Step 6: Using the commutative and associative properties of multiplication to

Question

simplify multiplication of two rational numbers
a) The event coordinator asks you to determine how many students participated in the
track-and-field day. The total number of students in seventh and eighth grade
combined is 584;
5
8
of them are seventh graders and 3
8
of them are eighth graders. If 4
5
of the seventh graders participated in track-and-field day and 7
8
of the eighth graders
participated, about how many total students participated? Describe the process you
used to find your answer. (4 points)

b) Each team earns 1 point for each person who participated in the track-and-field day.
Who wins the meet—the eighth graders or the seventh graders? Explain. (2 points)

Answer 1

To solve this problem, we'll break it down step by step.

a) Finding the Total Number of Students Who Participated

1. Calculate the number of seventh graders: We know that \(\frac{5}{8}\) of the total students (584) are seventh graders. \[ \text{Number of seventh graders} = \frac{5}{8} \times 584 \] To simplify, first multiply: \[ \frac{5 \times 584}{8} = \frac{2920}{8} = 365 \]

2. Calculate the number of eighth graders: Similarly, for eighth graders, we know that \(\frac{3}{8}\) of the total students are eighth graders. \[ \text{Number of eighth graders} = \frac{3}{8} \times 584 \] So we multiply: \[ \frac{3 \times 584}{8} = \frac{1752}{8} = 219 \]

3. Calculate the number of seventh graders who participated: We know that \(\frac{4}{5}\) of the seventh graders participated in the track-and-field day. \[ \text{Seventh graders participated} = \frac{4}{5} \times 365 \] To simplify: \[ = \frac{4 \times 365}{5} = \frac{1460}{5} = 292 \]

4. Calculate the number of eighth graders who participated: We also know that \(\frac{7}{8}\) of the eighth graders participated. \[ \text{Eighth graders participated} = \frac{7}{8} \times 219 \] To simplify: \[ = \frac{7 \times 219}{8} = \frac{1533}{8} = 191.625 \] Since the number of participants must be whole, we round 191.625 to 192.

5. Calculate the total number of students who participated: \[ \text{Total students participated} = 292 + 192 = 484 \]

Answer: About 484 students participated in the track-and-field day.

b) Determining the Winning Team

1. Points for the seventh graders: The seventh graders had 292 participants, which means they earned 292 points.

2. Points for the eighth graders: The eighth graders had 192 participants, which means they earned 192 points.

3. Comparing points: Since 292 (seventh graders) is greater than 192 (eighth graders), the seventh graders win the meet.

Conclusion: The seventh graders win the meet by 100 points (292 vs 192).