If the four numbers, 39, 117, 17 and y are in proportion, then find the value of y.

The correct answer is 51.

When four numbers are said to be in proportion, it means they follow the rule:

\frac{a_1}{a_2}=\frac{a_3}{a_4}

In other words, the ratio of the first number to the second equals the ratio of the third number to the fourth. This is written as:

a_1:a_2::a_3:a_4

Here, the four numbers given are 39, 117, 17, and y, so we set up the proportion:

39:117::17:y

Step 1: Apply the Cross-Multiplication Property (Product of Extremes = Product of Means)

In any proportion a:b::c:d, we have the fundamental rule:

a\times d=b\times c

The extremes are the outer terms (39 and y), and the means are the inner terms (117 and 17). Applying the rule:

39\times y=117\times 17

Step 2: Calculate the right-hand side

117\times 17=1989

So the equation becomes:

39\times y=1989

Step 3: Solve for y

y=\frac{1989}{39}=51

Verification: Let's confirm by checking the ratios are equal:
\frac{39}{117}=\frac{1}{3} and \frac{17}{51}=\frac{1}{3}
Both ratios equal \frac{1}{3}, confirming the proportion holds. ✓

Therefore, the value of y = 51.