When four numbers \(\(a\)\), \(\(b\)\), \(\(c\)\), and \(\(d\)\) are in proportion, it means that the ratio of the first to the second is equal to the ratio of the third to the fourth (\(\(a/b = c/d\)\)).
Here we're given four numbers 2, 7, 17, and 37, and we want to find a constant number \(\(x\)\) such that when added to each number, the resulting four numbers are in proportion:
\(\[
(2+x) / (7+x) = (17+x) / (37+x)
\]\)
We need to find the value of \(\(x\)\) that satisfies this equation. To solve this, we will cross-multiply to eliminate the denominators:
\(\[
(2+x) \cdot (37+x) = (7+x) \cdot (17+x)
\]\)
Expanding both sides gives us:
\(\[
74 + 2x + 37x + x^2 = 119 + 7x + 17x + x^2
\]\)
Now, combine like terms:
\(\[
74 + 39x + x^2 = 119 + 24x + x^2
\]\)
Since \(\(x^2\)\) appears on both sides of the equation, they cancel each other:
\(\[
74 + 39x = 119 + 24x
\]\)
Subtract \(\(24x\)\) from both sides:
\(\[
74 + 15x = 119
\]\)
Now, subtract 74 from both sides:
\(\[
15x = 45
\]\)
Finally, divide by 15 to find \(\(x\)\):
\(\[
x = 45 / 15
\]\)
\(\[
x = 3
\]\)
So the constant number we need to add to each of 2, 7, 17, and 37 to make them in proportion is 3. To check it, let's add 3 to each number and see if both ratios are equal:
\(\[
(2+3) / (7+3) = (17+3) / (37+3)
\]\)
\(\[
5 / 10 = 20 / 40
\]\)
\(\[
1/2 = 1/2
\]\)
Both ratios are equal, confirming that our solution is correct.
