Enzo has fish in two aquariums. In one aquarium, the ratio of the number of guppies to the number of goldfish is 2:3. In the other, this ratio is 3:5. If Enzo has 20 guppies in total, the least number of goldfish that he could have is
(A) 29
(B) 30
(C) 31
(D) 32
(E) 33
Enzo has fish in two aquariums. In one aquarium, the ratio of the number of guppies to the number of goldfish is 2:3. In the other, this ratio is 3:5. If Enzo has 20 guppies in total, the least number of goldfish that he could have is
(A) 29
(B) 30
(C) 31
(D) 32
(E) 33

## Solution 1

The following tables give the possible numbers of fish in each aquarium.

The three lines join the results which give a total of 20 guppies, namely $2+18,8+12$ and $14+6$. The corresponding numbers of goldfish are 33, 32 and 31. The least number of goldfish that he could have is 31 .

| 1 st aquarium |  | 2nd aquarium |  |
| :---: | :---: | :---: | :---: |
| Number of  guppies | Number of  goldfish | Number of  guppies | Number of  goldfish |
| 2 | 3 | 3 | 5 |
| 4 | 6 | 6 | 10 |
| 6 | 9 | 9 | 15 |
| 8 | 12 | 12 | 20 |
| 10 | 15 | 15 | 25 |
| 12 | 18 | 18 | 30 |
| 14 | 21 |  |  |
| 16 | 24 |  |  |
| 18 | 27 |  |  |

## Solution 2

In the first aquarium, the ratio of the number of guppies to goldfish is 2:3 so let the actual number of guppies be $2 a$ and the actual number of goldfish be $3 a$. In the second aquarium, if we apply the same reasoning, the actual number of guppies is $3 b$ and the actual number of goldfish is $5 b$. In total, there are 20 guppies so we now

| $2 a+3 b$ | $a$ | $b$ | $3 a+5 b$ |
| :---: | :---: | :---: | :---: |
| 20 | 1 | 6 | 33 |
| 20 | 4 | 4 | 32 |
| 20 | 7 | 2 | 31 |

have the equation, $2 a+3 b=20$. We consider the different possibilities for $a$ and $b$ also calculating $3 a+5 b$, the number of goldfish. From the chart, we see that the smallest possible number of goldfish is 31 .

ANSWER: (C)