Directly/Inversely Proportional: is k = 1?

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1

<p>Hey,</p>

<p>I'm going through some BlueBook SAT math sections and am looking at Practice Test 3; Section 5; Question 6.</p>

<p>Here it is:
If x ≠ 0 and x is inversely proportional to y, which of the following is directly proportional to 1/(x^2) ?</p>

<p>a) -1/(y^2)
b) 1/(y^2)
c) 1/y
d) y
e) y^2</p>

<p>I understand that x being inversely proportional to y means that one needs to use
xy = k.
Thus, x = k/y
1/x = y/k
1/(x^2) = (y^2)/(k^2)</p>

<p>and finally here lies the problem.</p>

<p>The answer is e, y^2. Does this mean that I just assume that k = 1? If so, is this always the last step in a direct/inverse proportionality question? This might be too much to ask, but would somebody have any other examples of this if this is the case?</p>

<p>Thanks so much for any help you can provide.</p>

<p>(If the ≠ symbol doesn't show up on your browser, it's just a 'does not equal' sign.)</p>

2

<p>I’m not completely sure of this but I got the right answer so… here’s what i did.</p>

<p>So you got 1/(x^2) = (y^2)/(k^2).
The question says, “which of the following is directly proportional to 1/(x^2)”,
which means answer / (1/(x^2)) = some constant. In the problem, that constant is k^2 because it doesn’t contain any variables. You isolate the constant and you get
y^2 / (1/(x^2)) = k^2. Therefore answer is y^2…</p>

<p>Lol sorry if it doesn’t make sense. Sometimes I suck at explaining!</p>

3

<p>Yes, inversely proportional means xy = k. k does not necessarily equal 1, but it’s always constant. x and y, when multiplied together, always equal k, so y increases when you decrease x and vice versa.</p>

<p>Directly proportional is (just to use different variables) a/b = c, where a and b are variables like x and y, and c is a constant like k. </p>

<p>Anyway, you’re given that x is inversely proportional to y, that is, xy = some constant k. You’re trying to find what form of “y” (like y^2, y, y^3 maybe, etc.) is directly proportional to 1/(x^2). Let the form of “y” you are looking for be equal to the variable a. </p>

<p>Then the question becomes what is “a” in a / (1/(x^2)) = c (this is the definition of a direct proportion a / b = c)? Rearrange, and you get what is “a” in (a)(x^2) = c, where c is just any constant?</p>

<p>The easiest way is to square the equation given (xy = k) like the person above said to get x^2 y^2 = k^2. Since k is a constant, we can just let k^2 be the arbitrary constant “c”. So x^2 y^2 = c. This takes the form (a)(x^2) = c, and a = y^2. </p>

<p>[hide]An unnecessarily long explanation for something really quite simple. Hope this helps![/hide]</p>

<p>Shortened version:</p>

<p>xy = k
x^2 y^2 = k^2 = c
y^2 / (1/(x^2)) = c</p>

<p>Edit: I realize I didn’t really answer your question. k can be any number (except 0 for practicality’s sake): 943, 54353, .43985, -234938, 3/32
xy = 234 = k
xy = -9.38 = k
You don’t assume k = 1. You’re supposed to assume k is a constant, which it is.</p>

4

<p>Without k:
If x is inversely proportional to y then
1/x is directly proportional to y, and
(1/x)^2 is directly proportional to y^2, that is
1/x^2 is directly proportional to y^2.</p>

<p>With k:
Yes, for the sake of simplicity you can assume in this question k=1 (but not on some others where specific values are given). The reason: since it’s an SAT question, you don’t need to prove that the answer will be the same for any value of k.
xy=1
y=1/x
y^2 = 1/x^2
(1/x^2)/y^2 = 1, or
(1/x^2)/y^2 = const, therefore
1/x^2 is directly proportional to y^2.</p>

5

<p>Great! Thanks for all the responses. You’ve completely answered my question and I think I’ll find direct and inverse proportionality questions a lot easier from now on.</p>

6

<p>Here’s related question from the BB (p.906 q.12):

</p>

<p>In this case we are not free to choose k (constant of proportionality) because k is determined by given numbers.
But here’s the good news: we need not even bother finding k. All we need to know is that</p>

<p>If m is directly proportional to n, then
m/n = const, or
m1/n1 = m2/n2.</p>

<p>x^2 / y = const, so plugging given values
(1/2)^2 / (1/8) = x^2 / (9/2)
x^2 = 9
x = 3.
Answer D.</p>

<p>==================================================</p>

<p>Try using this technique on p.950 q.7:

Just remember that
If m is inversely proportional to n, then
mn = const, or
(m1)(n1) = (m2)(n2)</p>

<p>==================================================</p>

7

<p>Perfect - thank you so much.</p>

<p>I copied down the first problem and tried it, and I got it right but I first found what k was and then worked from there. Your way of making it into a proportion is much faster, and I used it on the 2nd question and had no trouble getting the right answer.</p>

<p>Thanks again! :D</p>