-1
$\begingroup$

The Problem

A duck has two legs. When a duck folds one leg, only one leg is visible. When a duck is sitting, neither of its legs is visible.

When Roman went to the lake, there were 33 ducks. He counted all the visible legs and found 32 visible legs. How many ducks folded one leg, assuming that the number of sitting ducks was half the number of the ducks that were either standing on one leg or two legs?

My Approach

I reasoned that if the total number of legs is 33×2=66, and 32 visible legs were counted, then the remaining 34 legs must belong to the sitting ducks. Since sitting ducks don't show their legs, I thought that these 34 legs must be the legs of the sitting ducks, and I tried to divide them to figure out how many sitting ducks there are.

However, I'm not getting the correct solution.

Where I Think I Went Wrong

I thought that the 34 unseen legs belong to sitting ducks, but I’m not sure if that’s the right approach. Could someone help me understand where I went wrong, and how to properly solve this problem?

Thank you in advance for any help

New contributor
user205312 is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
$\endgroup$
5
  • 2
    $\begingroup$ I would try s=sitting ducks, t1 = standing 1 leg, t2 = standing 2 legs and set up some equations. $\endgroup$ Commented 2 hours ago
  • $\begingroup$ BTW it may easily be that several visible legs were those of one leg folded ducks $\endgroup$ Commented 2 hours ago
  • 1
    $\begingroup$ If the sitting ducks are half the other ducks then they are $11$ ($1/3$ of the total ducks). Now, the total number of legs if all the other ducks were standing on both feet would be $44$. So it must be that $44-32 = 12$ ducks are lifting one leg. The remaining $10$ ducks sit on both legs. $\endgroup$ Commented 2 hours ago
  • $\begingroup$ I want to understand my mistake, not the correct answer $\endgroup$ Commented 1 hour ago
  • $\begingroup$ As you pointed out yourself, you attributed all the "missing legs" to sitting ducks. But the problem statement says that those ducks are only half of all the other ducks. This implies that $1/3$ of the ducks are sitting, and the remaining $2/3$ are standing either on one leg or on two legs. $\endgroup$ Commented 1 hour ago

1 Answer 1

Reset to default
-1
$\begingroup$

I want to understand my mistake, not the correct answer

We are trying to tell you that simply by solving the problem properly and obtaining the answer, you would have been able to deduce where you are wrong.

Let $n$ be the number of ducks sitting down so that no leg is showing, $f$ for the number of ducks folding one leg and $s$ for the standing ducks with both legs visible.

Then $$ \begin{align} \tag1n+f+s&=33\\ \tag2f+2s&=32\\ \tag3f+s&=2n \end {align} $$ the solution is $$ \begin{align} \tag43n&=33\quad\implies\quad n=11\quad\implies\quad f+s=22\\ \tag5\implies\quad s&=10\quad\implies\quad f=12 \end {align} $$ Your argument is $$ \begin{align} \tag62n+2f+2s&=66\qquad\text{(correct)}\\ \tag72n&=66-32=34 \end {align} $$ Your Equation (7) did the subtraction correctly but is manifestly wrong.

The reasoning is just wrong because you forgot that the folded legs are also not seen.

$\endgroup$

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.