# PSAT Math : How to use the inverse variation formula

## Example Questions

### Example Question #3 : Direct And Inverse Variation

The square of  varies inversely with the cube of . If  when , then what is the value of  when

Possible Answers:

Correct answer:

Explanation:

When two quantities vary inversely, their products are always equal to a constant, which we can call k. If the square of x and the cube of y vary inversely, this means that the product of the square of x and the cube of y will equal k. We can represent the square of x as x2 and the cube of y as y3. Now, we can write the equation for inverse variation.

x2y3 = k

We are told that when x = 8, y = 8. We can substitute these values into our equation for inverse variation and then solve for k.

82(83) = k

k = 82(83)

Because this will probably be a large number, it might help just to keep it in exponent form. Let's apply the property of exponents which says that abac = ab+c.

k = 82(83) = 82+3 = 85.

Next, we must find the value of y when x = 1. Let's use our equation for inverse variation equation, substituting 85 in for k.

x2y3 = 85

(1)2y3 = 85

y3 = 85

In order to solve this, we will have to take a cube root. Thus, it will help to rewrite 8 as the cube of 2, or 23.

y3 = (23)5

We can now apply the property of exponents that states that (ab)c = abc.

y3 = 23•5 = 215

In order to get y by itself, we will have the raise each side of the equation to the 1/3 power.

(y3)(1/3) = (215)(1/3)

Once again, let's apply the property (ab)c = abc.

y(3 • 1/3) = 2(15 • 1/3)

y = 25 = 32

The answer is 32.

### Example Question #4 : Direct And Inverse Variation

varies inversely as both the square of  and the square root of . Assuming that  does not depend on any other variable, which statement is true of  concerning its relationship to ?

Possible Answers:

varies directly as the fourth root of .

varies directly as the fourth power of .

varies inversely as the fourth root of .

varies inversely as the fourth power of .

varies inversely as .

Correct answer:

varies inversely as the fourth power of .

Explanation:

varies inversely as both the square of  and the square root of , meaning that for some constant of variation ,

.

Square both sides, and the expression becomes

takes the role of the new constant of variation here, and we now have

,

meaning that  varies inversely as the fourth power of .

### Example Question #4 : Direct And Inverse Variation

varies directly as the square of  and inversely as  and . Assuming that  does not depend on any other variables, which of the following gives the variation relationship of  to  ?

Possible Answers:

varies directly as the fourth power of .

varies directly as .

varies inversely as the seventh power of .

varies inversely as .

varies inversely as the fourth power of .

Correct answer:

varies directly as .

Explanation:

varies directly as the square of  and inversely as ; therefore, for some constant of variation ,

Setting  and , the formula becomes

Setting  as the new constant of variation, we have a new variation equation

,

meaning that  varies directly as .