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Working together, Rocco and Giulia can paint a room in 3 hours. It would have taken Rocco 7 hours to do the job alone. Which equation can be used to determine r, Giulia’s rate of work in parts per hour?

A table showing Rate in part per hour, Time in hours, and Part of Room Painted. The first row shows Rocco and has StartFraction 1 Over 7 EndFraction, 3, and StartFraction 3 Over 7 EndFraction t. The second row shows Giulia, and has r, 3, and 3 r.

+ 3r = 7
+ 3r = 1
= r
= 3r

1 answer

To determine Giulia's rate of work (r) in parts per hour, we start by analyzing the information given:

  1. Rocco's rate of work: Since Rocco can paint the room in 7 hours, his rate is: \[ \text{Rocco's rate} = \frac{1 \text{ room}}{7 \text{ hours}} = \frac{1}{7} \text{ rooms per hour} \]

  2. Combined rate of Rocco and Giulia: Together, they can paint the room in 3 hours, so their combined rate is: \[ \text{Combined rate} = \frac{1 \text{ room}}{3 \text{ hours}} = \frac{1}{3} \text{ rooms per hour} \]

  3. Giulia's rate of work: Let \( r \) be Giulia's rate in rooms per hour. Therefore: \[ \text{Giulia's rate} = r \text{ rooms per hour} \]

  4. The equation for their combined work is: \[ \text{Rocco's rate} + \text{Giulia's rate} = \text{Combined rate} \] Substituting the rates we found: \[ \frac{1}{7} + r = \frac{1}{3} \]

  5. To solve for \( r \), we subtract \( \frac{1}{7} \) from both sides: \[ r = \frac{1}{3} - \frac{1}{7} \]

Now, calculating the right-hand side, we need a common denominator, which is 21: \[ r = \frac{7}{21} - \frac{3}{21} = \frac{4}{21} \]

So Giulia's rate \( r \) is \( \frac{4}{21} \text{ rooms per hour} \).

From the equations provided in the options, the one that fits our derived equation is: \[ 3r = 1 - 3\left(\frac{1}{7}\right) = 1 - \frac{3}{7} = \frac{4}{7} \] which leads to \( 3r = \frac{4}{3} \).

Thus, the equation that can be used to determine Giulia’s rate \( r \) is: \[ \frac{1}{7} + r = \frac{1}{3} \rightarrow 3r = 1 - 3\left(\frac{1}{7}\right) \rightarrow 3r = 1 - 3. \]

The correct equation is: \[ 3r = 1 - \frac{3}{7} = \frac{4}{7} \]

So, the direct answer to your question for Giulia's effecient equation operationally is: \[ 3r = 1 - \frac{3}{7} = 3r = \frac{4}{7}. \] However, the basic understanding is simply: \[ \frac{1}{7} + r = \frac{1}{3}. \]