A Piggy Bank Bet

My piggy bank has been my trusted repository for some time. I have used it exclusively to save those little extra pennies, nickels, and dimes that I have received as change when I purchase things.

 

One day I had need of some cash to buy some school supplies before going to classes, so I took “piggy” off my shelf, preparing to make a “withdrawal”. Just then my pesky little sister, Charlene, entered my room and said, “Oh, money! Gimme some, will ya’?”

 

Being the kind brother that I am, I said, “I’ll make you a deal. If you can guess how much is inside, with less than a 10% error, I’ll give you a quarter. How ’bout that?”

 

With nothing to lose if she was wrong, “Charlie” shook the bank to hear the clink-clank of the coins, thought a moment, then said, “Ok. I bet there is $3.50 inside.”

 

After counting out the money, we found that her guess was pretty good, though not perfect. (But I gave her the quarter anyway; fair is fair, you know.)

 

I would like for you to tell me how many of each type of coin was in the bank and how much I was left with after paying off my debt to her. You have these facts to work with:

 

  • 14 coins were of one of the three types;
  • 16 2/3 % of them were of another type; and
  • 3/5 of them were of the third type.

 

As always, before writing out your answer, please check our Guidelines for Writing POW Answers

http://mathforum.com/algpow/images/writing_answers.html.

A Nutty Problem

Mama Squirrel found a bag of nuts in the driveway of the house near her tree home. She decided to separate the nuts into equal portions for her children.When she tried putting 6 nuts on each plate, one squirrel baby was left out and received no nuts at all. (So there was one sad little squirrel now.)

But when she tried putting 5 nuts per plate, there was one nut left over. (At least nobody was unhappy this way.)

What is the sum of the number of the nuts and the number of squirrel children?


NOTE: Before writing out your answer, please check our Guide lines for Writing POW Answers.

 

Shanell’s Holiday Treats

Shanell’s favorite hobby is baking cakes and cookies to serve her friends and family during holiday occasions. She often buys cashews, pecans, almonds, and walnuts to use in her recipes.

One week, she bought one pound of cashews and two pounds of pecans, and paid $2.40. The next week she bought four pounds of almonds and one pound of cashews, paying $3.60. A week later she bought three pounds of walnuts, one pound of cashews and one pound of almonds for $2.10.

How much would she have to pay on her next trip to the grocery store, if she bought one pound of each of the four types of nuts?


NOTE: Before writing out your answer, please check our Guide lines for Writing POW Answers.

What’s the Score?

Fred Flintstone and his buddy, Barney Rubble, are coaches in a football league for young kids during summer vacation. Fred’s team is called Fred’s Falcons and Barney’s is called Barney’s Bears. Rivalry is always intense between them as they are the best teams in the league. Whichever of these two teams wins the game against each other usually wins the league championship.

So the tension was high one Friday afternoon when the Falcons faced the Bears in a winner-take-all game. After the dust settled, the Falcons came out on top. But it was certainly a hard fought contest.
Lots of scoring plays were made. The Falcons scored in some manner 10 different times. This means TD’s (touchdowns), PATs (aka point-after-touchdown or extra point), and FG’s (field goals). (However, there were no “safeties” at all, by either team.)
Here are some additional facts for you to consider:
  • the Falcons missed 2 PAT’s;
  • the Falcons’ final score was an odd, composite number; and
  • the Falcons outscored the Bears by 18 points.
Here are the two questions for you to answer:
  1. What was the number of points scored by the unhappy Bears?
  2. What is the fewest number of scoring plays that the Bears could have made to achieve their point total?

 

 


If you wish to send me your answers and proof of solution, you can email me at either of these two addresses:

trottermath@gmail.com or ttrotter3@yahoo.com

Perhaps you aren’t sure about the value of the extra point in the story. Modern football has the 2-point option of running or passing the ball over the goal line. Just remember: this game was played in Bedrock times before the rules were changed. So a PAT is only worth 1 point.

Also, you may have wondered why the football is sometimes jokingly referred to as a “pigskin”. Well, as a service to WTM readers, we present here a picture of the ball used in the game of this story. Now you know!

 

 

Cats and More Cats

My neighbor, George, owns seven cats. The three oldest cats (Liberty, Lucky and Alexander) are 100% black and the only outdoor cats in the group. The sum of their weights equals the sum of the weights of the four indoor cats (Sassy, Moonlight, Bubba and Randy).Alexander weighs 2 oz. more than Lucky, who in turn weighs 6 oz. more than Liberty.

Randy, in turn, weighs 6 oz. more than Moonlight and 9 oz. more than Sassy. In addition Bubba weighs 28 oz. less than twice Randy’s weight.

 

Lucky and Randy are almost the same weight, but Lucky is the heavier of the two.

If their weights are all integers, what is the least that the cats could weigh?

Bonus: How many pounds do the 7 cats weigh in all?

 


Here is a variant on the story as given above. It has different answers…

My neighbor, George, owns seven kittens. The three oldest kittens are Liberty, Lucky and Alexander. The sum of their weights equals the sum of the weights of the four younger kittens (Sassy, Moonlight, Bubba and Randy).

Alexander weighs 2 oz. more than Lucky, who in turn weighs 6 oz. more than Liberty.

Bubba weighs 12 oz. more than Randy, who is 3 oz. more than Moonlight and 13 oz. more than Sassy.

Lucky and Bubba are almost the same weight, but Lucky is the heavier of the two.

What is the least that each kitten could weigh and still meet all of the given requirements?

Bonus: How many pounds do the 7 kittens weigh in all?

 


 

NOTE: Before writing out your answer, please check our Guide lines for Writing POW Answers.

 

The King’s Coins

The second half of a famous “Mother Goose” nursery rhyme goes something like this:

The King was in the counting-house,
Counting out his money,
The Queen was in the parlour,
Eating bread and honey,

The maid was in the garden,
Hanging out the clothes,
When down came a little bird
And snapped off her nose!

Now, I have no particular interest in the affairs of those two ladies. As the king’s Royal Mathematician, however, I do have a curiosity about how much money he was counting.

So I went to His Highness to inquire.

His reply to my question as to how many gold coins he had on His counting table was rather unexpected. He said:

“If I put all my coins in stacks of 25 each, I have 15 coins left over in another, shorter stack. If I then put them in stacks of 35 each, I still have 15 left over, just as before. But if I put them in stacks of 45, there are no coins left over. It comes out just right!”

Well, my friends, I’m not called the Royal Mathematician for nothing. I took out my quill, ink bottle and a piece of parchment, and set to work. In no time at all, I knew just how many gold coins the King had.

To answer this problem, tell me the number of stacks of coins the King had in His first counting method.

Remember: define your variables carefully, set up good equations, and use algebraic thinking. (And remember; there were no calculators nor computers in the King’s time!)

 

BONUS: How does the first half of this famous rhyme go?


NOTE: Before writing out your answer, please check our Guide lines for Writing POW Answers.

Susita’s Savings Account

At the beginning of January, Susita had some money in her savings account.During each month she was able to deposit enough from her allowance to double the amount currently in the account.

But she had a loan to pay off, requiring her to withdraw $10 from the account monthly.

She had to repeat this procedure each month.

At the end of May, she had $2 left in the account.

How much did Susita have at the beginning of January?

Fireball Frank

His real name was Frank Johnson, but everybody who followed baseball in the Tri-County Baseball League just called him “Fireball”. He was definitely the best pitcher on his team, the Bensonville Beagles, and probably the best in the whole 10-team league. He earned that nickname, and the respect of every batter who had to face him, due to his blazing fast ball pitch. Legend has it that the ball was nearly invisible as he threw it to the plate.

One day the Hatcherton Hawkeyes came to play on Bensonville’s home field. Fireball was going to pitch for the Beagles. The heckling got started early, led by Big Mark Stockdale, a pretty fair hitter of home runs on the Hawkeyes’ team.

“Hey, Fireball,” goaded Mark, “you think you’re a hot pitcher, I’d guess. You throw pretty hard to your catcher.”

“Well, reckon I do, ’cause he complains of a sore hand after games I pitch.” said Frank.

“So you’re good throwin’ to a batter. But just how good are you at throwin’ upwards to the sky?” was Mark’s challenge. “Straight up vertically, I mean. How long do ya’ think ya’ could keep the ball in the air?”

“Never really thought about that,” replied Frank. “Maybe 4 or 5 seconds, I s’pose.”

“I bet you a 20-dollar bill you can’t keep it up in the air for more than 5 seconds. What do ya say about that, Fireball?”

“I say take out your money, Mark.” Frank said.

Just then an umpire stepped up and said, “I’ll hold the money, boys. And I’ve got a stop watch here to do the timing.” Everyone agreed to that. The umpire tossed Frank a new baseball.

Frank walked confidently out to the middle of the infield, took off his cap, and tossed it about 6 feet away from where he was standing. “The ball will come down and land in my cap,” he boasted.

Frank gave the ball a good rub to soften the leather a bit. He went into his windup routine, then turned sideways and with a mighty effort, sent that ball skyward. Up, up it went. Quickly it was no larger than a dot in a “dot-com” address. Then just as quickly, it returned to earth – landing right in Frank’s cap! What a pitcher!

When the umpire announced the time, the Beagles fans went wild with happiness. Their Fireball had done another great performance! But as Frank reached out to receive the $40 from the umpire, he felt a strange pain in his pitching arm. “Gee, I hope I haven’t hurt my arm throwing that way.”

Your task is to tell me how long the ball was in the air if Frank released the ball from his hand 2 meters from ground level with an initial upward velocity of 30 meters per second. (Round your answer to the nearest 10th of a second.)

 


NOTE: Before writing out your answer, please check our Guide lines for Writing POW Answers.

And some technical help can be found in the Commentary for the problem at this site: Motorcycle Daredevil.

 

The Professors’ Ages

If you were solving AlgPoW problems from the Math Forum in the year 2001, and in particular the one titled “The Professors’ Primes“, (May 7) you met three of my favorite university professors: Drs. Ken Travers, Peter Braunfeld, and Wilson Zaring. Before I left them to finish their lunch in the cafeteria that day, I inquired as to what their ages would be when students would be returning to school for next year (September, 2001). They were kind enough to oblige; hence I have some data for this problem you are about to work now.Upon careful examination of their three ages, I have formulated the four following facts. I am using the initial letter from each man’s first name to represent his age, respectively.

 

  1. 60 < K < P < W < 80
  2. The sum of the older two professors’ ages is a square number.
  3. The sum of the oldest and youngest professors’ ages is the first palindrome less than the square.
  4. The sum of the younger two professors’ ages is the 2nd prime less than the palindrome (of statement #3).

With that information, give me the “digit sum” of the product of the three professors’ ages.

NOTE: show enough steps of work so that any reader can follow your reasoning easily. The variables for your equations are already given, but restate them for clarity as part of your solution process.

BONUS: What is the “digital root” of the product of the three professors’ ages?

Before you try writing up your solution, it is highly recommended that you check out the Guidelines for Writing PoW Answers.

You may send your answers via email to me at: trottermath@gmail.com or ttrotter3@yahoo.com. Or click on the icon below.

 

Guidelines for Writing Math Solutions

Writing a math answer for a Problem of the Week is very different from writing an essay in English class or a term paper in History class, so we would like to give you some guidelines. You write only one document, but we receive sometimes as many as 300-400 (or more) answers per week to read and analyze; when your presentation style is at its best, much time can be saved, a more efficient service can be provided, and everybody will be happier.


Readability

The first thing that would speed up the evaluation process can be called readability. Sometimes an individual sends us an answer in one long continuous paragraph, with equations embedded in it. Such paragraphs are very hard to read.

The solution is simple: just break the long paragraph up into several short ones, each one with its own concept, and leave a blank line between paragraphs.

Another matter regarding readability concerns polynomial expressions and equations. Notice the difference between these items:

    EASY TO READ                      HARDER TO READ
    
    x^2 + 2x + 1                      x^2+2x+1
    
    x^3 + 4x^2 - 6x + 10              x^3+4x^2—6x+10
    
    (3a + 4b)(3a - 4b)                (3a+4b)(3a-4b)

See how a space on either side of a plus or a minus sign makes the reading easier? (This is what good textbooks do.)

Similarly, when you show the steps in solving equations, add spaces and align the equals signs, like this (when you align text, never use tabs!):

    EASY TO READ                      HARDER TO READ
    
    2x + 48 = 58                      2x+48=58
                                      2x=10
         2x = 10                      x=5
    
          x =  5

Getting Off To a Good Start

After carefully reading a problem, it is essential to determine just what you need to find to answer the question posed. You should now select a letter, or letters, that will represent your unknown quantity, or quantities. This is the famous “Let” statement. Then, and only then, are you ready to begin forming your expressions or equations.

Be careful here, however. Many times “Let” statements aren’t clear. Examples:

GOOD ==> Let x = the number of apples in the basket

BAD ==> Let x = apples

In the latter case it’s not clear if we’re counting apples, or weighing pounds of apples. So be as precise as possible. It will save troubles later in your solving process.


Use of Guess-and-Check Procedure

In general, the method of “guess-and-check” is not allowed in AlgPoW as your primary strategy to solve the problem. This is not saying guess-and-check is not a good way to solve problems. In fact, it is often a good way to start to understand a problem, and therefore recommended for that. But for most of our problems, you must define variables or unknowns, then form equations to solve by logical steps.

Historically, it was the main way that problems were solved. But as advances were made in symbolic notation, mathematicians moved away from it and toward the more efficient and time-saving methods of step-by-step manipulations on equations.

One of our mentors advises students in the following way:

Guess and check is a valid problem-solving approach. However, it also one of the most difficult to explain. If you are going to use guess and check, you must list every guess, along with the reason that you know the answer is incorrect. You also must explain why you know your final answer is the only possible answer. In all, a pretty long process; however, since this is the Algebra Problem of the Week, you might want to try algebra. Please read the “Guidelines for Writing POW Answers.” The link is at the bottom of the problem.
So, unless otherwise indicated, please do not use guess-and-check as your principal solving procedure.


Writing a Complete Answer

The Problem of the Week (PoW) project here at the Math Forum, as you probably know, is a very unique one. Unlike other math tests in school or competitions (such as SAT), here we are not only interested in the right answer, but also how you arrived at it. This means, you must show your procedure and steps and thinking along with the final answer.

Even more so, your presentation must be explained well as you go from start to finish. Just imagine if you were to show your solution to a friend who was unfamiliar with the problem. Would that friend be able to read it and understand what you were saying?

ElemPoW has its own Guidelines document such as this one. Here is what is said there:
One good way to make sure you include enough information in your solution is to pretend you are explaining the problem to a friend who does not know anything about it. Imagine yourself leading your friend on a tour of your thinking as you solved the problem. How did you start? Where did you find the information you used? What were your calculations? How did you check your solution?

Math steps without a math explanation in words is much like watching a talented magician on stage. You see all the moves go by rapidly and you are “amazed”, but still you are left with the question, “How’d he do that?” Problem solving in PoW is not magic. Our goal is for everybody to understand as much as possible, according to his/her capacity.

Again, a thought from the ElemPoW service:

Our focus here at the Math Forum is not only on getting the correct answer, but also on communicating the steps involved in finding the correct answer.

To see the entire ElemPoW document about writing good answers, consult this page: How do you write a good math solution? There is much good advice to be found there.


E-Mail Notation

Sometimes we cannot write certain symbols (like exponents or square roots) in e-mail as we do using paper and pencil. Here are some examples:

Exponents

    It is standard now in e-mail to use the ^ (caret sign) found above the 6 on the keyboard for exponents. If we wish to say ‘four squared’, we write 4^2. For higher powers we do the same: ‘The volume of a cube is e-cubed’ would be V = e^3.

Square roots

    •                __                 _____
                    V64       or      \/a + b
  • Some people use the notation popular in spreadsheet applications, e.g. sqrt(16), to mean ‘the square root of 16′. This even applies in formulas; for the Pythagorean theorem, we can write:

     

    c = sqrt(a^2 + b^2).

    Other students ‘draw’ a square root symbol this way:

    [A few people try decimal or fractional exponents: 64^0.5 or 64^(1/2),
    but depending on the font this method can be difficult to read, so it is not recommended. However, there are occasions in which such exponents are better.]

Fractions

    •  1               15               3a  +  4b
      ---             -----            ------------
       2               25               5c  -  6d
      Two-fourths  2/4     five-sixths  5/6     etc.
      (3a + 4b)/(5c - 6d)
                 2
      Vertical: --- x      Horizontal: (2/3)x
                 3
                 2
      Vertical: ----       Horizontal: 2/(3x)
                 3x
  • Writing fractions is more complicated. There are two basic styles: vertical (sometimes called ‘stacked’) fractions, and horizontal fractions. Vertical fractions are what we are used to writing with pencil and paper, and are what you see in books. We can make them in e-mail as well; it just takes more effort and more keystrokes. But they are more readable when we need to write algebraic fractions.

    Horizontal fractions consisting only of numerals are easy to write, as these examples show:

    Even fractions that contain binomials, as shown above, can be written horizontally, if you employ parentheses. Observe:

    The difficulty arises when you need to express something like ‘two-thirds of x’. If you write this as 2/3x, it could be misinterpreted as 2 over 3x. Luckily we have ways of clarifying our meaning:

    Now if your intention really was 2 over 3x, you still have two options:

Subscripts

    • a1, a2, a3, a4, …
           y1 - y2
      m = ---------   instead of  m = (y_1 - y_2)/(x_1 - x_2)
           x1 - x2
      m = (y1 - y2)/(x1 - x2)
  • Unlike exponents, which go above the line (that’s why they’re sometimes called ‘superscripts’), subscripts go below the line. Unfortunately, the standard keyboard doesn’t have a true subscript key. Some people write a_1 for ‘a-sub-one’, but since many cases that need subscripts occur in sequences, we could write the following:

    to stand for a sequence of terms (a-sub-one, a-sub-two, …). In this context there is no real confusion with multiplying ‘a’ by 4. We universally write that as 4a.

    Notice how nice the slope formula can look using vertical fractions with this subscript style:

    The vertical equation looks almost like a line from a textbook, but even a horizontal equation like this one would be preferable:

Quadratic Formula

The quadratic formula is often needed in algebra problems. Here are two good ways to write it in email answers:

x = (-b +/- sqrt(b^2 - 4ac))/(2a)
                            -b +/- sqrt(b^2 - 4ac)
                       x = -----------------------
                                      2a

Determinants

    When you are using determinants to solve a system of equation by Cramer’s
    Rule, they may be nicely formed as shown here:
        |   3      5  |
    D = |             | = (3)(6) - (-1)(5) = 18 - (-5) = 18 + 5 = 23
        |  -1      6  |
    For a 3-by-3 case, the same idea applies:
        | a    b    c |
        |             |
    D = | d    e    f | = aei + dhc + gbf - gec - dbi - ahf
        |             |
        | g    h    i |

 

The method you use will often depend on the needs of the specific problem you are working; these comments should be understood as suggestions and general guidelines only.