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