- Identify the starting amount \(\frac{5}{6}\), the amount used \(\frac{1}{5}\), and the amount added \(\frac{1}{3}\).
- Find a common denominator for the fractions, which is 30.
- Convert the fractions to \(\frac{25}{30}, \frac{6}{30},\) and \(\frac{10}{30}\).
- Calculate the final amount as \(\frac{25-6+10}{30} = \boxed{\frac{29}{30}}\).
Explanation
- Problem Analysis
Joe starts with \(\frac{5}{6}\) cup of paint. He uses \(\frac{1}{5}\) cup, which means we subtract this amount from the initial total. Finally, he adds \(\frac{1}{3}\) cup, so we add this to the remaining amount. The goal is to find the total volume of paint currently in the container.
- Calculating the Final Amount of Paint
To find the total amount of paint, we calculate the sum of the initial paint, the amount subtracted (the paint he used), and the amount added. We express this as:
\[F = \frac{5}{6} - \frac{1}{5} + \frac{1}{3}\]
To perform the addition and subtraction, we must find a common denominator for the fractions 6, 5, and 3. The least common multiple of 6, 5, and 3 is 30. We convert each fraction to an equivalent one with a denominator of 30:
\[\frac{5}{6} = \frac{5 \times 5}{6 \times 5} = \frac{25}{30}\]
\[\frac{1}{5} = \frac{1 \times 6}{5 \times 6} = \frac{6}{30}\]
\[\frac{1}{3} = \frac{1 \times 10}{3 \times 10} = \frac{10}{30}\]
- Final Result and Conclusion
Now that we have the fractions with a common denominator, we can perform the arithmetic:
\[F = \frac{25}{30} - \frac{6}{30} + \frac{10}{30}\]
\[F = \frac{25 - 6 + 10}{30} = \frac{29}{30}\]
Thus, Joe has \(\frac{29}{30}\) cups of paint now. The final answer is \(\boxed{\frac{29}{30}}\).
Examples
This type of calculation is very useful in everyday life, such as when cooking or baking. If a recipe calls for a certain amount of liquid or flour, and you use a part of it and add more, you need to track the remaining amount. Understanding how to work with fractions with different denominators ensures you measure ingredients accurately, maintaining the correct proportions in your dishes.