Tag Archives: algebra

Tire Wear

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Thanks to Futility Closet I discovered a new source of math puzzles: A+Click.

“A+ Click helps students become problem solvers. Free, without ads, no calculators, and without signing-up. The website features a graduated set of 16,000+ challenging problems for students in grades one through twelve, starting from the very simple to the extremely difficult. … The questions concentrate on understanding, spatial reasoning, usefulness, and problem solving rather than math rules and theorems. The problems include a short description and an illustration to help problem solvers visualize the model. The problems can be solved within one minute and without using a calculator.”

My only quibble with “The questions concentrate on understanding, spatial reasoning, usefulness, and problem solving rather than math rules and theorems.” is that by keeping explicit math notation  and concepts to a minimum, the use of symbolic algebra and calculus is muted and there is a whiff of the medieval reliance on mental verbal agility rather than the power of the new mathematics. 

Still the problems are imaginative and challenging.  Here is a good example.

“The rear tires of my car wear out after 40,000 miles, while the front tires are done after 20,000 miles.  Estimate how many miles I should drive before the tires (front and rear) are rotated to drive the maximal distance.

Answer Choices:    15,000 miles     12,000 miles     13,333 miles     16,667 miles”

(I admit solving these under a minute is a challenge, at which I often failed.  Ignoring time constraints allows for greater care and a more thorough mulling over the intricacies of the problem.  Yes, those who have mastered math can solve problems faster than those who have not, but real mastery of math requires an inordinate attention to details, and that requires time.)

Answer.

See Tire Wear for solutions.

Mystery Quadratic

Presh Talwalkar has an interesting new problem.

“Students and teachers found a recent test in New Zealand to be confusing and challenging for covering topics that were not taught in class.

For the equation below, find the value of k for which the equation has numerically equal but opposite signs (for example, 2 and –2):”

The problem didn’t mention how old the students were, but the solution to another problem on the test indicates they needed to know calculus.

Answer.

See Mystery Quadratic for a solution.

Air Travel

This is a nice problem from Five Hundred Mathematical Challenges.

“Problem 62. A plane flies from A to B and back again with a constant engine speed.  Turn-around time may be neglected.  Will the travel time be more with a wind of constant speed blowing in the direction from A to B than in still air?  (Does your intuition agree?)”

Answer

See Air Travel for a solution.

Language Students Puzzle

This is a slightly challenging problem from Dan Griller.

“Every pupil at the Euler Academy studies French or Spanish.  At the start of the year, one third of the French students also studied Spanish, and 2 fifths of the Spanish students also studied French.  After one term, six of the double-linguists dropped French, so that now only a quarter of the French students study Spanish.  How many pupils are at the Euler Academy?”

Just to be clear, “French students” means Euler Academy pupils studying French, and similarly for “Spanish students.”

Answer.

See the Language Students Puzzle for solution.

Al the Chemist I

This is a relatively simple problem from the inventive Raymond Smullyan in the “Brain Bogglers” section of the 1996 Discover magazine.

“AL THE CHEMIST—not an alchemist, though his name might suggest it­—one day partially filled a container with some concoction or other. He knew the volume of fluid in the container, as well as the volume of empty space, and real­ized that two-thirds of the former was equal to four-fifths of the latter. Was the container then less than half full, more than half full, or exactly half full?”

Answer.

See Al the Chemist I for solution.

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