For four numbers to be in proportion, the ratio of the first to the second must equal the ratio of the third to the fourth. Mathematically, if $a$, $b$, $c$, and $d$ are in proportion, then:

$\frac{a}{b} = \frac{c}{d}$

Given the numbers 39, 117, 17, and $y$, they are in proportion, so:

$\frac{39}{117} = \frac{17}{y}$

To find $y$, we first simplify the ratio $\frac{39}{117}$:

$\frac{39}{117} = \frac{1}{3} \quad \text{(since 39 and 117 have a common factor of 39)}$

Now, set the two ratios equal to each other:

$\frac{1}{3} = \frac{17}{y}$

Next, cross-multiply to solve for $y$:

$1 \times y = 3 \times 17$

$y = 51$

So, the value of $y$ is 51.

Would you like more details or have any questions about this solution?

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Here are 5 related questions you might find interesting:

1. If $y$ were unknown and $17$ were given, how would the proportion be affected if $y$ were larger than 51?
2. What if the numbers were given as decimals instead of integers? How would you approach finding the missing value?
3. How would the problem change if it stated that the four numbers formed a geometric sequence instead of a proportion?
4. What if $39$ and $y$ were swapped? How would you solve for the new value of $y$?
5. How can you verify that four given numbers are in proportion without solving for $y$?

Tip: When dealing with proportions, always simplify the ratios if possible, as this makes cross-multiplication easier and reduces the chances of error.