Legs at the Party

Combinatorics · Easy · Free problem

A variety of multi-legged critters show up to a function. Wasps have $6$ legs each, horses have $4$ legs each, and ducks have

$ legs each. There are five times as many ducks as horses, and twice as many wasps as ducks. The total number of legs in the room is $888$. How many wasps showed up?

Hints

  1. All animal counts can be expressed as multiples of the number of horses -- let $n$ denote the number of horses.
  2. Write the total leg count as a linear equation in $n$: each animal type contributes (number of that animal) times (legs per animal).
  3. With $n$ horses, there are $5n$ ducks and
    0n$ wasps, giving $4n + 10n + 60n = 74n = 888$.

Worked Solution

How to Think About It: Everything in this problem is expressed as a multiple of the number of horses, so let $n$ be the number of horses and express everything else in terms of $n$. Once you have a single linear equation in $n$, solve it and scale up to get the wasp count.

Quick Estimate: The wasp-to-horse ratio is

0:1$, and wasps have the most legs ($6$ each), so they dominate the leg count. With
0n$ wasps contributing $60n$ legs, $5n$ ducks contributing
0n$ legs, and $n$ horses contributing $4n$ legs, total legs $= 74n$. At $888$ total legs, $n \approx 888 / 74 \approx 12$. So roughly
20$ wasps.

Approach: Set up one linear equation in $n$ and solve exactly.

Formal Solution:

Let $n$ = number of horses. Then: - Ducks: $5n$ (five times as many as horses) - Wasps:

\times 5n = 10n$ (twice as many as ducks)

Total legs: $4n + 2(5n) + 6(10n) = 4n + 10n + 60n = 74n = 888$

Solving: $n = 12$.

Number of wasps:

0n = 10 \times 12 = 120$.

Answer: $\boxed{120}$ wasps.

Intuition

The key move in these multi-variable word problems is to reduce everything to one variable as quickly as possible. When every quantity is stated as a ratio of some base quantity -- here, all counts are multiples of the number of horses -- you should immediately set $n$ to be that base and express the others in terms of $n$. One equation, one unknown.

This pattern appears constantly in interview brain teasers: the problem looks like it has three unknowns, but the ratio constraints collapse it to one. Recognizing that structure early is the sign of a clear thinker under time pressure.

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