CHAPTER 6 / WHAT THE SAT MATH IS REALLYTESTING 259
test choice (B) next. If Ellen weighs 120 pounds,
then Jim weighs 290 − 120 =170 pounds and Ria
weighs 230 − 120 =110 pounds. In total, they
weigh 120 + 170 + 110 =400 pounds. Bingo! The
answer is (B).
Method 2: Use algebra. Let e=Ellen’s weight,
j=Jim’s weight, and r=Ria’s weight. Translate
the problem into equations: e+j= 290
e+r= 230
e+j+r= 400
Add first two equations: 2 e+j+r= 520
Subtract third equation: −(e+j+r=400)
e= 120- Method 1: Use algebra. Let xbe the number of gal-
lons of paint that she starts with. Translate the
problem into an equation: x−(1/4)x−(1/3)x= 10
Simplify: (5/12)x= 10
Multiply by 12/5: x= 24
The answer is (B).
Method 2: Test the choices. Look at the problem
carefully. She uses one-fourth of her paint, then
one-third of her paint, and is left with 10 gallons
of paint, a whole number.This suggests that she
started with a quantity that is divisible by 3 and 4.
Since 24 is divisible by 3 and 4, it’s a good choice
to test. One-fourth of 24 is 6, and one-third of 24
is 8. This means she would be left with
24 − 6 − 8 =10. Bingo! - Method 1: Plug in. Let s=35, so r=35/5 =7 and
t=4. (Check that they “fit.”) Since the question
asks for the value of s,write down 35 and circle it.
Plugging these values into the choices gives
(A) 140 (B) 35 (C) 136 (D) 124 (E) 280
The answer is (B).
Method 2: Use algebra. Solve the first equation
for s: s= 5 r.Then solve the second equation for r:
r=(7/4)t.Then substitute: s=5(7/4)t= 35 t/4.
Concept Review 6
- There are many situations in which this is possi-
ble, but perhaps the most common is where you’re
asked to find the solution of an equation, and the
choices are ordinary numbers. - Because the answer choices are usually presented
in numerical order. If choice (C) doesn’t work, you
may be able to tell whether it is too big or too
small, and thereby eliminate two other answers as
well. This way, you will only need to “test” one
more choice to get the answer. - When it is not easy to tell whether the choice is
“too big or too small,” or when there is no pattern
to the choices.
4. Usually, when the answer choices contain un-
knowns or represent ratios of unknowns; also
when the problem contains more unknowns than
equations.
5. (1) Check that the values satisfy any given equa-
tions or other restrictions, (2) write down the
values you are plugging in for each unknown,
(3) solve the problem numericallyand write down
this number, and (4) plug in the values to every
choice and eliminate those that don’t give the
right answer. If more than one choice gives the
right answer, plug in different numbers.
6. Because the two methods can provide a “check”
against one another: if they both give the same an-
swer, you are almost certainly right!
Answer Key 6: Finding Alternatives
SAT Practice 6
- Method 1: The problem asks you to solve for xin
terms of mand n.Notice that every choice con-
tains the expression m−n.
By substitution: m−n=(2x−5) −(x+7)
Simplify: m−n=x− 12
Add 12: m−n+ 12 =x
So the answer is (D).
Method 2: Just plug in simple values for the un-
knowns. If x=1, then m=(2)(1) − 5 =−3 and n=
(1) + 7 =8. Since the problem asks for x,write
down its value, 1, and circle it. Then plug in m=
−3 and n=8 to every choice, and simplify:
(A)−4.5 (B) − 9 (C) 0.5 (D) 1 (E) 2
So the answer is (D). - Method 1: Plug in numbers. Let abe 100. If bis 20%
greater than a,then b=120. If cis 25% greater than
b,then c=150. The area of the largest square, then,
is (150)^2 =22,500, and the area of the smallest
square is (100)^2 =10,000. The percent difference is
(22,500 −10,000)/10,000 =1.25 =125% (D).
Method 2: Use algebra. b=1.2aand
c=(1.25)(1.2a) =1.5a.So the area of the smallest
square is a^2 and the area of the largest square is
(1.5a)^2 =2.25a^2. Since 2.25a^2 −a^2 =1.25a^2 , the area
of the bigger square is 125% larger. - Method 1: Test the choices, starting with (C).
If Ellen weighs 130 pounds, then Jim weighs
290 − 130 =160 pounds and Ria weighs
230 − 130 = 100 pounds. All together, their weight
would be 130 + 160 + 100 =390 pounds. Close,
but too small. This means that our guess for
Ellen’s weight is too big(because increasing
Ellen’s weight decreases both Jim and Ria’s
weight by the same amount, for a net decrease).
This lets us eliminate choices (C), (D), and (E).
Since our guess wasn’t far off, it makes sense to