Fractions: mixed operations word problems | Worsheets library - Free Printable

Problem Analysis:

The worksheet contains three word problems involving fractions and mixed operations. Let's solve each problem step by step.

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Problem 1:

Question: Ellen planned to interview some applicants for a position in her office. If she scheduled $\frac{3}{4}$ of an hour to meet each of them, how much time did she schedule for all 6 applicants?

#### Solution:
1. Time per applicant: Ellen schedules $\frac{3}{4}$ of an hour for each applicant.
2. Number of applicants: There are 6 applicants.
3. Total time calculation: To find the total time, multiply the time per applicant by the number of applicants:
$$
\text{Total time} = \frac{3}{4} \times 6
$$
4. Perform the multiplication:
$$
\frac{3}{4} \times 6 = \frac{3 \times 6}{4} = \frac{18}{4}
$$
5. Simplify the fraction:
$$
\frac{18}{4} = \frac{9}{2} = 4\frac{1}{2} \text{ hours}
$$

#### Final Answer for Problem 1:
$$
\boxed{4\frac{1}{2} \text{ hours}}
$$

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Problem 2:

Question: Instead of $\frac{3}{4}$ of an hour, Ellen spent $\frac{1}{2}$ of an hour to meet each applicant. How much more time did she spend meeting all the applicants?

#### Solution:
1. Time per applicant (original plan): $\frac{3}{4}$ of an hour.
2. Time per applicant (actual plan): $\frac{1}{2}$ of an hour.
3. Difference in time per applicant: Subtract the actual time from the original time:
$$
\text{Difference per applicant} = \frac{3}{4} - \frac{1}{2}
$$
4. Find a common denominator: The least common denominator of 4 and 2 is 4. Rewrite $\frac{1}{2}$ as $\frac{2}{4}$:
$$
\frac{3}{4} - \frac{1}{2} = \frac{3}{4} - \frac{2}{4} = \frac{1}{4}
$$
5. Total difference for all applicants: Multiply the difference per applicant by the number of applicants (6):
$$
\text{Total difference} = \frac{1}{4} \times 6 = \frac{6}{4} = \frac{3}{2} = 1\frac{1}{2} \text{ hours}
$$

#### Final Answer for Problem 2:
$$
\boxed{1\frac{1}{2} \text{ hours}}
$$

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Problem 3:

Question: After the interview, Ellen decided to hire two of the applicants and reject the other. Ellen’s assistant called back the applicants to tell them the result. She spent $\frac{1}{10}$ of an hour calling each of the rejected applicants and $\frac{1}{5}$ of an hour calling each of the hired applicants. How much time did the assistant spend making these calls?

#### Solution:
1. Number of rejected applicants: There is 1 rejected applicant.
2. Time spent on each rejected applicant: $\frac{1}{10}$ of an hour.
3. Total time for rejected applicants:
$$
\text{Time for rejected applicants} = 1 \times \frac{1}{10} = \frac{1}{10} \text{ hours}
$$

4. Number of hired applicants: There are 2 hired applicants.
5. Time spent on each hired applicant: $\frac{1}{5}$ of an hour.
6. Total time for hired applicants:
$$
\text{Time for hired applicants} = 2 \times \frac{1}{5} = \frac{2}{5} \text{ hours}
$$

7. Total time spent on all calls: Add the time spent on rejected applicants and hired applicants:
$$
\text{Total time} = \frac{1}{10} + \frac{2}{5}
$$
8. Find a common denominator: The least common denominator of 10 and 5 is 10. Rewrite $\frac{2}{5}$ as $\frac{4}{10}$:
$$
\frac{1}{10} + \frac{2}{5} = \frac{1}{10} + \frac{4}{10} = \frac{5}{10} = \frac{1}{2} \text{ hours}
$$

#### Final Answer for Problem 3:
$$
\boxed{\frac{1}{2} \text{ hours}}
$$

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Final Answers:

1. $\boxed{4\frac{1}{2} \text{ hours}}$
2. $\boxed{1\frac{1}{2} \text{ hours}}$
3. $\boxed{\frac{1}{2} \text{ hours}}$