First, calculate the length Liam painted: \(72 \times \frac{3}{8} = 9 \times 3 = 27\) feet. Next, find the length of the remaining portion: \(72 - 27 = 45\) feet. Then, calculate the length Chloe painted, which is \(\frac{1}{3}\) of the remaining part: \(45 \times \frac{1}{3} = 15\) feet. Finally, find the difference between the lengths they painted: \(27 - 15 = 12\) feet. Liam painted 12 more feet than Chloe.

First, calculate the total area of the painting by multiplying its length and width. Convert mixed numbers to improper fractions: \(3\frac{1}{2} = \frac{7}{2}\) and \(2\frac{1}{4} = \frac{9}{4}\). Area = \(\frac{7}{2} \times \frac{9}{4} = \frac{63}{8}\) square feet. Next, find \(\frac{2}{3}\) of this area: \(\frac{63}{8} \times \frac{2}{3} = \frac{126}{24}\). Simplify by dividing the numerator and denominator by their greatest common divisor, 6: \(\frac{21}{4}\). Convert this to a mixed number: \(5\frac{1}{4}\) square feet.

Let C be the total capacity. The problem states that \(\frac{3}{5}\) of the total capacity is 45 gallons, so \(\frac{3}{5}C = 45\). To find C, divide 45 by \(\frac{3}{5}\): \(C = 45 \div \frac{3}{5} = 45 \times \frac{5}{3} = \frac{225}{3} = 75\). The total capacity is 75 gallons.

First, calculate the sale price. A discount of \(\frac{1}{4}\) means the price is \(1 - \frac{1}{4} = \frac{3}{4}\) of the original. Sale price = \($320 \times \frac{3}{4} = $240\). The coupon gives an additional \(\frac{1}{5}\) off this sale price, so the customer pays \(1 - \frac{1}{5} = \frac{4}{5}\) of the sale price. Final price = \($240 \times \frac{4}{5} = $192\).

To find the distance traveled, multiply the total length of the trail by the fraction completed. First, convert the mixed number to an improper fraction: \(12\frac{1}{2} = \frac{25}{2}\). Now multiply: \(\frac{25}{2} \times \frac{2}{5}\). You can simplify by canceling the 2s and dividing 25 by 5: \(\frac{25}{2} \times \frac{2}{5} = \frac{5}{1} \times \frac{1}{1} = 5\). The hiker has traveled 5 kilometers.

To find the number of batches, divide the total amount of sugar by the amount needed per batch. First, convert the mixed number to an improper fraction: \(13\frac{1}{2} = \frac{27}{2}\). Now, divide: \(\frac{27}{2} \div \frac{3}{4} = \frac{27}{2} \times \frac{4}{3}\). Simplify by cross-cancellation: \(27 \div 3 = 9\) and \(4 \div 2 = 2\). The calculation becomes \(\frac{9}{1} \times \frac{2}{1} = 18\). The baker can make exactly 18 full batches.

This is a division problem involving mixed numbers and fractions. When you need to find how many times one quantity fits into another, you divide the total by the individual amount.

To find the number of laps, you need to divide the total distance by the length of one lap: $$3\frac{1}{2} \div \frac{1}{4}$$

First, convert the mixed number to an improper fraction: $$3\frac{1}{2} = \frac{7}{2}$$

Now divide: $$\frac{7}{2} \div \frac{1}{4}$$

Remember that dividing by a fraction is the same as multiplying by its reciprocal: $$\frac{7}{2} \times \frac{4}{1} = \frac{28}{2} = 14$$

The runner completed 14 laps, making D correct.

Looking at the wrong answers: A) $$\frac{7}{8}$$ laps results from incorrectly multiplying $$3\frac{1}{2} \times \frac{1}{4}$$ instead of dividing—this gives you a fraction of a lap, which doesn't make sense when someone ran multiple miles. B) 12 laps comes from converting $$3\frac{1}{2}$$ incorrectly to $$\frac{6}{2}$$ and then calculating $$\frac{6}{2} \times 4 = 12$$. C) 16 laps results from converting $$3\frac{1}{2}$$ to 4 (rounding up) and then multiplying by 4.

When solving "how many times" problems, always ask yourself whether your answer makes logical sense. If someone ran $$3\frac{1}{2}$$ miles on a $$\frac{1}{4}$$-mile track, they must have completed more than 3 laps but fewer than 20, which helps you verify that 14 is reasonable.

First, determine the fraction of the entire band that plays the trumpet. This is \(\frac{3}{5} \times \frac{1}{3} = \frac{3}{15} = \frac{1}{5}\). So, \(\frac{1}{5}\) of the total band members are trumpet players. If T is the total number of members, then \(\frac{1}{5}T = 12\). To find T, multiply 12 by 5: \(12 \times 5 = 60\). There are 60 members in the band.

First, find the average of the three test scores: \((80 + 90 + 100) \div 3 = 270 \div 3 = 90\). This average counts for \(\frac{2}{3}\) of the final grade, so its contribution is \(90 \times \frac{2}{3} = 60\) points. The final exam score is 84, and it counts for \(\frac{1}{3}\) of the grade, so its contribution is \(84 \times \frac{1}{3} = 28\) points. The final grade is the sum of these two parts: \(60 + 28 = 88\).

This is a two-step problem working backwards. Let L be the weight of the leftover pizza. David ate \(\frac{2}{3}\) of L, which weighed 6 ounces. So, \(\frac{2}{3}L = 6\). To find L, calculate \(6 \div \frac{2}{3} = 6 \times \frac{3}{2} = 9\) ounces. So, the leftover pizza weighed 9 ounces. This leftover amount was \(\frac{1}{4}\) of the original pizza's weight, P. So, \(\frac{1}{4}P = 9\). To find P, calculate \(9 \div \frac{1}{4} = 9 \times 4 = 36\) ounces.