Niall has three large fish tanks and would like to put some tetras and guppies in each tank. He currently has \(19\) tetras and \(18\) guppies and doesn’t want any of his tanks to contain more guppies than tetras. Also, he would like each tank to have at least \(5\) tetras and \(3\) guppies.
Determine the largest number of fish that can be in one of his fish tanks.
Since each fish tank must have at least \(5\) tetras and \(3\) guppies, we will start by placing \(5\) tetras and \(3\) guppies into each fish tank. The number of fish now in each tank is summarized in the table below.
| Tank 1 | Tank 2 | Tank 3 | |
|---|---|---|---|
| Number of tetras | \(5\) | \(5\) | \(5\) |
| Number of guppies | \(3\) | \(3\) | \(3\) |
We have placed \(5 \times 3 = 15\) tetras and \(3 \times 3=9\) guppies, so we have \(19-15=4\) tetras and \(18-9=9\) guppies left to place.
In order to place the largest number of fish in a tank, as many as possible of the remaining fish should be placed in one particular tank. We will start by adding all of the remaining tetras to Tank \(1\). The number of fish now in each tank is summarized in the table below.
| Tank 1 | Tank 2 | Tank 3 | |
|---|---|---|---|
| Number of tetras | \(9\) | \(5\) | \(5\) |
| Number of guppies | \(3\) | \(3\) | \(3\) |
Since Niall doesn’t want any of his tanks to contain more guppies than tetras, Tank \(1\) can contain at most \(9\) guppies. Since we want Tank \(1\) to contain the largest number of fish, we will add \(6\) guppies to Tank \(1\).
Then we have \(9-6=3\) guppies left to place. We cannot add them to Tank \(1\). However, we can add \(2\) guppies to Tank \(2\), and \(1\) guppy to Tank \(3\). The number of fish now in each tank is summarized in the table below.
| Tank 1 | Tank 2 | Tank 3 | |
|---|---|---|---|
| Number of tetras | \(9\) | \(5\) | \(5\) |
| Number of guppies | \(9\) | \(5\) | \(4\) |
Therefore, the largest number of fish that can be in one of Niall’s fish tanks is \(18\).