Question

1. a printer cartridge with 3\frac{2}{3} milliliters of ink will print off \frac{2}{4} of a box of paper. how many milliliters of ink will it take to print an entire box?
2. a cookie recipe called for 3\frac{1}{2} cups of sugar for every \frac{5}{6} cup of flour. if you made a batch of cookies using 1 cup of flour, how many cups of sugar would you need?
3. a container with 3\frac{1}{5} liters of weed killer can spray \frac{1}{4} of a lawn. how many liters would it take to spray 1 entire lawn?
4. a bucket of water was \frac{1}{2} full, but it still had 2\frac{4}{5} gallons of water in it. how much water would be in one fully filled bucket?
5. a bike tire was \frac{1}{2} full. it took a small air compressor 3\frac{1}{3} seconds to fill it up. how long would it have taken to fill an empty tire?
6. it takes 2\frac{1}{2} yards of thread to make \frac{4}{6} of a sock. how many yards of thread will it take to make an entire sock?
7. a machine made 2\frac{2}{3} pencils in 2\frac{1}{4} minutes. how many pencils would the machine have made after 5 minutes?
8. a carpenter goes through 2\frac{4}{5} boxes of nails finishing 3\frac{1}{3} rooves. how much would he use finishing 4 rooves?
9. it takes 3\frac{1}{4} spoons of chocolate syrup to make 2\frac{1}{5} gallons of chocolate milk. how many spoons of syrup would it take to make 3 gallons of chocolate milk?
10. a bag with 3\frac{4}{6} quarts of peanuts can make 2\frac{3}{6} jars of peanut butter. how many quarts of peanuts would you need to make 5 jars?

Explanation:

Step1: Analyze the problem (Problem 1)

We know that \( 3\frac{2}{3} \) milliliters of ink prints \( \frac{2}{4} \) (which simplifies to \( \frac{1}{2} \)) of a box. We need to find the ink for 1 box. Let \( x \) be the ink for 1 box. We can set up a proportion: \( \frac{3\frac{2}{3}}{\frac{1}{2}}=\frac{x}{1} \). First, convert \( 3\frac{2}{3} \) to an improper fraction: \( 3\frac{2}{3}=\frac{11}{3} \). Then the proportion becomes \( \frac{\frac{11}{3}}{\frac{1}{2}} = x \). Dividing by a fraction is multiplying by its reciprocal, so \( x=\frac{11}{3}\times2=\frac{22}{3}=7\frac{1}{3} \).

Step2: Analyze the problem (Problem 2)

The ratio of sugar to flour is \( 3\frac{1}{2} \) cups of sugar for \( \frac{5}{6} \) cup of flour. Let \( x \) be the sugar for 1 cup of flour. Set up the proportion \( \frac{3\frac{1}{2}}{\frac{5}{6}}=\frac{x}{1} \). Convert \( 3\frac{1}{2} \) to \( \frac{7}{2} \). Then \( x = \frac{7}{2}\div\frac{5}{6}=\frac{7}{2}\times\frac{6}{5}=\frac{42}{10}=4\frac{1}{5} \).

Step3: Analyze the problem (Problem 3)

\( 3\frac{1}{5} \) liters sprays \( \frac{1}{4} \) of a lawn. Let \( x \) be liters for 1 lawn. Proportion: \( \frac{3\frac{1}{5}}{\frac{1}{4}}=\frac{x}{1} \). Convert \( 3\frac{1}{5} \) to \( \frac{16}{5} \). Then \( x=\frac{16}{5}\times4=\frac{64}{5}=12\frac{4}{5} \).

Step4: Analyze the problem (Problem 4)

\( \frac{1}{2} \) full bucket has \( 2\frac{4}{5} \) gallons. Let \( x \) be full bucket. Proportion: \( \frac{2\frac{4}{5}}{\frac{1}{2}}=\frac{x}{1} \). Convert \( 2\frac{4}{5} \) to \( \frac{14}{5} \). Then \( x=\frac{14}{5}\times2=\frac{28}{5}=5\frac{3}{5} \).

Step5: Analyze the problem (Problem 5)

\( \frac{1}{2} \) full tire takes \( 3\frac{1}{3} \) seconds to fill the remaining \( \frac{1}{2} \). So to fill empty tire (which is 2 times \( \frac{1}{2} \)), time is \( 3\frac{1}{3}\times2 \). Convert \( 3\frac{1}{3} \) to \( \frac{10}{3} \). Then \( \frac{10}{3}\times2=\frac{20}{3}=6\frac{2}{3} \) seconds.

Step6: Analyze the problem (Problem 6)

\( 2\frac{1}{2} \) yards for \( \frac{4}{6} \) (simplifies to \( \frac{2}{3} \)) of a sock. Let \( x \) be yards for 1 sock. Proportion: \( \frac{2\frac{1}{2}}{\frac{2}{3}}=\frac{x}{1} \). Convert \( 2\frac{1}{2} \) to \( \frac{5}{2} \). Then \( x=\frac{5}{2}\div\frac{2}{3}=\frac{5}{2}\times\frac{3}{2}=\frac{15}{4}=3\frac{3}{4} \).

Step7: Analyze the problem (Problem 7)

First, find the rate of pencils per minute. Machine makes \( 2\frac{2}{3} \) (which is \( \frac{8}{3} \)) pencils in \( 2\frac{1}{4} \) (which is \( \frac{9}{4} \)) minutes. Rate \( r=\frac{\frac{8}{3}}{\frac{9}{4}}=\frac{8}{3}\times\frac{4}{9}=\frac{32}{27} \) pencils per minute. In 5 minutes, number of pencils \( n = r\times5=\frac{32}{27}\times5=\frac{160}{27}=5\frac{25}{27} \).

Step8: Analyze the problem (Problem 8)

First, find rate of boxes per rooves. \( 2\frac{4}{5} \) (which is \( \frac{14}{5} \)) boxes for \( 3\frac{1}{3} \) (which is \( \frac{10}{3} \)) rooves. Rate \( r=\frac{\frac{14}{5}}{\frac{10}{3}}=\frac{14}{5}\times\frac{3}{10}=\frac{42}{50}=\frac{21}{25} \) boxes per roove. For 4 rooves, \( n = \frac{21}{25}\times4=\frac{84}{25}=3\frac{9}{25} \) boxes.

Step9: Analyze the problem (Problem 9)

Rate of syrup per gallon: \( 3\frac{1}{4} \) (which is \( \frac{13}{4} \)) spoons for \( 2\frac{1}{5} \) (which is \( \frac{11}{5} \)) gallons. Rate \( r=\frac{\frac{13}{4}}{\frac{11}{5}}=\frac{13}{4}\times\frac{5}{11}=\frac{65}{44} \) spoons per gallon. For 3 gallons, \( n=\frac{65}{44}\times3=\frac{195}{44}=4\frac{19}{44} \) spoons.

Step10: Analyze the problem (Problem 10)

Rate of peanuts per jar: \( 3\frac{4}{6} \) (simplifies to \( 3\frac{2}{3} \), which is \( \frac{11}{3} \)) quarts for \( 2\frac{3}{6} \) (simplifies to \( 2\frac{1}{2} \), which is \( \frac{5}{2} \)) jars. Rate \( r=\frac{\frac{11}{3}}{\frac{5}{2}}=\frac{11}{3}\times\frac{2}{5}=\frac{22}{15} \) quarts per jar. For 5 jars, \( n=\frac{22}{15}\times5=\frac{22}{3}=7\frac{1}{3} \) quarts.

Answer:

1. \( 7\frac{1}{3} \) milliliters
2. \( 4\frac{1}{5} \) cups
3. \( 12\frac{4}{5} \) liters
4. \( 5\frac{3}{5} \) gallons
5. \( 6\frac{2}{3} \) seconds
6. \( 3\frac{3}{4} \) yards
7. \( 5\frac{25}{27} \) pencils
8. \( 3\frac{9}{25} \) boxes
9. \( 4\frac{19}{44} \) spoons
10. \( 7\frac{1}{3} \) quarts