426 McGRAW-HILL’S SAT
- B It’s probably easiest to set up the equation,
then choose simple values for the volume and
pressure, and then “experiment.” Since the vol-
ume varies inversely as the pressure, the product
of the volume and the pressure is a constant:vp= k.
Now choose simple values for vand p,such as 2
and 4: vp= (2)(4) = 8 = k.Therefore, in this case,
the product of the volume and the pressure is
always 8. If the pressure is increased 50%, then
it grows to 1.5(4) = 6. Now solve for the corre-
sponding value of v:
v(6) = 8
Divide by 6:
Therefore, the volume has decreased from 2 to.
To calculate the percent decrease, use the “percent
change” formula from Chapter 7, Lesson 5:
Percent change =
4
32
2
100
2
3
2
100
1
3
100 33
1
3
−
×=
−
%%%%×=−×=
4
3v=4
3
Concept Review 4
- y= kx^2
- product
- quotient or ratio
- It is a straight line passing through the origin with
a slope equal to k, the constant of proportionality.
For every point on the line, the ratio of the ycoor-
dinate to the xcoordinate is equal to k. - Write the general variation equation: w= kv^3
Substitute w= 16 and v= 2: (16) = k(2)^3
Simplify: 16 = 8k
Divide by 8: 2 = k
Write the specific variation equation: w= 2v^3
Substitute v= 3: w= 2(3)^3
Simplify: w= 54- Write the variation equation: y= k/x^2 or x^2 y= k
Choose any values for xand y: x^2 y= (1)^2 (3) = 3 = k
Write the specific variation equation: x^2 y= 3
Double the original value of x:(2)^2 y= 3
Simplify: 4y= 3
Solve for y: y= 3/4So what was the effect on ywhen you doubled the
value of x? It went from 3 to 3/4, therefore, it was
divided by 4 or multiplied by 1/4.- If avaries inversely as b, then ab= k, where kis a
constant. If b= 0.5 when a= 32, then
k= (0.5)(32) = 16. Therefore, in any ordered pair
solution (a, b), the product of aand bmust be 16.
The only solutions in which aand bare both pos-
itive integers are (1, 16), (2, 8), (4, 4), (8, 2), and
(16, 1), for a total of five ordered pairs. - If xvaries directly as the square root of yand
directly as z, then. First, substitute the
values x= 16, y= 64, and z= 2 to find k:
Simplify: 16 = 16k
Divide by 16: 1 = k
Substitute y= 36 and x= 60:
Simplify: 60 = 6z
Divide by 6: 10 = z60 1 36= z16 =k()( 2 64 )xkzy=Answer Key 4: Variation
SAT Practice 4
- E Recall from the lesson that whenever two vari-
ables vary inversely, they have a constant product.
The product of 4 and 6 is 24, so every other correct
solution for pand qmust have a product of 24
also. Choice (E) is the only one that gives values
that have a product of 24. - D It helps first to notice from the table that as m
increases, ndecreases, so any variation relation-
ship must be an inversevariation. Therefore, only
choices (B) and (D) are possibilities. If nvaried in-
versely as m, then the two variables would always
have the same product, but this is not the case:
1 4 = 4, 2 1 = 2, and 4 .25 = 1. However, if n
varied inversely as the squareof m, then nand m^2
would always have the same product. This is true:
12 4 = 4, 2^2 1 = 4, and 4^2 .25 = 4. Therefore,
the correct answer is (D). - D You are given that f(a, b) = a^2 b^3 = 10. Using the
definition, f(2a, 2b) = (2a)^2 (2b)^3 = (4a^2 )(8b^3 )
= 32a^2 b^3. Substituting a^2 b^3 = 10, you get 32a^2 b^3
= 32(10) = 320.