Sat Math Hard Practice Quiz Worksheet With Answer Key Page 14
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18SAT Math Hard Practice Quiz Answers
13. A
(Estimated Difficulty Level: 4)
16. 75/2 or 37.5
(Estimated Difficulty Level: 5)
A good “skip-the-algebra” way to do this problem is to
To make this problem more concrete, make up a number
for the circumference of the racetrack. It doesn’t really
use the answers by plugging them into x until the two
given equations work. Using answer A, you should find
matter what number you use; I’ll use 75 feet. Since
2
2
2
2
that ( 3+1)
= ( 2)
= 4 and ( 3 1)
= ( 4)
= 16,
speed is distance divided by time, the speed of car A
is 75/15 = 5 feet per second, and the speed of car B is
so answer A is correct.
75/25 = 3 feet per second. (I picked 75 mostly because
You must have the algebraic solution, you say? Try
it is divided evenly by 15 and 25.) Every second, car A
taking the square root of both sides of the equations,
gains 2 feet on car B. To pass car B, car A must gain
but don’t forget that there are two possible solutions
75 feet on car B. This will require 75/2 = 37.5 seconds.
when you do this. The first equation gives: x + 1 = ±2
so that x = 1 or x =
3. The second equation gives
You may be thinking, “Whoa, tricky solution!” Here
is the mostly straightforward but somewhat tedious al-
x
1 = ±4 so that x = 5 or x =
3. The only solution
that works for both equations is x =
3.
gebraic solution. Once again, I’ll use 75 feet for the
circumference of the track. Suppose that you count
time from when car A first passes car B. Then, car A
travels a distance (75/15)t = 5t feet after t seconds.
(Remember that distance = speed × time.) For ex-
14. 96
(Estimated Difficulty Level: 5)
ample, after 15 seconds, car A has traveled a distance
5 · 15 = 75 feet, and after 30 seconds, car A has traveled
Since the tick marks correspond to consecutive integers,
a distance 5 · 30 = 150 feet. Similarly, car B travels a
and it takes four “steps” to go from x/12 to x/8, we
distance (75/25)t = 3t feet after t seconds. When the
know that x/8 is four greater than x/12. (Or, think of
two cars pass again, car A has traveled 75 feet more
the spaces between the tick marks: there are four spaces
than car B: 5t = 3t + 75. Solving for t gives: 2t = 75,
and each space is length 1, so the distance from x/12
or t = 75/2 = 37.5 seconds.
to x/8 is 4.) In equation form:
x
x
=
+ 4.
8
12
Multiplying both sides by 24 gives: 3x = 2x + 4 · 24 so
that x = 96.
15. B
(Estimated Difficulty Level: 5)
First, recall that if y is proportional to x, then y = kx
for some constant k. So, “y increased by 12 is directly
proportional to x decreased by 6” translates into the
math equation: y + 12 = k(x
6). Plugging in y = 2
and x = 8 gives 14 = k · 2 so that k = 7. Our equation
is now: y + 12 = 7(x
6). Plugging in 16 for y gives
28 = 7(x
6) so that x
6 = 4, or x = 10.
pg. 14
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