First, find the fraction of paint remaining after painting the wall: \(1 - \frac{1}{3} = \frac{2}{3}\). Next, find the amount of paint used for the door, which is \(\frac{1}{2}\) of the remaining \(\frac{2}{3}\): \(\frac{1}{2} \times \frac{2}{3} = \frac{2}{6} = \frac{1}{3}\). The amount of paint left is the amount that remained after painting the wall minus the amount used for the door: \(\frac{2}{3} - \frac{1}{3} = \frac{1}{3}\). So, \(\frac{1}{3}\) of the original can is left.

If Leo sold \(\frac{2}{5}\) of his comic books, the fraction he has left is \(1 - \frac{2}{5} = \frac{3}{5}\). The problem states that this remaining fraction is equal to 18 comic books. So, \(\frac{3}{5}\) of the original total is 18. To find the original total, we can set up the equation \(\frac{3}{5} \times \text{Total} = 18\). Solving for the total gives \(\text{Total} = 18 \div \frac{3}{5} = 18 \times \frac{5}{3} = 6 \times 5 = 30\). Leo originally had 30 comic books.

First, find the total number of non-apple fruits. Since 6 oranges make up \(\frac{1}{2}\) of the non-apple fruits, the total number of non-apple fruits is \(6 \times 2 = 12\). Next, determine what fraction of the total fruits are non-apples. If \(\frac{2}{3}\) are apples, then \(1 - \frac{2}{3} = \frac{1}{3}\) of the fruits are non-apples. We know that this \(\frac{1}{3}\) is equal to 12 fruits. To find the total number of fruits, solve \(\frac{1}{3} \times \text{Total} = 12\). The total is \(12 \div \frac{1}{3} = 12 \times 3 = 36\).

First, calculate the amount of juice poured out: \(\frac{1}{4} \times 2 = \frac{1}{2}\) gallon. Next, find the amount of juice remaining: \(2 - \frac{1}{2} = 1\frac{1}{2}\) gallons. Sara and her two friends make a total of 3 people. They share the remaining juice equally, so divide the remaining amount by 3: \(1\frac{1}{2} \div 3 = \frac{3}{2} \div 3 = \frac{3}{2} \times \frac{1}{3} = \frac{3}{6} = \frac{1}{2}\) gallon. Each person gets \(\frac{1}{2}\) gallon of juice.

First, determine the fraction of girls who do not have brown hair. If \(\frac{1}{4}\) of the girls have brown hair, then \(1 - \frac{1}{4} = \frac{3}{4}\) of the girls do not. Next, find what fraction this group represents of the entire school. Since girls make up \(\frac{3}{5}\) of the school, multiply this by the fraction of girls without brown hair: \(\frac{3}{5} \times \frac{3}{4} = \frac{9}{20}\). Thus, \(\frac{9}{20}\) of the students are girls without brown hair.

To find the number of pieces, divide the total length of the ribbon by the length of each piece. First, convert the mixed number to an improper fraction: \(5\frac{1}{2} = \frac{11}{2}\). Now, divide \(\frac{11}{2}\) by \(\frac{1}{4}\): \(\frac{11}{2} \div \frac{1}{4} = \frac{11}{2} \times \frac{4}{1} = \frac{44}{2} = 22\). Therefore, 22 full pieces can be cut.

To find the number of shelves, divide the total length of the board by the length of one shelf. First, convert the mixed number to an improper fraction: \(2\frac{1}{4} = \frac{9}{4}\). Now, divide 12 by \(\frac{9}{4}\): \(12 \div \frac{9}{4} = 12 \times \frac{4}{9} = \frac{48}{9}\). To find how many full shelves can be cut, convert the improper fraction to a mixed number: \(\frac{48}{9} = 5\frac{3}{9} = 5\frac{1}{3}\). This means he can cut 5 full shelves and will have a piece of wood left over.

When you see a question asking "how many full cups can be filled," you're dealing with a division problem where you need to find how many smaller units fit into a larger unit.

To solve this, divide the total amount of water by the capacity of each cup: $$6 \div \frac{3}{8}$$. When dividing by a fraction, multiply by its reciprocal instead: $$6 \times \frac{8}{3}$$. This gives you $$\frac{6 \times 8}{3} = \frac{48}{3} = 16$$ full cups.

Let's examine why the other answers are incorrect. Choice A (2¼ cups) likely comes from incorrectly multiplying $$6 \times \frac{3}{8} = \frac{18}{8} = 2\frac{1}{4}$$, but this tells you how much water would be in 6 cups, not how many cups you can fill. Choice B (24 cups) might result from confusing the fraction and thinking each cup holds $$\frac{1}{4}$$ liter, leading to $$6 \div \frac{1}{4} = 24$$. Choice C (18 cups) could come from multiplying 6 by the numerator 3, ignoring the denominator entirely.

Remember: when a problem asks "how many of X fit into Y," you're dividing Y by X. The key trap here is remembering to flip and multiply when dividing by fractions, and being careful not to multiply when you should divide. Always check if your answer makes sense—16 cups holding $$\frac{3}{8}$$ liter each should give you close to 6 liters total.

First, find the initial number of adults: \(\frac{2}{3} \times 60 = 40\) adults. Next, find the number of adults who left: \(\frac{1}{4} \times 40 = 10\) adults. Finally, subtract the number of adults who left from the initial number of adults to find how many remain: \(40 - 10 = 30\) adults.

To find the number of smaller lots, divide the total area of the land by the area of each lot. The calculation is \(10 \div \frac{2}{5}\). To divide by a fraction, multiply by its reciprocal: \(10 \times \frac{5}{2} = \frac{50}{2} = 25\). He can create 25 lots.