Pat can shelve a cart full of books in the library in 24 minutes and Kevin can do it in 30 minutes. How long will it take them if they work together? Equation:

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To determine how long it will take Pat and Kevin to shelve the cart of books together, we can use the concept of **work rates**. Here's how you can set up and solve the equation:

### Step 1: Define the Work Rates

- **Pat's Work Rate:** Pat can complete the task in 24 minutes. Therefore, Pat's work rate is:
  
  $$
  \frac{1 \text{ cart}}{24 \text{ minutes}} = \frac{1}{24} \text{ cart per minute}
  $$

- **Kevin's Work Rate:** Kevin can complete the task in 30 minutes. Therefore, Kevin's work rate is:
  
  $$
  \frac{1 \text{ cart}}{30 \text{ minutes}} = \frac{1}{30} \text{ cart per minute}
  $$

### Step 2: Combine the Work Rates

When Pat and Kevin work together, their combined work rate is the sum of their individual work rates:

$$
\text{Combined Work Rate} = \frac{1}{24} + \frac{1}{30}
$$

### Step 3: Set Up the Equation

Let $$ t $$ be the time in minutes it takes for both to complete the task together. Since together they complete $$ 1 $$ cart:

$$
\left( \frac{1}{24} + \frac{1}{30} \right) \times t = 1
$$

### Step 4: Solve for $$ t $$

1. **Find a Common Denominator:**

The least common denominator (LCD) of 24 and 30 is 120.

$$
   \frac{1}{24} = \frac{5}{120}, \quad \frac{1}{30} = \frac{4}{120}
   $$

2. **Add the Work Rates:**

$$
   \frac{5}{120} + \frac{4}{120} = \frac{9}{120} = \frac{3}{40}
   $$

3. **Set Up the Equation:**

$$
   \frac{3}{40} \times t = 1
   $$

4. **Solve for $$ t $$:**

$$
   t = \frac{40}{3} \text{ minutes} \approx 13.\overline{3} \text{ minutes}
   $$

### Final Answer

**It will take Pat and Kevin approximately 13 and $$ \frac{1}{3} $$ minutes to shelve the cart together.**

### Summary of the Equation

$$
\frac{1}{24} + \frac{1}{30} = \frac{1}{t}
$$

Solving this equation gives:

$$
t = \frac{40}{3} \text{ minutes} \approx 13.\overline{3} \text{ minutes}
$$
It will take Pat and Kevin approximately 13 and 1/3 minutes to shelve the cart together.

Pat can shelve a cart full of books in the library in 24 minutes and Kevin can do it in 30 minutes. How long will it take them if they work together? Equation:

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