Okay, let's see. The problem is about Louis and Jack sharing a pie. Louis eats 1/5 of the pie, and Jack eats 2/3 of the pie. The question is asking how much of the pie do they eat altogether. Hmm, so I need to add these two fractions together, right? But wait, adding fractions with different denominators requires finding a common denominator. Let me think about how to approach this step by step.

First, let me recall how to add fractions. When adding fractions, if they have the same denominator, you just add the numerators and keep the denominator. But here, the denominators are different: 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 15, since 15 is the smallest number that both 5 and 3 can divide into without a remainder. 

So, to convert 1/5 and 2/3 to fractions with denominator 15, I need to multiply both the numerator and the denominator of each fraction by the necessary number. For 1/5, multiplying numerator and denominator by 3 gives (1*3)/(5*3) = 3/15. For 2/3, multiplying numerator and denominator by 5 gives (2*5)/(3*5) = 10/15. 

Now that both fractions have the same denominator, I can add them together. Adding the numerators: 3 + 10 = 13. So the sum is 13/15. 

Wait, let me double-check that. If I convert 1/5 to 3/15 and 2/3 to 10/15, then adding them gives 3/15 + 10/15 = 13/15. That seems right. But let me verify the conversions again. 

1/5: 5 times 3 is 15, so multiply numerator 1 by 3 to get 3. Correct. 2/3: 3 times 5 is 15, so multiply numerator 2 by 5 to get 10. Correct. So 3/15 + 10/15 is indeed 13/15. 

Is there another way to approach this problem? Maybe by converting the fractions to decimals first? Let's try that as a check. 

1/5 as a decimal is 0.2. 2/3 is approximately 0.6667. Adding them together: 0.2 + 0.6667 = 0.8667. Now, converting 13/15 to decimal: 15 divided by 13. Let's see, 13 divided by 15. 15 goes into 13 zero times. Add a decimal point, 130 divided by 15 is 8 (15*8=120), remainder 10. Bring down a zero: 100 divided by 15 is 6 (15*6=90), remainder 10. Bring down another zero: 100 divided by 15 is again 6. So it's 0.8666..., which is approximately 0.8667. That matches the decimal sum. So 13/15 is correct. 

Alternatively, if I didn't use the LCD, could I still add the fractions? Let's see. If I tried to add 1/5 and 2/3 without converting, I would need to find a common denominator. The denominators are 5 and 3. The denominators' product is 15, which is the LCD. If I didn't use the LCD, maybe I could use another common denominator, but 15 is the smallest. If I used a larger denominator, like 30, then 1/5 would be 6/30 and 2/3 would be 20/30, adding to 26/30, which simplifies to 13/15. So even if I use a larger common denominator, I still end up with the same result. So that's a good check. 

Another way to think about this: imagine the pie is divided into 15 equal parts. Louis eats 3 parts (since 1/5 of 15 is 3), and Jack eats 10 parts (since 2/3 of 15 is 10). Together, they eat 13 parts out of 15. That makes sense. 

Wait, let me visualize this. If the pie is cut into 15 slices, each slice is 1/15 of the pie. Louis eats 1/5, which is 3 slices. Jack eats 2/3, which is 10 slices. Total slices eaten: 3 + 10 = 13 slices. So 13/15 of the pie is eaten. That's another way to see it. 

Is there a possibility that I made a mistake in the conversion? Let me check again. For 1/5: 1 divided by 5 is 0.2. 3/15 is 0.2. 10/15 is approximately 0.6667, which is 2/3. So adding 0.2 and 0.6667 gives 0.8667, which is 13/15. Correct. 

Alternatively, if I tried to add the fractions without converting, maybe by multiplying numerator and denominator incorrectly. For example, if someone thought 1/5 + 2/3 = (1+2)/(5+3) = 3/8, which is incorrect. But that's a common mistake, so it's good that I avoided that by properly finding the LCD. 

Another check: 1/5 is less than 1/4 (which is 0.25), and 2/3 is more than 1/2 (0.5). So together, their total should be more than 0.5 + 0.25 = 0.75, which is 3/4. 13/15 is approximately 0.8667, which is indeed more than 0.75. That seems reasonable. 

If the answer were less than 1/5 + 1/3, which is 8/15, but that's not the case here. Wait, no, 1/5 + 2/3 is more than 1/5 + 1/3, which is 8/15, but 13/15 is much larger. Wait, 1/5 is 3/15, 2/3 is 10/15, so 3 + 10 = 13. So that's correct. 

I think that's thorough. All checks lead to the same answer. So I'm confident that 13/15 is the correct amount of pie eaten altogether by Louis and Jack.
Therefore, the total amount of pie that Louis and Jack eat altogether is $\boxed{\frac{13}{15}}$ of the pie.
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