<p>If x is not equal to 0, and x is inversely proportional to y, what is directly proportional to 1/x^2?<br>
a)-1/y^2
b)1/y^2
c)1/y
d)y
e)y^2</p>
<p>Inversely is xy=k, right? What is directly proportional then? Answer is E by the way.</p>
<p>xy = k is inversely proportional
x/y = k is directly proportional</p>
<p>Wait...i thought the answer should be B? I'm confused...</p>
<p>choice e is correct</p>
<p>Could you please explain?</p>
<p>it is inversely proportional, so xy = k. It also equals y = 1/x. </p>
<p>But this is asking for the y value, when it = 1/x^2, so since the x is squared now, the y has to be squared since it is asking for the y value when it is directly proportional. So if you square the x then you have to square the y, so the answer is E. y^2 </p>
<p>...I think I explained that right?</p>
<p>EDIT to above (to clear any confusion), xy = k is equal to y = k/x. But in this problem k = 1.</p>
<p>
[quote]
If x is not equal to 0, and x is inversely proportional to y, what 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
[/quote]
</p>
<p>(i) x is not equal to 0: x <> 0
(ii) x is inversely proportional to y: x = 1/y
(iii) 1/x^2 = ?</p>
<p>From (ii) and (iii): </p>
<p>1/x^2 = 1/(1/y)^2 ....because x = 1/y from (ii)
1/x^2 = 1/((1^2)/(y^2)) ...distribute ^2
1/x^2 = 1/(1/y^2) ...simplify
1/x^2 = y^2 ...simplify again answers (iii) </p>
<p>Ans is (e).</p>
<p>Taking the square root of the relationship 1/x^2 = y^2 should also respect the propostion (ii)... and it does: </p>
<p>SQRT(1/x^2) = SQRT(y^2)
1/x = y
x = 1/y ...which is proposition (ii)</p>
<p>OK thanks guys!</p>
<p>If
x is inversely proportional to y, then</p>
<p>1/x is directly proportional to y,</p>
<p>(1/x)^2 is directly proportional to y^2, </p>
<p>1/x^2 is directly proportional to y^2.
Choice E.</p>
<p>How do you go from 1/x2 is inversely proportional to Y to 1/x is directly proportional to Y?</p>
<p>^^RahoulVA
You probably meant
from "x is inversely proportional to y" to "1/x is directly proportional to y".</p>
<p>Two definitions:
1. P is directly proportional to Q if P=(const)x(Q),
or P/Q=const for non-zero Q;</p>
<ol>
<li>P is inversely proportional to Q if PxQ=const</li>
</ol>
<p>(const is the proportionality constant).</p>
<p>===========================
The following statements ( <1>, <2>, <3>, <4>)
are equivalent to each other:</p>
<p>X is directly proportional to Y <1></p>
<p>1/X is directly proportional to 1/Y <2></p>
<p>X is inversely proportional to 1/Y <3></p>
<p>1/X is inversely proportional to Y <4></p>
<p>========================
<1> means X/Y = const.</p>
<p>It follows
(1/X) / (1/Y) = Y/X = const, so <2> is true.</p>
<p>X(1/Y) = X/Y = const, so <3> is true.</p>
<p>(1/X)Y = Y/X = const, so <4> is true.</p>
<p>==========================
Also,
if X is directly proportional to Y,
then
X^2 is directly proportional to Y^2:
X^2 / Y^2 = (X/Y)^2 = const.</p>
<p>To make it less cluttered (I hope),
here's a nifty rule:</p>
<p>In
"P is directly proportional to Q"
you can replace one of the variables with its reciprocal AND change directly to inversely,
or
replace both of the variables with their reciprocals keeping directly .</p>
<p>Respectively,
in
"P is inversely proportional to Q"
you can replace one of the variables with its reciprocal AND change to inversely to directly ,
or
replace both of the variables with their reciprocals keeping inversely.</p>
<p>Of course, both P and Q are not =0.</p>
<p>Another thread worth of brining up with a good example from the Blue Book.</p>
<p>To remind two definitions:
1. P is directly proportional to Q if P/Q=const for non-zero Q;
2. P is inversely proportional to Q if (P)(Q)=const.</p>
<p>p.794, q.12.
(x^2) / y = const
((1/2)^2) / (1/8) = (x^2) / (9/2)
x^2 = (2) (9/2)
x^2 = 9
x = 3 (a positive root).
Answer D.</p>