“Working Together” Problem

[The title of this post in the old blog was actually “if there’s cold beer in the fridge.” It’s dumb, but I did end the post with “Forgive me for being such a delinquent, but my last day of school is not until June 21. I’m losing it.” So I have an excuse for everything.]

Heya, back-to-back post about a problem from Five Triangles mathematics.


When I tweeted how much I love this problem, a few people did not feel the same at all. Here are my reasons for appreciating this problem:

  1. It’s a notched up “working together” problem that I have not seen before.
  2. It has percentages and fractions.
  3. I can use rectangles to solve this. (I was asked on Twitter how I would solve this using rectangles, hence this post.)
  4. I had to work on this problem. This is a big reason for me. We should assume that if we’re teaching a particular math subject — Geometry, Statistics, or Calculus — that we’re able to easily do all the exercises in the textbook. A set of exercises allows us to practice a particular skill. But a problem should require us to think. I hope I’ve encouraged problem-solving enough with my students that they value a problem more when they have to struggle with it, when they don’t know immediately how to start it, when they get stuck and become frustrated, when they seek others for help, when they can leave the problem and come back to it another day.

While I’m at it, I also love the site Five Triangles in general for a couple of reasons:

  1. The Geometry problems are simply stated and interesting. They make me pause and think, very few have been automatic gimmes.
  2. The solutions are not posted. I really appreciate this because if they were, we might be tempted (mainly due to lack of time) to check the answers too early before we allow ourselves a chance to work through the problem and perhaps struggle with it. “Anticipating” is the first of 5 Practices that gives us insight on how students might solve the problem.

I did, however, retype the question above so it’s easier to read and track information. I also numbered the paragraphs for quicker reference.


How we worked through this problem. Colors and all.

Draw a rectangle to represent the task. It has an area of 80 square units because that’s the LCD of the three fractions in the problem.


Because this grid represents the task, we use it to fill in the amount of work done. Paragraph [3] is the first concrete piece of information that allows us to do this.


We continue to fill in the work done as described in paragraph [4].


Paragraph [5] is the first piece of information that allows us to figure out C’s rate. Knowing that C can do 16 boxes in 8 hours means C can do the task — 80 boxes — in 40 hours.


With C’s rate, we can now take on paragraph [2]. We know from the last step that C’s hourly rate working alone is 2 boxes per hour or 10 boxes in 5 hours. But when working with A, C’s rate is 40% faster, therefore instead of getting just 10 boxes done, C can get 14 boxes done in 5 hours when working with A.

From picture above in green, we know A and C did 24 boxes in 5 hours, and since C was responsible for 14 of those, the remaining 10 boxes were done by A.

Then A’s hourly rate when working with C is 2 boxes per hour. Because this hourly rate represents a 20% increase than if A were to work alone, the math we need to do is 2 boxes divided by 1.2 to get 5/3 boxes. Solving for x in the proportion below gives us the answer that A completes the task in 48 hours.


Lastly we use paragraph [1] to figure out rate for B. We know A’s alone rate is 5/3 boxes per hour, but when working with B, A’s rate is 40% faster. Thus we multiply 5/3 by 1.4 to get 7/3. If A can do 7/3 in 1 hour, then A can do 35/3 in 5 hours when working with B.

The yellow boxes show that A and B can do 25 boxes in 5 hours, so subtracting 35/3 from 25, we see that B did 40/3.

To get B’s alone rate, we divide 40/3 by 1.2 (because B is 20% faster when working with A) to get 100/9. Solving the proportion below gives us the answer of B completing the task in 36 hours.


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