Percentage Problems

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Problem with percentages.  These are not trick problems.  This kind of data and/or situations actually occur.  Maybe they are not so simple, however they do require some clarity of thought to resolve them.

• Jim make 75% of what Sally makes, and Mary makes 62% of what Jim makes.  However, it is known that Jim and Mary together make 1.5 times what Sally makes.  How much does each make?
  
  Answer: We do know that Mary makes 62% of 75% of what Sally makes, or in decimals 0.62 * 0.75 of Sally's salary.  But... This is an impossible problem as together Jim and Mary make (0.62 * 0.75+0.75) =1.215 of what Sally makes.  So, together they cannot make 1.5 what Sally makes.  In addition, we can never know how much each makes unless we know how much at least of the make.  There is the certain confusion of the percentages (62% and 75%) given and the factor 1.5.  Unless the student knows the 1.5 means 150%  he/she has little hope of solving the problem.
  
  Answer alternative:  What one student could say is that because we don't know what any of the three makes we cannot possibly know what all make.  This is a little superficial because the problems is missed completely - although the answer is correct.
  
  Curiously when multiplying percentages (of the same thing) the result is a percentage, not percentage-squared.  This is unlike the product of length measurements, where you do indeed get length * length = length-squared, or area.  You can also multiply area by length and get volume, as you know.  Yet, multiplying areas doesn't give much of anything, including areas-squared, which has no meaning.  No wonder kids have math problems with incredibly basic problems. They don't teach this, I think.
• From actual data...
  In 1961 71% of American men were married, when just last year 51% of American men were married.  So, the teacher said we have a drop of married men over this period of 21%. Sally Ann, from the last row said she thought is was more like 30%. Who is correct?
  
  Answer: In fact there is a 21% drop in the number of men actually married - in absolute quantities, but the drop in percentage is just like the drop in any quantities.  You take
  
  (previous - now)/previous * 100%
  to get the result, which is about 30%.  The confusion is with percentages and what they mean, and what "drop" means.
• At camp Wilderness, 45% of the campers got poison ivy while 70% had mosquito bites.  The teacher asked how many had mosquito bites and not poison ivy?   And how many had one or the other?  Bobby answered the first by subtracting to come of with 25%.  Karen answered the second by summing to come up with 115%.  Ken volunteered his ideas by saying both solutions are incorrect but gave no numbers.  Which answers are correct?
  
  Answer:  The issue here is with "number-banging," that is combining numbers in some fashion.  The tip off for Karen is that adding the percentages gives a result that is total unreal, i.e more than 100%.  The problem with Bobby is that he doesn't know how the populations of the mosquito and poison ivy folks interact.  Fundamentally, what we haven't considered is that the percentages are relative and not absolute. Ken is correct, and not giving the answers is correct - it cannot be done with the information at hand.
• Alternative...
  At camp Wilderness, 25% of the campers got poison ivy while 70% had mosquito bites.  The teacher asked how many had mosquito bites and not poison ivy?   And how many had one or the other?  Bobby answered the first by subtracting to come of with 45%.  Karen answered the second by summing to come up with 95%.  Ken volunteered his ideas by saying both solutions are incorrect but gave no numbers.  Whose is correct?
  Answer: The result is the same as above, but this time the sum of 25% and 70% being 95% result, is less a tip-off for a problem with the math.  Ken is again correct.  We definitely need to know how many had both mosquito bites and poison ivy. 
• In Mason City there are 1,000,000 TV sets.  What is known is that at a given time 44% of all TV's turned on were tuned to Channel 3.  It is also known that 56,000 TVs turned on were not tuned to Channel 3.  How many TV's were tuned to Channel 3?
  Answer:  The red herring here is the total number of TV's in Mason City.  This value is not needed.   So, we need to focus on the 56,000 TVs on but not turned to Channel 3.  Since this number is 56% (coincidence) of the number of TVs (i.e. 100%-44%), we know there must have been 100,000 TVs turned on.  Therefore, there were 44,000 TVs tuned to Channel 3.  What is the meaning of the 1,000,000 TVs?  It is to make the problem solvable.  See below.
  
  Alternate problem. In Mason City there are 75,000 TV sets.  What is known is that at a given time 44% of all TV's turned on were tuned to Channel 3.  It is also known that 56,000 TVs turned on were not tuned to Channel 3.  How many TV's were tuned to Channel 3?
  Answer:  This problem cannot be resolved because if there are 56,000 TVs turned to other channels, this represents a fraction of more than 56/75 TVs tuned to other channels, and this value exceeds 56% of the total.  Therefore, it is not possible that 44% of TVs turned on are tuned to Channel 3.   In this problem, we have inconsistent conditions given.

Why the Dozen?

Fact: 12, 24, 60, these are the big numbers of your everyday life.  These are surely different from the nickel, dime, quarter, and dollar.  Just why is that?

You may have questioned why do we, users of a base 10 system of numbers, seem so dedicated to the dozen (12), the 24 hour clock ( = 2 x 12), the minute (= 60 seconds), and the hour (= 60 minutes).  The answer lies in divisibility.

Clearly, a dozen (12) is divisible by 2, 3, 4, and 6.  This means you can ask for a quarter dozen, third of a dozen, half a dozen, etc.  The math is simple.  If the basic multiple were ten (10 = 2 x 5) we could only do a half or a fifth.  That would be it.

For the day (24 hours), we can divide it by 2, 3, 4, 6, 8, and 12.  This allows a half day, a third of a day (typical shift), a sixth of a day (= 4 hours which is the morning shift), and more.

For an hour (60 minutes) we can divide by 2, 3, 4, 6, 12, 20, 30.  This allows a half hour, a quarter hour, a third of an hour, and more.  The divisibility makes the calculation convenient and easy.  Imagine an hour of 100 minutes.  Then we could divide it into only half, fourth, fifth, tenth, twentieth, and a couple of others. There is less flexibility.

The ancient Babylonians used a base 60 system of enumeration for good reason.  Why? It's all in the divisibility.  Also, it made dealing with fractions a whole lot easier.   BTW, handling fractions were a problem for all the ancient civilizations.   Naturally, the multiplication tables were horrific.  But they had tricks to make calculations easy (well, easier than memorizing all products x times y for x and y in the range from 1 to 59).  Isn't it interesting that in the schools, the multiplication tables taught are for all products x times y for x and y in the range from 1 to 12.  There is that number, 12, once again. 

The year of 365 1/4 days, we can do nothing about.  It is what it is. But note, we have conveniently divided the year into 12 months.  Sure, the numbers of days differ a bit, but calculations are easy in this context.  On the other hand, imagine a year with ten months.  Divisibility is only possible into halves and fifths.  This is the same as if the basic multiple unit consisted of ten items, just like the discussion about the dozen.

Fingers and toes.  These are the basis of our monetary and measurement systems.  This means tens and twenties.  It is an apparent explanation of one system in contradistinction to the more facile base 12, 24, and 60 systems of everyday counting.  Go figure.