<p>I always mess these problems up on the math section… </p>
<p>For example, if a problem says: If height is directly proportional to length, and a piece of board is w long and x high, what is the height of a piece of board that is y long? </p>
<p>When it says directly proportional, I always think the setup is (w)(x)=(y)(z)</p>
<p>However, it’s actually: (w)/(x)=(y)/(z)</p>
<p>Same problem with inverse proportions </p>
<p>Is there a way to straighten this problem out? To me, it’s really hard to remember, because when you say inverse, you think of dividing, when you are actually multiplying. How do you remember this rule? (because it’s driving me insane :P)</p>
<p>Direct variation comes in the form of y=kx. This is where the notion of associating multiplying with direction variation comes from. However, k is the constant, so if we want to relate the variables, we have k=y/x.</p>
<p>Same with indirect variation; it comes in the form of y=k/x, so naturally one would associate dividing with indirect variation. But again, to directly compare the variables we need it in this form: k=yx.</p>
<p>That is where the confusion comes in.</p>
<p>So, in your problem, since we’re doing direct variation, we want k=y/x. In this case, y1=w, x1=x, y2=y, and x2=z. Since the k is the same, we can set them equal to each other:</p>
<p>y1/x1=y2/x2, or w/x=y/z.</p>
<p>Here’s another way to think about it; you said you originally thought of multiplying w and x: k=wx. Well is this direct variation? No because if we increase w (keeping k constant), x decreases, which is the definition of inverse variation.</p>
<p>AoPS if you have it explains this concept pretty well.</p>
<p>For example, the number of eggs that hens lay is directly proportional to number of hens </p>
<p>e/h = k, where k is a constant.</p>
<p>Think about it. If the farmer needs more eggs, he needs to buy more hens. If e (the numerator) gets bigger, the denominator (h) must also to maintain a constant value. You can also represent direct variation with e = kh. If h goes up, so must e, and vice versa.</p>
<p>Inverse variation is the opposite. If one factor goes up, the other must go down. Say the area of a rectangular coffee table must remain A square units. The formula for the table’s area is thus lw = A. If l is increased, w must decrease in order to maintain a constant area. In other words, l = A/w. If w, the denominator, gets bigger, the quotient gets smaller and thus l gets smaller. If w gets smaller, the quotient gets bigger and thus l gets bigger.</p>
<p>Direct: use the “zig zag method”
X: (B x C)/ A
start with X, draw a line to the variable across it, draw a line to the letter above X, and finally across to the last.
The first two variables are multiplied and divided by the last.</p>
<p>Inverse: use “circle method”
X= (C x A)/ B</p>
<p>start from X, draw a line up, across the top, and back down the other side. The first two variables are multiplied and divided by the last.</p>
<p>draw it out, it’ll make sense. after you get this down, it’ll take you like 3 seconds to solve the question.</p>
<p>hey antimmy, that’s pretty neat! are you setting up the equation as x/y=x1/y1? That’s what I’m assuming, because it wouldn’t really make sense to put like variables on different sides of the fraction bar :P</p>