<Question>
Louis and Jack are sharing a pie.  Louis eats $\frac{1}{5}$ of the pie, and Jack eats $\frac{2}{3}$ of the pie.  How much of the pie do Louis and Jack eat altogether?
</Question>

<think>
<Plan_1>
Okay, so Louis and Jack are sharing a pie, and I need to figure out how much of the pie they eat altogether. Let me start by understanding the problem. Louis eats 1/5 of the pie, and Jack eats 2/3. I have to add these two fractions together to find the total amount they consumed. Hmm, adding fractions. Let me recall how to add fractions with different denominators.

First, I remember that to add fractions, they need to have the same denominator. The denominators here are 5 and 3. So I need to find a common denominator for these two fractions. The least common denominator (LCD) of 5 and 3 would be their least common multiple (LCM).

<prompt_1.1> Find the least common denominator (LCD) for the fractions 1/5 and 2/3. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
The multiples of 5 are 5, 10, 15, 20, 25, etc. , and the multiples of 3 are 3, 6, 9, 12, 15, 18, etc. The smallest common multiple is 15. So the LCD is 15.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Next, I need to convert each fraction to an equivalent fraction with the denominator of 15. For Louis's portion, which is 1/5, I multiply both the numerator and denominator by 3 to get (1×3)/(5×3) = 3/15. For Jack's portion, 2/3, I multiply both the numerator and denominator by 5 to get (2×5)/(3×5) = 10/15.

Now, adding these two fractions together: 3/15 + 10/15.
Since the denominators are the same, I can simply add the numerators: 3 + 10 = 13. So the sum is 13/15.

Wait a second, 13/15 is less than the whole pie, which makes sense because Louis and Jack are sharing the pie, so their total consumption should be less than the whole pie. Let me verify if 13/15 is correct.

Alternatively, maybe I can check by converting the fractions to decimals. Let's see, 1/5 is 0.2, and 2/3 is approximately 0.666... Adding those together: 0.2 + 0.666... = 0.866...

<prompt_2.1> Verify the result by converting the fractions 1/5 and 2/3 to decimals and adding them. </prompt_2.1>

Wait, but maybe I made a mistake in converting the fractions. Let me double-check. 1/5 to 15ths: 1/5 = 3/15.

<prompt_2.2> Double-check the conversion of fractions and the addition process. </prompt_2.2>

Alternatively, maybe there's another way to check. If I take the original fractions: 1/5 + 2/3. Let me use the formula for adding fractions: (a/b + c/d) = (ad + bc)/bd.

<prompt_2.3> Use an alternative method to add the fractions 1/5 and 2/3 and verify the result. </prompt_2.3>

Wait, but let me make sure that 13/15 is the correct answer. Let me visualize the pie. If the pie is divided into 15 equal slices, Louis eats 3 slices (since 1/5 of 15 is 3), and Jack eats 10 slices (since 2/3 of 15 is 10). Together, they eat 3 + 10 = 13 slices.
So yes, 13/15 of the pie is eaten by both of them.

Is there any chance that they might have eaten more than the whole pie? Well, 1/5 is 0.2 and 2/3 is approximately 0.666...., adding them gives 0.866...., which is less than 1, so that's okay. The total can't exceed 1 because they are sharing the pie, so their combined consumption is a part of the whole pie.

Alternatively, maybe I can think of another way to verify. Suppose the pie is divided into 15 parts. Louis eats 1/5, which is 3 parts, and Jack eats 2/3, which is 10 parts. So 3 + 10 = 13 parts. Therefore, 13/15 of the pie is eaten.
Another way: Let's say the pie is 15 units. Then Louis eats 1/5 * 15 = 3 units, Jack eats 2/3 * 15 = 10 units. Total eaten is 3 + 10 = 13 units. So 13/15 of the pie.

I think that's thorough. I considered multiple methods: converting to like fractions, decimal conversion, using the formula for adding fractions, and visualizing the pie. All methods lead to the same answer.
</Plan_2>

<Execution_2>
<execution_2.1>
Which is approximately 0.866...., and 13 divided by 15 is approximately 0.866...., so that matches. So that seems correct.
</execution_2.1>

<execution_2.2>
So 3/15 + 10/15 = 13/15. That seems right. And 13/15 is approximately 0.866...., which matches the decimal addition.
</execution_2.2>

<execution_2.3>
So for 1/5 + 2/3, that would be (1*3 + 2*5)/(5*3) = (3 + 10)/15 = 13/15. Yep, that's the same result. So that's correct.

Therefore, the total amount of pie that Louis and Jack eat altogether is 13/15.
</execution_2.3>
</Execution_2>

<Final_answer>
Therefore, the total amount of the pie that Louis and Jack eat altogether is $\boxed{\frac{13}{15}}$.
</Final_answer>
</think>