Problems of Convenience

The candy that my sons received while trick-or-treating all had names with references to various disciplines:

• Baby Ruth – history; named for the daughter of President Grover Cleveland
• Snickers – sports; named after Frank C. Mars’s favorite horse
• 3 Musketeers – literature; named for Athos, Porthos, and Aramis from Alexandre Dumas’s novel
• Milky Way – astronomy; named after the galaxy
• 5th Avenue – geography; named after 5th Avenue in Reading, PA, where the candy bar was originally made

While there was no candy with references to advanced mathematics, several at least had numbers in the names. In addition to 3 Musketeers and 5th Avenue, there were also:

• Zero
• Take 5
• 100 Grand

I visited several local convenience stores to find other candy bars with numbers or math in the name. Sadly, my search yielded no others. Luckily, interesting things always happen when I’m in convenience stores…

A woman walks into a 7-Eleven and takes four items to the cash register. The clerk informs her that the register is broken, but he can figure the total using his calculator. The clerk then proceeds to multiply the prices together and declares that the total is $7.11. Although the woman knows the prices should have been added, not  multiplied, she says nothing — as it turns out, the result would have been $7.11 whether the four prices were added or multiplied.

There was no sales tax. What was the cost of each item?

Of course, you may be thinking, “If the four prices were multiplied together, the total would actually be 7.11 dollars⁴.” And you would be correct. But for the sake of the problem, it’s best not to introduce “quartic dollars” as a unit of measure. I’ll ask that you please suspend disbelief, at least until you’ve solved the problem.

The problem above involves four items, and finding its solution is quite difficult. To reduce the level of difficulty, I wondered if an analogous problem could be created that involves only three items. After an hour of playing with Excel, I was able to create such a problem.

A woman walks into a 6-Sixty and takes three items to the cash register. The clerk informs her that the register is broken, but he can figure the total using his calculator. The clerk then proceeds to multiply the prices together and declares that the total is $6.60. Although the woman knows the prices should have been added, not  multiplied, she says nothing — as it turns out, the result would have been $6.60 whether the three prices were added or multiplied.

There was no sales tax. What was the cost of each item?

The problem with only three items is not significantly less difficult than the problem with four items, however, it is helped by the fact that there are two different solutions. Still, I wondered if an analogous problem could be created that involves only two items. Sure enough, one could.

A woman walks into an 8-Forty-One and takes two items to the cash register. The clerk informs her that the register is broken, but he can figure the total using his calculator. The clerk then proceeds to multiply the prices together and declares that the total is $8.41. Although the woman knows the prices should have been added, not  multiplied, she says nothing — as it turns out, the result would have been $8.41 whether the two prices were added or multiplied.

There was no sales tax. What was the cost of each item?

This last problem is far less difficult than the other two. Enjoy!