Question

The number of ducks in a pond is equal to the square root of the number of fish, and the number of turtles is equal to the cube root of the number of fish. The number of fish in the pond is more than 1 but less than 100. There are no fractional or negative numbers of ducks, fish, or turtles.

There are fish, ducks, and turtles in the pond.

Answer 1

There are 64 fish, 8 ducks, and 4 turtles in the pond.

Explanation:

Let D denote the number o ducks, F the number of fish, and T the number of turtles. The following equations can be modeled from the given information:

`D=\sqrt F\\T=\sqrt[3]{F}`

If F is an integer between 1 and 100, it needs to be both a perfect square and a perfect cube.

The perfect squares from 1 to 100 are:

4, 9, 16, 25, 36, 49, 64, 81, 100.

The perfect cubes from 1 to 100 are:

8, 27, 64.

As we can see, 64 is the only number that fits both criteria, therefore F = 64 and:

`D=\sqrt 64\\D=8\\T=\sqrt[3]{64}\\T=4`

There are 64 fish, 8 ducks, and 4 turtles in the pond.