To find out how many gallons of water the large fishbowl holds, we first need to determine the capacity of the small fishbowl.
The capacity of the small fishbowl is given as: \[ 2 \frac{2}{5} \text{ gallons} \]
To convert this mixed number to an improper fraction, we can do the following:
- Multiply the whole number (2) by the denominator (5): \[ 2 \times 5 = 10 \]
- Add the numerator (2): \[ 10 + 2 = 12 \] So, the small fishbowl can hold: \[ \frac{12}{5} \text{ gallons} \]
Next, we find the capacity of the large fishbowl which holds up to \(2 \frac{1}{3}\) times as much water as the small fishbowl. First, we need to convert \(2 \frac{1}{3}\) to an improper fraction:
- Multiply the whole number (2) by the denominator (3): \[ 2 \times 3 = 6 \]
- Add the numerator (1): \[ 6 + 1 = 7 \] So, \(2 \frac{1}{3}\) is: \[ \frac{7}{3} \]
Now, we calculate the capacity of the large fishbowl by multiplying the capacity of the small fishbowl by \(\frac{7}{3}\): \[ \text{Capacity of large fishbowl} = \left(\frac{12}{5}\right) \times \left(\frac{7}{3}\right) \]
Carrying out the multiplication: \[ \frac{12 \times 7}{5 \times 3} = \frac{84}{15} \]
Next, we simplify \(\frac{84}{15}\). First, we find the greatest common divisor (GCD) of 84 and 15:
- 84 can be factored as \(2^2 \times 3 \times 7\).
- 15 can be factored as \(3 \times 5\). The common factor is 3.
Now we divide both the numerator and the denominator by 3: \[ \frac{84 \div 3}{15 \div 3} = \frac{28}{5} \]
Next, we convert \(\frac{28}{5}\) into a mixed number:
- Divide 28 by 5 to find the whole number part: \[ 28 \div 5 = 5 \quad \text{(with a remainder of } 3\text{)} \] So the mixed number is: \[ 5 \frac{3}{5} \]
Thus, the large fishbowl holds: \[ \boxed{5 \frac{3}{5}} \] gallons of water.