The Dogs-and-Ducks Problem

Recently, while reviewing some grade 5 olympiad questions for a local company, which specializes in conducting olympiad math programs in elementary schools, I came across the following nonroutine dogs-and-ducks question.



The number of ducks is 10 fewer than the number of dogs. 
The dogs have three times as many legs as the ducks. 
How many ducks and how many dogs are there?



When I realized that my mental calculated answer differs from the one given by the problem poser, I thought maybe my incorrect solution might serve as a "logic exercise" to tickle others' mathematical bones to point out the flawed reasoning. My faulty argument followed something along these lines:

Since there are 10 fewer ducks, and each duck has 4 legs, there are 10 × 4 = 40 fewer duck legs.



From the model drawing,



2 units = 40


1 unit = 20



Number of duck legs = 1 unit = 20


Number of ducks = 20 ÷ 2 = 10



Number of dog legs = 3 units = 3 × 20 = 60


Number of dogs = 60 ÷ 4 = 15



A quick check shows that the answers don't satisfy the given conditions in the question. Although the dogs have three times as many legs as the ducks, however, the number of ducks is only 5 fewer than the number of dogs, which is expected to be 10.




When the why is harder than the how



The lesson here is to identify the illogical step, although seasoned problem solvers would probably not commit this kind of error or blunder. 

In general, it's one thing not to make an error, and get the correct answer; it's another thing to be able to explain someone else's error.

At a deeper level, it's sometimes conceptually harder to explain someone's flawed reasoning to an incorrect answer than just know how to solve a problem. Think of mathematical fallacies and paradoxes, where in a number of cases the illegal step is anything but obvious.




A fabricated faux solution

Alternatively, if we argued that a dog has 2 more legs than a duck, we'd then have a model that looks like this:

In this case, students would easily realize that the answers are also incorrect, when they find out that the number of dogs isn't an integer.


Two quick-and-dirty solutions

Why don't you try solving the dogs-and-ducks question on your own first, before comparing your solution with the ones I've worked out? If yours is an alternative (or elegant) solution, the mathematical brethren couldn't wait to reading it!

Method 1

Since a dog has twice as many legs as a duck, and there are 3 times as many dog legs as duck legs, there are 3/2 times as many dogs as ducks.

A model depicting the above relationship may be drawn as follows:

From the model drawing,

1 unit = 10

Number of ducks = 2 units = 2 × 10 = 20

Number of dogs = 3 units = 3 × 10 = 30

Check: Ducks: 20 × 2 = 40 legs

Dogs: 30 × 4 = 120 legs = 3 × 40 legs

Method 2

Given: Number of dogs = Number of ducks + 10

Number of duck legs = 2 × Number of ducks

Number of dog legs = 4 × Number of dogs

Given: Number of dog legs = 3 × Number of duck legs

A quick-and-dirty model representing the above information may be drawn as follows:

From the model drawing,

3 × 2 units = 6 units = 4 units + 10 + 10 + 10 + 10 

2 units = 10 + 10 + 10 + 10

1 unit = 10 + 10 = 20

1 unit + 10 = 30

Therefore, there are 20 ducks and 30 dogs.

Method 3 (Sakamoto method)

Those of you who are versed with the three-step Sakamoto method in solving word problems, may proceed as follows:

1. Grasp the relation

Let ④ represent the number of duck legs.

Duck legs           Dog legs

④                     ⑫

__________________________

Number of ducks     Number of dogs

④ ÷ 2 = ②                 ⑫ ÷ 4 = ③

                                             – 10

__________________________

1              :              1

                             + 10

2. Diagram

3. Number sentences

③ – ② = ① = 10

② = 2 × 10 = 20 (ducks)

③ = 3 × 10 = 30 (dogs)

The number of ducks is 20, and the number of dogs is 30.

Two bonus problems

If you like being challenged by similar dogs-and-ducks or chickens-and-rabbits problems, may I direct you to solving two more questions at Yan's One Minute Math Blog?

Happy Mathematical Problem Solving!


© Yan Kow Cheong, March 3, 2013