Some Problems With Two Unknowns

All of these problems are easily solved by “textbook” methods. But, the textbook is not a 6th grade textbook. If you can solve them by “textbook” methods, feel free to do so.

6th Grade Math Challenge 2010, Individual Round #1, Problem #1

Two notebooks and one pen cost $3.17.
One notebook and two pens cost $3.97

How much do 5 notebooks and 5 pens cost?

Hints: Let N be the price of a notebook and P be the price of a pen. Confession: I wasted time solving for N and P. Then I thought: what are they really asking for?

An equation always says: left side = right side. Since you can always add or subtract the same thing to each side of an equation, you can add or subtract equations. In this case, if you write two equations and add them, you nearly have what they are asking for.

5th Grade Math Challenge 2007, Tie Breaker, Problem #1. Note: Tie Breakers are usually challenging problems.

1. The owner of a bicycle store had a sale on bicycles (two-wheelers) and tricycles (three-wheelers). When he counted the total number of pedals of the cycles on sale, he got 50. When he counted the total number of wheels of the cycles on sale, he got 64.

How many tricycles were offered in the sale?

(Note: each cycle has two pedals.)

Hints: Let T = the number of tricycles and B = the number of bicycles. Write two equations. Since each cycle has two pedals, you know the number of cycles on sale.  If you solve the second equation for B in terms of T, you know that T is an even number.

Guessing a few even numbers and checking the total number of cycles will lead you to the right answers. (They ask for the number of tricycles.)

5th Grade Math Challenge 2007, Team Round #1, Problem #3. Note: Team Round Problems  are usually challenging.

You took a 20-question exam that was scored in this way: 10 points are awarded for each correct answer and 5 points are deducted for each incorrect answer. You answered all 20 questions and received a score of 125 points.

How many questions did you answer incorrectly?

Hints: You know there are 20 questions. Let C be the number answered correctly and W be the number answered incorrectly, or Wrong. (I used W instead of I because I is easily confused with 1.)

Use C and W to write an equation for score, which you know is 125. Solve for W in terms of C. You will find that C has to be equal to or greater than 13 (otherwise, W would be negative) and must be less than 20 (if you had 20 correct answers you would have no wrong answers and would have scored 200.) There are only 7 possible guesses, and if you start at 13, you will get the correct answer after only a few guesses.

Guess and Check Method for Simultaneous Equations: Shown to Me by Kai Alton.

Here is a typical problem with two unknowns:

The gym has 2 kg and 5 kg disks for weight lifting. There are 14 disks in all.

The total weight of the 2 kg disks is the same as the total weight of the 5 kg disks.

What is the total weight of all the disks?

The first step is to read the problem and turn the English into mathematics. Part of this is: give the unknown quantities one-letter names that are meaningful. Also: identify what they are asking for. In this problem, it is the total weight of all the disks.

The two unknowns are the number of 2 kg disks and the number of  5 kg disks. Let’s call these T for number of  Two kg disks and F for number of Five kg disks.

The first English sentence to turn into mathematics is: “there are 14 disks in all.”

In mathematics, it becomes T + F = 14.

The second English sentence to turn into mathematics is: “the total weight of the 2 kg disks is the same as the total weight of the 5 kg disks.” In algebra, if we write a number next to a letter, it means “multiply.”

So the sentence becomes: 2 T = 5 F.

There are formal mathematical methods for solving the two equations simultaneously, but I would like to show you a method shown to me by a Seward student, Kai Alton. It works best where the total of the two unknowns is given, as in T + F = 14.

First:  subtraction <= undoes => addition (read it forwards or backwards).
Also: division <= undoes => multiplication (read it forwards or backwards).

A summary of the method is:

1. Make a smart guess for one unknown.
2. Solve for the other unknown in terms of your smart guess, using the “undoing” rules.
3. Check to see if the two (now known) unknowns add up to the given total.
4. If not, make another smart guess until the total is right.
5. Use the (now known) unknowns to give them what they are asking for.

Using the “undo” on 2 T = 5 F, divide both sides of the equation by 5, undoing one multiplication.

The result is F = (2/5) T . Both F and T are whole numbers, so a smart guess for T is a whole number divisible by 5, for example, guess T = 5. Then, F = 2 and the total number of weights is 7. The total should be 14, so a better guess is T = 10, which gives F = 4 and a total of 14. Checking,  5 F= 5 x 4 = 20 kg  and 2 T = 2 x 10 = 20 kg (the total weight of the 5 kg weights equals the total weight of the 2 kg weights) and what they asked for, the total weight of all the disks, is 20 + 20 = 40 kg.

The problem is solved by smart guessing, checking, and arithmetic.

Test Post

This is a test post.

The arithmetic of the election

The 2012 Presidential election will take place on Tuesday, November 6.

It is then that you will hear about electoral votes.

There are 538 electoral votes. Minnesota has 10.

The electoral system was created by the United States Constitution, Article II, modified by the 12th and 23rd amendments. The 23rd amendment was added in 1961 and gives 3 electoral votes to The District of Columbia; this leads to the present total of 538 electoral votes.

The number 538 is divisible by 2 (its last digit is the even number 8). Half of 538 is 269, hence, if either candidate gets 270 votes or more, he wins. Because 538 is divisible by 2, a tie is possible; this happened twice, in the elections of Thomas Jefferson and John Quincy Adams. In those days, the number of electoral votes was smaller and a tie more likely.

In the case of a tie, the President is chosen by the House of Representatives and the Vice President by the Senate.

There are many arguments for and against the use of electoral votes but one is surely true; it preserves the influence of states with small populations. You may have been in some of those states; North Dakota, South Dakota, Wyoming and Montana have small populations but each has 3 electoral votes.

You could add up the populations of the 4 states above and find that their combined population is less that of Minnesota, but they have 12 electoral votes to Minnesota’s 10.

To read more about the electoral vote system, go to:

http://en.wikipedia.org/wiki/U.S._Electoral_College

This is a long article, but if you stay up until the election is decided, you may have time to read it.

The electoral votes will not be officially counted until December, 2012.

Solution to An Old Problem About Shopping and Taxes

I am going to use rot13 to give answers:

Copy the encrypted text below: (use the Copy command) which is Ctrl-C on Windows computers and Command-C on Apple computers.

Zl Erprvcg sebz Cnaren Oernq

SBBQ: fvk qbyynef naq 59 pragf
GNK: sbegl guerr pragf
GBGNY: frira qbyynef naq gjb pragf
PHFGBZRE CNVQ: frira qbyynef naq svir pragf
PUNATR QHR: guerr pragf

Go to:  http://www.rot13.com

Paste the encrypted text into the box, using (Ctrl–V or Command-V) Click on “Cypher.” You should see the decoded answer.

Answer to Units Conversion As A Joke

I am going to use rot13 to give answers:

Copy the encrypted text below: (use the Copy command) – it is Ctrl-C on Windows computers and Command-C on Apple computers.

Wbxr nafjre: ur pbhyq svaq bhg ol purpxvat uvf fcrrqbzrgre.

Havgf pbairefvba:

60 srrg   3600 frpbaq   1 zvyr
------- k      ------ k ------ =   sbegl cbvag avar zvyrf cre ubhe
frpbaq         ubhe    5280 srrg

Ur vf fcrrqvat ohg whfg n yvggyr ovg.

Go to: http://www.rot13.com

Paste  using (Ctrl–V or Command-V) the encrypted text into the box. Click on “Encrypt.” You should see the decoded answer.

Units conversion as a joke, corrected

Dr. Steve is driving his car. He is traveling at 60 feet per second and the speed limit is 40 miles per hour. Is Dr. Steve speeding?

Answer in next blog.

Units Conversion As A Joke

Dr. Steve is driving his car. He is traveling at 60 feet per second and the speed limit is 40 miles per hour. I Dr. Steve speeding?

Answer in next blog.

An Old Problem About Shopping and Taxes

On February 10, 2007 “Steve” went to Panera Bread in Eagan MN and bought a “U Pick 2” (soup, sandwich and an apple) for $6.59.

Sales tax in Eagan was 6.5% back then. In Minnesota, tax is rounded to the nearest cent.

Three questions:

What was the tax?

What was the total, including tax?

How much change would Steve get if he presented a $5 bill, a $2 bill, and a nickel (five cents)?

-------------------------------------------

Note: November 2012

The sales tax in Eagan has gone up to 7 1/8 %  (7.125%). The actual ancient receipt will be posted, somewhere in Seward.

Crypt Arithmetic Problems That Are Easy To Solve, Revised

Students in 5th grade math challenge at Seward.

Please perform both checks and write on paper.
Write your name and your E2 teacher's name on
the paper and hand in to Karen Utter during your normal Monday or Wednesday class, but as soon as possible.

We want to verify that you are reading this blog.
Please tell your friends about it.

The assignment may seem trivial but we just want to verify that you are reading this blog.

2012 6th Grade Math Challenge Problem #1
162
+XD
-----
2D7
D must equal 5 so we have:
162
+X5
-----
257
When we add X to 6, we must get 5 and a carry of 1.
So, 6 + X = 15, or X = 9
162
+95
-----
257

Check: Do the above with a calculator and verify that it is correct.

2012 6th Grade Math Challenge Problem #5
ON + ON + ON + ON = GO
Adding ON 4 times is the same as multiplying by 4.
ON
x4
-----
GO
Try various values for N.
N can't be zero because then, N and O would be the same.
If N is 1, O = 4 and 4 x O would be 16, generating a carry and giving a three-digit answer.
If N is 2, O = 8 and 4 x O would be 32, again generating a carry and giving a three-digit answer.
If N is 3, 4 x 3 = 12. That makes O = 2 with a carry of 1.
G = 4 x 2 + 1, or 9.
This gives:
23
x4
----
92
Check: 23 + 23 + 23 + 23 = 92
Verify this with a calculator.