% + 10% and that those are convenient (easy) percents since they represent a half and one-tenth respectively. Thus,

    (60% of 20) = (50% of 20) = (10% of 20) = (1/2 of 20) + (1/10 of 20) = 10 + 2 = 12 (Check this answer!)

    Mechanical solution would write x = (60%)·(20) = (0.6)·20 = 12,

    which is even shorter, but students whose finger slips or misses typing in part of this or the conversion to 0.6, very often blindly repeat the calculator's (wrong) answer, and clearly the above conceptual method is not just fun but pretty fast -- it only takes 2 or 3 extra seconds' time -- and it is, importantly, easy to mentally check its correctness.

    (c) 12 is 15% of ____

    This takes a bit more conceptual thinking but it's not too difficult once we realize that if 12 is 15% of W (the whole) then one-third, or 4 is one-third as much (or 5%) of that same W.

    Thus 4 is 5% of W. And 5% is 1/20 of a number. So, 4 = (1/20)·W.

    So W = 4·20 = 80. Check:

    5% of 80 is (80/20) = 4 so 15% of 80 is 3 times as much as that 5%, namely it equals 3· (5% of 80) = 3 · (4) = 12.

    Another method: 4 is 5% of W. So 8 is 10% of W (why?) Thus W = 80.

    A Warning:

    Not every way of trying to "break things up" into the "sum of simpler pieces" will work! You must always ask yourself whether your 'breaking-up' method makes good conceptual sense. For example, the following false 'method' is tempting for many to try but it does not work

    The following 'method' does NOT work!
    "12 is 10% of 120,

    and 12 is 5% OF 240

    therefore it follows that 12 is 15% of 360" (crossed off because this step is false)

    No Way! We can't add the 10% and the 5% since they are percents "of" two very different numbers (of 120 versus of 240). It's like trying to add 1% of your money to 1% of some rich person's (Bill Gates') money...you don't get "2%" of either your nor his money! The cartoon below reminds us of this fact

    However below is another way that does work:

    We know that 15% is "one and a half times as much" as the (very convenient) 10% so, what is 12 "one and a half times as much"? It's one and a half times as much as 8. So if 12 is 15% of the "whole", then 8 must be 10% of that whole, so 8 = 10% of W = W÷10 so W = 80.

    What if the "one and a half times as much" wasn't obvious to you though? The beauty of conceptual methods is that they allow you to arrive at the same, correct final answer in many ways. If we're willing to have it take a few more steps, we can arrive at the same final answer starting from whatever numbers/methods are most convenient for us. For many people, looking at smaller numbers helps, and below is one more conceptual method.

    Before looking at another conceptual way that works, one more Warning: if 9 is 5% of 180 then it is 15% of what? not of 3·180 but rather of 180÷3 !! (Why? Think about this for a moment before you continue!)

    Now we can see one more conceptual method for solving "12 is 15% of ____"

    Suppose we would prefer to use smaller numbers than 12, like 9 and 3. Take the smallest and least intimidating number, 3 for example. Suppose we know that 3 is 15% of some number A. That would make 3 a smaller percent, 5% of a larger number 3·A (see above). Hmmm, 5% is 1/20 so 3 is 1/20 of what? Of 3·20=60. So 3 is 15% of a number that's as much smaller than 60 as 5% is of 15%, namely 3 times smaller, so 3 is 15% of 60÷3 = 20 (check this! Convince yourself that 3 is 15% of 20. You can confirm this also using the '15% is one-and-a-half times bigger than 10%' idea above).

    Now let's play the same game with 9. If 9 is 15% of B, then 9 is 5% of 3·B. This would make 9 a 5% (or 1/20) portion of 3·B [Algebraically, 9 = (1/20)· (3·B) can be solved, giving us 3 = (1/20)·B or B = 60. Or avoid algebra as follows] We know 9 is 5% (or 1/20th) of 20·9 = 180. And 9 is 15% (a larger percent) of a proportionately smaller number than 180, namely (1/3)·180 = 60. We now can add the pieces:

    12 = 9 + 3 = (15% of 60) + (15% of 20) = (15% of 80)

    This is not a "made up" or contrived method: college students not particularly strong at math (namely elementary education majors) as well as other students have come up with the above method! If you "see" it, that's great, if not, that's ok since there are so many conceptual methods, some will be "obvious" to you and others "seem hard". The key thing is to improve your comfort level with conceptual methods by at least being able to follow each such method. A final Question For Thought: Why does adding work in this case, the "(15% of 60) + (15% of 20) = (15% of 80)" case above?

    [Suggestion, why does 15% of A plus 15% of B always equal 15% of (A+B) for any numbers A and B. For that matter, why does 1% of (A+B) equals ((1% of A) + (1% of B)) Can you convince yourself of this? Can you convince a fellow student who is (especially after seeing the above methods that don't work) skeptical? Once you see why 1% of (A+B) equals ((1% of A) + (1% of B)), you can of course switch the equality to get that ((1% of A) + (1% of B))? equals 1% of (A+B)]

    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

    Answer: Yes. Does this always happen? Yes, it does. One way to see is to notice that (20% of 80) = (0.2)·(80) = [(2) · (1/10)] · (80), however using the Associative Property,
    [(2) · (1/10)] · (80) = (2) · [(1/10) · (80)] = (2) · (0.8)

    Which by the Commutative Property equals (0.8)·20 which equals 80% of 20. Using "A" and "B" instead of 20 and 80 (and (A/10) instead of 0.2 and so forth), and using the Associative and Commutative properties, you can prove the pattern seen here "always happens".

    Warning: We are not saying that "$80 increased by 20%" is equal to "$20 increased by 80%" (see section below on percent increases/decreases) and those are indeed not equal.

    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:

    1. "45 is P% of 20"
    2. "35 is P% of 20" (Hint: 35 = 20 + 10 + 5)
    3. 33 is Q% of 18 (hint: write 33 = 18 + ...)

    Solution:

  • 45 = 20 + 20 + 5 = (100% of 20) + (100% of 20) + (25% of 20) = 225% of 20
  • 35 = 20 + 10 + 5 = (100% of 20) + (50% of 20) + (25% of 20) = 175% of 20.

    Check it! (175% of 20) = ((175/100)·20) = (1.75)·(20) which your calculator (which you just proved you don't really need if you can carry out these conceptual steps) will nevertheless confirm equals 35.

  • 33 = 18 + ? = 18 + 9 + ? = 18 + 9 + 6 = (100% of 18) + (50% of 18) + (1/3 of 18) = (150% of 18) + (33.333... percent of 18) = 183-point- repeating-three percent, or 183 and one-third percent of 18. Check it: (1.833333...)·18 does equal 33 (or 32.9999... the more 3's you put in the more 9's you'll get here).

    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:

    Exercises:

    1. If we increase 20 by 25% what would we get?
    2. If we increase 25 by 20% what would we get?
    3. If we decrease 25 by 20% what would we get?
    4. If we decrease 25 by 40% what would we get?
    5. If we increase 24 by 20% what would we get?
    6. If we decrease 24 by 20% what would we get?
    7. $A increased by 33 and 1/3 percent is 16. What is A?
    8. The old price of $2 became $5.50 what was the percentage increase?

    9. 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?

    10. Mathematically analyze and explain the following cartoon: Explanation:_______________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

    Solution:

    1. If we increase 20 by 25% what would we get? 20 + (25% of 20) = 20 + 5 = 25

    2. If we increase 25 by 20% what would we get? 25 + (20% of 25) = 25 + 5 = 30.

    3. If we decrease 25 by 20% what would we get? 25 - 5 = 20

    4. If we decrease 25 by 40% what would we get? 25 - 5 - 5 = 15 (why does this work? Or think mentally to yourself: (25 - (40% of 25)) = (25 - (20% of 50)) = (25 - (10% of 100)) = (25 - 10) = 15. Why does this work?).

    5. If we increase 24 by 20% what would we get? 24 + (10% of 24) + (10% of 24) = 24 + (2.4) + (2.4) = 24 + (4.8) = 28.8 easy, and without a calculator!

    6. If we decrease 24 by 20% what would we get? We already know what 20% of 24 is (4.8) so the answer must be 24 - 4.8 = 20 - 0.8 = 19.2

    7. $A increased by 33 and 1/3 percent is 16. What is A? We are told that (A + (1/3 of A)) = 16. Thus ((4/3) of A) = 16, or (4/3)·A = 16 thus (1/3)·A = 4, so A = 3·4 = 12. Verify that $12 increase by a third (12 + (12/3)) indeed is 16.

    8. The old price of $2 became $5.50 what was the percentage increase? 5.50 = 5.5 = 2 + 2 + 1 + 0.5 = 2 + (2 + 1 + 0.5) = 2 + ( (100% of 2) + (50% of 2) + (25% of 2)) = 2 + (175% of 2) so 2 was increased by an additional 175%, and the answer is 175%. Notice that $5.50 is 275% of the quantity $2 but $2 need only be increased by (an additional) 175% to produce $5.50

    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     		1,200,0001,800,000

    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?

    Solution: The tax rate increased by 2 percentage points (from 5% to 7%). There was however a 40% increase in the sale tax (from 5 cents on the dollar to 7 cents on the dollar, we have 7÷5 = 14÷10 = 1.4 so 7 is 140% of 5 -- and thus, 5 needs to increase by 40% to arrive at 7). The real-world meaning of the 40% is that this is how much extra you would be paying: In the past, for example, you would have paid $50 in sales tax on a $1,000 purchase, and you will now pay $70 on such a purchase. This extra $20 is the additional 40% more you will be paying in taxes, versus the $50. The real-world meaning of the "2 percentage points" is less clear-cut, but reasonable answers might be that if consumer sales stay constant, then an additional 2 percent of that sales total, is how much extra revenue your state will have raised for its annual budget.

    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











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