Here are a few problems for an Algebra 1 class that would take some thought, in addition to Algebra 1 skills.  I intend to add more by and by.  Feel free to send your favorite Algebra 1 problems, and if I like them I’ll post them and credit you.

BTW, in y = mx + b, where does the m come from?  I’m told that it stands for the French word montee, meaning “gradient” or “up grade”, from the verb monter, to climb.

1.  (using formulas:  distance-rate-time)  Two cars are driving around a circular track in opposite directions.  The red car goes at 30 mph and the blue car at 40 mph.  They pass each other every 2 minutes.  How many miles around is the track?

2.  (solve linear equations)  Two cars are driving around a circular track 8 miles in circumference in opposite directions.  The blue car goes 10 mph faster than the red car.  They pass each other every 2 minutes.  How fast is the red car traveling?

3.  (unit conversions -- a hard one, for after they’ve been doing these awhile)  Mr. Thompson lives in the U.S.  He buys gasoline at US$1.10 per gallon and finds that it costs him US$0.11 (for fuel) to drive one mile in his car.  Mr. Johnson lives in Canada.  He buys gas at Canadian $  per liter, and finds it costs him Canadian $ (for fuel) to drive one kilometer.  Whose car gets better gas mileage?  (Show your work finding the fuel efficiency of each car.)  Use:  Canadian $1 = US $0.79, 1 gallon = 3.785  liters, 1 km = 0.6214 miles.

4.  (multiplying fractions; understanding ratios; problem-solving)  Two-thirds of the argles are bornleys.  Five sixths of the bornleys are argles.  Three-fifths of the chumps are bornleys.  One-half of the bornleys are chumps, and one-half of the argles are chumps.  There are 25 chumps in all.  How many argles are there?

6.  Can you find a solution to the equivalent problem in three dimensions?  Using equal numbers of white and black cubes, make a rectangular parallelopiped of white cubes surrounded by a layer of black cubes one cube deep.  (There are several solutions.  One such is given at the bottom of this page.)

7.  (understanding and using ratios and percents)  The Nutmeg Thrift Shop (in New Hampshire, which has no sales tax) is having a sale:  75% off the marked price for all items over a dollar, and 50% off the marked price for all items $1.00 and under.  Mr. Thompson bought a coffee mug and a napkin holder.  The marked price of the coffee mug was one-third less than the marked price of the napkin holder, but the sale price of the mug was ten cents more than the sale price of the napkin holder!  He paid for both items with a dollar.  How much change did he get back?

8.  Factor the expression:  32x² - 245x - 88.  This looks incredibly tedious, but using an insight from simple number theory, you can go directly to the solution.

(Some) Solutions below

Some Solutions:

5.  6x8 works, and so does 5x12.

6.  8x10x12 works.

7.  If M=mug’s original price, N=napkin holders’s original price, then M = (2/3)N.  Clearly M<1.00 and N>1.00.  So the equation describing the sale prices is (M/2)+.10 = (N/4), which leads to $1.20 for N, $.80 for M, so with the sale prices Thompson got 30 cents in change.

8.  The prime factorization of (32)(88) consists of a great many 2’s and a single odd prime.  245 is odd, and (if there is a factoring over integers) represents the difference between a pair of factors of the product of (32)(88).  Ergo these two factors must have exactly one odd number between them, so the only possibility is 11 and 256.  This works, so we get (32x² + 256x) - (11x + 88), = (32x - 11)(x + 8).

Questions, corrections, comments:  Send me e-mail at  markthom@flash.net