764 MCGRAW-HILL’S SAT
6.B A little common sense should tell you that they
will not need a full hour to clean the pool, because
Stephanie can clean it in an hour all by herself, but
Mark is helping. Therefore, you should eliminate
choices (C), (D), and (E) right away. You might also
notice that it can’t take less than 30 minutes, because
that is how long it would take if they both cleaned
one pool per hour (so that the two working together
could clean it in half the time), but Mark is slower, so
they can’t clean it quite that fast. This eliminates
choice (A) and leaves (B) as the only possibility.
But you should know how to solve this problem if it
were not a multiple-choice question, as well:
Stephanie’s rate for cleaning the pool is one pool per
hour. Mark’s rate for cleaning the pool is one pool ÷
1.5 hours =^2 ⁄ 3 pools per hour. Combined, they can
clean 1 +^2 ⁄ 3 =^5 ⁄ 3 pools per hour. Set up a rate equation
using this rate to determine how much time it would
take to clean one pool:
1 pool =(^5 ⁄ 3 pools per hour)(time)
Divide by^5 ⁄ 3 :^3 ⁄ 5 hours to clean the pool
Multiply by 60:^3 ⁄ 5 (60) =36 minutes
(Chapter 9, Lesson 4: Rate Problems)
7.A Change each expression to a base-10 exponential:
(A) =((10^2 )^3 )^4 = 1024
(B)=((10^2 )^5 )((10^2 )^6 ) =(10^10 )(10^12 ) = 1022
(C)=((10^4 )^4 ) = 1016
(D)=(((10^2 )^2 )((10^2 )^2 ))^2 =((10^4 )(10^4 ))^2 =(10^8 )^2 = 1016
(E)=(10^6 )^3 = 1018
(Chapter 8, Lesson 3: Working with Exponentials)
8.B Consider the points (0, 2) and (3, 0) on line l.
When these points are reflected over the x-axis, (0, 2)
transforms to (0, −2) and (3, 0) stays at (3, 0) because
it is on the x-axis. You can then use the slope formula
to find the slope of line m:
It’s helpful to notice that whenever a line is reflected
over the x-axis (or the y-axis, for that matter—try it),
its slope becomes the opposite of the original slope.
(Chapter 10, Lesson 4: Coordinate Geometry)
9.C (a+b)^2 =(a+b)(a+b)
FOIL: a^2 +ab+ab+b^2
Combine like terms: a^2 + 2 ab+b^2
Plug in 5 for ab: a^2 +2(5) +b^2
Simplify: a^2 +b^2 + 10
Plug in 4 for a^2 +b^2 :4 + 10 = 14
(Chapter 8, Lesson 5: Factoring)
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10.D The total area of the patio to be constructed is
24 × 12 =288 ft^2. The slab shown in the figure has an
area of 8 ft^2. Therefore, to fill the patio you will need
288 ÷ 8 =36 slabs.
(Chapter 10, Lesson 5: Areas and Perimeters)11.D The prize money ratio can also be written as
7 x:2x:1x.Because the total prize money is $12,000,
7 x+ 2 x+ 1 x=12,000
Combine like terms: 10 x=12,000
Divide by 10: x=1,200
The first place prize is 7x=7(1,200) =$8,400.
(Chapter 7, Lesson 4: Ratios and Proportions)12.E Always read the problem carefully and notice
what it’s asking for. Don’t assume that you must solve
for xand yhere. Finding the value of 6x− 2 yis much
simpler than solving the entire system:
2 x+ 3 y= 7
4 x− 5 y= 12
Add straight down: 6 x− 2 y= 19
(Chapter 8, Lesson 2: Systems)13.D Think carefully about the given information
and what it implies, then try to find counterexamples
to disprove the given statements. For instance, try to
disprove statement I by showing that scan be even.
Imagine s=2:
s+ 1 = 2 r
Substitute 6 for s: 6 + 1 = 2 r
Combine like terms: 7 = 2 r
Divide by 2: 3.5 =r(nope)
This doesn’t work because rmust be an integer. Why
didn’t it work? Because 2rmust be even, but if sis
even, then s+1 must be odd and cannot equal an even
number, so smust always be odd and statement I is
true. (Eliminate choice (B).)
Statement II can be disproven with r=1:
s+ 1 = 2 r
Substitute 1 for r: s+ 1 =2(1)
Subtract 1: s=1 (okay)
Since 1 is an integer, we’ve proven that ris not nec-
essarily even, so II is false. (Eliminate choices (C)
and (E).)
Since we still have two choices remaining, we have to
check ugly old statement III. Try the values we used
before. If r=1 and s=1, then , which
is an integer. But is it always an integer? Plugging in
more examples can’t prove that it will ALWAYS be an
integer, because we can never test all possible solu-
tions. We can prove it easily with algebra, though.
Since s+ 1 = 2 r:
Divide by r:Distribute:
s
rr+=
1
2
s
r+
=
1
2
s
rr