To solve this problem, we can find the rates at which Dominik and Katarina work and then calculate how much of the project Dominik completes when both work together.
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Dominik's Rate: If Dominik can complete the project by himself in 8 hours, his rate of work (part of the project completed per hour) will be: \[ R_D = \frac{1 \text{ project}}{8 \text{ hours}} = \frac{1}{8} \text{ of the project per hour} \]
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Combined Rate: When Dominik and Katarina work together, it takes them 5 hours to complete the project. Their combined rate will be: \[ R_{D+K} = \frac{1 \text{ project}}{5 \text{ hours}} = \frac{1}{5} \text{ of the project per hour} \]
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Katarina's Rate: To find Katarina's rate, we can set up the equation: \[ R_{D+K} = R_D + R_K \] Plugging in the values we have: \[ \frac{1}{5} = \frac{1}{8} + R_K \]
To find \( R_K \), we can isolate it: \[ R_K = \frac{1}{5} - \frac{1}{8} \]
To perform the subtraction, we need a common denominator. The least common multiple of 5 and 8 is 40: \[ \frac{1}{5} = \frac{8}{40} \quad \text{and} \quad \frac{1}{8} = \frac{5}{40} \] So, \[ R_K = \frac{8}{40} - \frac{5}{40} = \frac{3}{40} \] This means Katarina completes \(\frac{3}{40}\) of the project per hour.
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Part of the Project Dominik Completes: In the time they are working together for 5 hours, we can calculate how much of the project Dominik has completed: \[ \text{Part completed by Dominik} = R_D \times \text{Number of hours} = \frac{1}{8} \times 5 = \frac{5}{8} \]
The missing value from the table representing the part of the project Dominik will complete when working together with Katarina for the 5 hours is \(\frac{5}{8}\).
Thus, if we denote the part of the project Dominik completes as \(r\), then:
- If \(r\) represents the part of the project completed, the answer is: \[ r = \frac{5}{8} \] In terms of the options you presented ("\((5)\)", "8r", "8(5)", "r"), it seems the representation might be numerical and not directly matching standard fractional forms. However, you can imply that \(r\) represents the \( \frac{5}{8}\) in the context of what you're asking for.
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