Example 3 demonstrates how to set up a work rate equation to find out how long Julie needs to finish a cake after working with Marty.

Problem Statement:

Julie can complete a wedding cake in 8 hours, and Marty can complete the same cake in 10 hours. If Julie and Marty work together for 4 hours, how long will it take Julie to finish the job alone?

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Step-by-Step Solution:

#### Step 1: Determine the work rates of Julie and Marty
- Julie's work rate is the fraction of the cake she can complete per hour. Since she completes the cake in 8 hours, her work rate is:
$$
\text{Julie's work rate} = \frac{1}{8} \text{ cakes per hour}
$$

- Marty's work rate is the fraction of the cake he can complete per hour. Since he completes the cake in 10 hours, his work rate is:
$$
\text{Marty's work rate} = \frac{1}{10} \text{ cakes per hour}
$$

#### Step 2: Calculate the combined work rate when Julie and Marty work together
When Julie and Marty work together, their combined work rate is the sum of their individual work rates:
$$
\text{Combined work rate} = \frac{1}{8} + \frac{1}{10}
$$

To add these fractions, find a common denominator. The least common multiple of 8 and 10 is 40:
$$
\frac{1}{8} = \frac{5}{40}, \quad \frac{1}{10} = \frac{4}{40}
$$
$$
\text{Combined work rate} = \frac{5}{40} + \frac{4}{40} = \frac{9}{40} \text{ cakes per hour}
$$

#### Step 3: Calculate the amount of work completed by Julie and Marty in 4 hours
If Julie and Marty work together for 4 hours, the amount of work they complete is:
$$
\text{Work done in 4 hours} = \text{Combined work rate} \times \text{Time}
$$
$$
\text{Work done in 4 hours} = \frac{9}{40} \times 4 = \frac{36}{40} = \frac{9}{10} \text{ of the cake}
$$

#### Step 4: Determine the remaining work
The total work required to complete the cake is 1 (representing the whole cake). After working together for 4 hours, the remaining work is:
$$
\text{Remaining work} = 1 - \text{Work done in 4 hours}
$$
$$
\text{Remaining work} = 1 - \frac{9}{10} = \frac{1}{10} \text{ of the cake}
$$

#### Step 5: Calculate the time Julie needs to complete the remaining work alone
Julie's work rate is $\frac{1}{8}$ cakes per hour. To find the time she needs to complete the remaining $\frac{1}{10}$ of the cake, use the formula:
$$
\text{Time} = \frac{\text{Remaining work}}{\text{Julie's work rate}}
$$
$$
\text{Time} = \frac{\frac{1}{10}}{\frac{1}{8}} = \frac{1}{10} \times \frac{8}{1} = \frac{8}{10} = \frac{4}{5} \text{ hours}
$$

#### Step 6: Convert the time from hours to minutes
Since 1 hour = 60 minutes, convert $\frac{4}{5}$ hours to minutes:
$$
\frac{4}{5} \text{ hours} = \frac{4}{5} \times 60 = 48 \text{ minutes}
$$

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Final Answer:

$$
\boxed{48 \text{ min}}
$$