Category Archives: Problem Solving

Activities to help students learn how to solve math problems

Marathon Mitch

My nephew, site Mitch, pharm loves to run competitively, like cross country races, marathons, or whatever. Recently, he competed in a special for-charity race in a small Kansas town near where he lives: the Midtown City Marathon.

On this occasion, he paced himself in the following way: he ran the first 50% of the distance at his normal, preferred running speed. Then for the next third of the distance he increased his tempo by 25%. For the final portion of his run (3 km), he increased his previous speed by another 20%.

Given that he finished this marathon in the fantastic time of 1 hour, 19 minutes, state now his running speed (i.e. in kilometers per hour) for the final portion of the event.

Extra: How many minutes did Mitch spend running at each of the 3 speeds?

A Quartet of Integers

I have a quartet of integers with which I’ll present a little number problem for you to solve. Here are some facts to work with:

  1. Twice the first integer, minus the second, equals 7.
  2. Twice the second integer, minus the third, equals 4.
  3. Twice the third integer, minus the fourth, equals 3.

Now there are an infinite number of quartets of integers that share those three facts, so I will now add an additional condition:

When you subtract the first integer of my particular group from twice the fourth integer, you get the minimum positive integral value for this operation of all possible cases.

What is the sum of my four integers?

Two Differences to a Sum

Two positive numbers are such that their difference is 7 and the difference of their squares is 105. What is their sum?
Your task:

1. Show your kid brother (or sister) what this means by solving it in an arithmetical manner. Be patient.
2. Show your parents how much you’ve learned in Algebra 1 this year by solving it with basic algebra. This means, you just demonstrate what the sum is, without determining what the two numbers are.

Alvin’s Theorem

While Ms. Powers was leading a class discussion about square numbers, Absent-minded Alvin was in another “world”, looking for interesting patterns in the topic. Shortly, he raised his hand and said, “Ms. Powers, I’ve found something rather nice. Look. If I take 2 consecutive squares and subtract them, the difference is always the sum of 2 consecutive integers.””Show the class what you mean by that, Alvin,” said the teacher.

Alvin wrote the following on the board:

49 – 36 = 13 and 13 = 6 + 764 – 49 = 15 and 15 = 7 + 8

Turning to the class, he shyly said, “I call this ‘Alvin’s Theorem‘.”

Ms. Powers smiled and said, “Very good, but if you want to call it a theorem, you must be able to prove it is always true for all numbers, using algebra.”

Alvin replied, “Oh yes, I can do that too. Here’s how.”

What did Alvin write on the board now?

Later, Alvin investigated the matter of the difference of consecutive “even” squares. What do you imagine he discovered this time?

Hiking Along the Hypotenuse

Cassie and her big brother, ask Charlie, illness like right triangles. One day they decided to walk along the legs and hypotenuse of a right triangle, viagra doing so in the following manner.

Starting at the same point, Cassie walked 5 kilometers straight north, while Charlie walked 12 kilometers straight east. Then each turned and walked towards each other along the hypotenuse, stopping when they met. It is also important to note that they walked at the same speed.

Your task is to determine the coordinates of the point where they are now standing.

Hint: Consider their starting point as the origin (0, 0) of a coordinate grid.

Extra: If Charlie walked twice as fast as his little sister, thereby covering twice as much distance, what would be the coordinates of their meeting point now?

A Real Estate Dealer

(Here is a problem from a math book published in 1900, there titled The New Higher Arithmetic, thumb by A. W. Rich.)

A dealer has land valued at $40, $60, $72, and $80 per acre. How many acres must he sell of each kind that it may average him $68 per acre? Prove your results correct. Can there be another result?

Can you solve this problem from over 100 years ago?

Hint: the average number of acres sold at these prices is 1.75.

Then I’d Rather Not Know

Mrs Holmann could not take it any longer that everyone kept asking her about her son’s age. This time she made her answer complicated enough for people never to ask again… neither her son’s age, nor her own! She said: “If you square my age and subtract the square of my son’s, the number you get is exactly 2 times the square of the difference between our ages. Also, if you square his age and you add it to the square of mine, the number you get is exactly 7776 less than 4 times the square of the sum of our ages!”

Can you work out how old they are?

[Note: this problem was written by Zsuzsanna Sukosd, a 15-year-old high school math student from Denmark.]

A New Spirit for Your Investments

     The year 2002 was not a good one for Wall Street and the investors in the stock market. It was certainly a tough time for many people. Then I saw this clever joke on the internet.


If you had bought $1000.00 worth of Nortel stock one year ago, it would now
be worth $49.00.

With Enron, you would have $16.50 of the original $1,000.00.

With WorldCom, you would have less than $5.00 left.

If you had bought $1,000.00 worth of Budweiser (the beer, not the stock)
one year ago, drank all the beer, then turned in the cans for the 10 cent
deposit, you would have $214.00.

Based on the above, our current investment advice is to drink heavily and

     But, once my chuckles subsided, my math mind kicked in, thinking: Gee, I wonder if that’s really true? I mean the part about earning $214.00 recycling the beer cans.

     Then my teacher side took over. Wouldn’t this make a nice middle school math problem? To verify the statement in the joke would require some research about the price of beer in various supermarkets or mini-marts, etc. Then there is the matter of the deposit one would receive by turning in the empty cans. Maybe it isn’t 10 cents everywhere. And so forth.

     Surely it would spark a lot of discussion.

     And in the case it wouldn’t be politically correct to discuss alcoholic beverages in the school classroom, well, it’s simple to substitute one’s favorite soft drink instead. In fact, a class could investigate different brands of colas. Afterall, it’s essentially the math that’s important here, and recycling the aluminum cans, not what was inside them originally, right?

     Well, think about it, won’t you? And report back to WTM what you discover, okay? Thanks.

Chocolate Chips

     Everyone knew that Charlie had a “sweet tooth”, ailment especially for anything with chocolate in it. So no one was very surprised to hear of his recent project. The only surprise was the manner he conducted it.

     He decided to eat one chocolate chip on the first day, search then 2 chips on the second day, 3 on the 3rd day. And so on. Each day he ate one more than the day before.

     By the last day of this current month, he will have consumed a total of 1035 chips.

     When did Charlie start this sweet journey?

Extra: On what date did Charlie reach the half-way point in his project? And which chip (1st, 2nd, 3rd, etc.) was it that day?

[Note: Since this is an “algebra” problem, it is expected that you will use normal algebra to solve it. This means that no brute force methods, such as spreadsheets or using a calculator to merely add up the numbers 1, 2, 3, 4, and so on, will be accepted. There is a famous formula that you should know about that automatically finds the sum of an expression like:

1 + 2 + 3 + … + n = ?

We hope you can find and use it.]

The Rub-a-dub-dub Restaurant

     In Mother Goose City, the most elegant restaurant where the elite meet to eat is the Rub-a-dub-dub Restaurant. It is owned and operated by those three wild and crazy guys: the butcher, the baker, and the candlestick maker. (Perhaps you will recall, they often traveled by tub!)

     When they decided to become partners in this expensive project, each promised to contribute as much money as he could, according to the funds he had in his bank account. The results can be described as follows:

  1. the ratio of the funds contributed by the butcher and baker was 3 to 5, respectively.
  2. the amount of money contributed by the candlestick maker was equal to twice the amount of the butcher less one-half that of the baker.

  3. the total amount of money raised by the three investors was greater than 100 mogolas and less than 125 mogolas. (A mogola is an informal unit of money in Mother Goose Land, similar to our use of “grand” to mean $1000.)

     Assuming that each man’s contribution was an integral number of mogolas, how much money did they have to start this culinary adventure?