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.