Handwritten solution setting up the equation 1/10 + 1/12 = 1/x for a combined work problem.

Problem Statement:

It takes Mariana 16 hours to pick forty bushels of apples, and Kayla can pick the same amount in 12 hours. How long would it take them if they worked together?

Solution:

#### Step 1: Understand the problem
We are given the individual rates at which Mariana and Kayla can pick apples. We need to determine how long it would take them to complete the task together.

#### Step 2: Define the rates
- Mariana can pick 40 bushels in 16 hours. Therefore, her rate is:
$$
\text{Mariana's rate} = \frac{40 \text{ bushels}}{16 \text{ hours}} = \frac{5}{2} \text{ bushels per hour}.
$$

- Kayla can pick 40 bushels in 12 hours. Therefore, her rate is:
$$
\text{Kayla's rate} = \frac{40 \text{ bushels}}{12 \text{ hours}} = \frac{10}{3} \text{ bushels per hour}.
$$

#### Step 3: Combine their rates
When Mariana and Kayla work together, their combined rate is the sum of their individual rates:
$$
\text{Combined rate} = \text{Mariana's rate} + \text{Kayla's rate} = \frac{5}{2} + \frac{10}{3}.
$$

To add these fractions, we need a common denominator. The least common multiple of 2 and 3 is 6. Convert each fraction:
$$
\frac{5}{2} = \frac{5 \times 3}{2 \times 3} = \frac{15}{6},
$$
$$
\frac{10}{3} = \frac{10 \times 2}{3 \times 2} = \frac{20}{6}.
$$

Now add the fractions:
$$
\text{Combined rate} = \frac{15}{6} + \frac{20}{6} = \frac{35}{6} \text{ bushels per hour}.
$$

#### Step 4: Determine the time to complete the task together
The total task is to pick 40 bushels. If their combined rate is $\frac{35}{6}$ bushels per hour, the time $T$ it takes for them to complete the task together is given by:
$$
T = \frac{\text{Total bushels}}{\text{Combined rate}} = \frac{40}{\frac{35}{6}}.
$$

To divide by a fraction, multiply by its reciprocal:
$$
T = 40 \times \frac{6}{35} = \frac{240}{35}.
$$

Simplify the fraction:
$$
\frac{240}{35} = \frac{48}{7} \text{ hours}.
$$

Convert $\frac{48}{7}$ to a mixed number:
$$
\frac{48}{7} = 6 \frac{6}{7} \text{ hours}.
$$

#### Final Answer:
$$
\boxed{\frac{48}{7}}
$$

This is the time it would take Mariana and Kayla to pick 40 bushels of apples together.