Question 1 :
If a/b = 2, what is the value of 4b/a?
(A) 0
(B) 1
(C) 2
(D) 4
Answer :
a/b = 2
Take reciprocal on both sides.
b/a = 1/2
Multiply both sides by 4.
4(b/a) = 4(1/2)
4b/a = 4/2
4b/a = 2
The correct answer choice is (C).
Question 2 :
If x > 3, which of the following is equivalent to
Answer :
The correct answer choice is (B).
Question 3 :
The price of a textbook this year is 20% greater than the price last year. If this year's price is p, what was last year's price in terms of p?
(A) p/5
(B) 4p/5
(C) 5p/6
(D) 6p/5
Solution :
Let be last year's price of the text book.
Given : The price of a textbook this year is 20% greater than the price last year and this year's price is p.
p = (100 + 20)% of x
or
(100 + 20)% of x = p
120% of x = p
1.2x = p
Divide both sides by 1.2.
x = p/1.2
x = 10p/12
x = 5p/6
The correct answer choice is (C) 5p/6.
Question 4 :
If k > 0 and x = 7 in the equation above, what is the value of k?
(A) 2
(B) 3
(C) 4
(D) 5
Answer :
Add x to both sides.
Square both sides.
2k^{2} + 17 = x^{2}
Substitute x = 7.
2k^{2} + 17 = 7^{2}
2k^{2} + 17 = 49
Subtract 17 from both sides.
2k^{2} = 32
Divide both sides by 2.
k^{2} = 16
Take square root on both sides.
√k^{2} = √16
k = ±4
k = 4 or k = 4
Since k > 0, k = 4.
The correct answer choice is (C).
Question 5 :
In the xy-plane above, line ℓ is parallel to line k. What is the value of p?
(A) 4
(B) 5
(C) 8
(D) 10
Answer :
Formula for slope of a line joining (x_{1}, y_{1}) and (x_{2}, y_{2}) :
Slope of line ℓ :
Substitute (x_{1}, y_{1}) = (-5, 0) and (x_{2}, y_{2}) = (0, 2) into the above formula.
Slope of line k :
Substitute (x_{1}, y_{1}) = (0, -4) and (x_{2}, y_{2}) = (p, 0) into the above formula.
Since line ℓ is parallel to line k, the slopes must be equal.
The correct answer choice is (D).
Question 6 :
The graph of a line in the xy-plane has slope 2 and contains the point (1, 8). The graph of a second line passes through the points (1, 2) and (2, 1). If the two lines intersect at the point (a, b), what is the value of a + b?
(A) 4
(B) 3
(C) -1
(D) -4
Solution :
Equation of the line which has slope 2 and contains the point (1, 8) :
y = mx + b
Substitute m = 2.
y = 2x + b
Since the line contains the point (1, 8), substitute x = 1 and y = 8 to solve for b.
8 = 2(1) + b
8 = 2 + b
Subtract 2 from both sides.
6 = b
Equation of the first line :
y = 2x + 6 ----(1)
Equation of the line passes through the points (1, 2) and (2, 1) :
Formula for slope of a line joining (x_{1}, y_{1}) and (x_{2}, y_{2}) :
Substitute (x_{1}, y_{1}) = (1, 2) and (x_{2}, y_{2}) = (2, 1) into the above formula.
Equation of a line in slope-intercept form :
y = mx + b
Substitute m = -1.
y = -x + b
Since the line passes through the point (1, 2), substitute x = 1 and y = 2.
2 = -1 + b
Add 1 to both sides.
3 = b
Equation of the second line :
y = -x + 3 ----(2)
Solve (1) and (2).
y = y
2x + 6 = -x + 3
Add x to both sides.
3x + 6 = 3
Subtract 6 from both sides.
3x = -3
Divide both sides by 3.
x = -1
Substitute x = -1 in (2).
y = -(-1) + 3
y = 1 + 3
y = 4
The point of intersection is (-1, 4).
Given : The two lines intersect at the point (a, b).
(a, b) = (-1, 4)
a = -1 and b = 4
a + b = -1 + 4
a + b = 3
The correct answer choice is (B).
Question 7 :
A website uses the formula above to calculate a seller’s rating, R, based on the number of favorable reviews, F, and unfavorable reviews, N. Which of the following expresses the number of favorable reviews in terms of the other variables?
Solution :
The correct answer choice is (A).
Question 8 :
A radioactive substance decays at an annual rate of 13 percent. If the initial amount of the substance is 325 grams, which of the following functions f models the remaining amount of the substance, in grams, t years later?
(A) f(t) = 325(0.87)^{t}
(B) f(t) = 325(0.13)^{t}
(C) f(t) = 0.87(325)^{t}
(D) f(t) = 0.13(325)^{t}
Answer :
Formula for exponential decay :
f(t) = a(1 - r)^{t}
a ----> Initial amount of the substance
r ----> Annual decay rate
t ----> time in years
f(t) ----> Amount of substance after t years
So, a = 325, r = 13% or 0.13.
f(t) = 325(1 - 0.13)^{t}
f(t) = 325(0.87)^{t}
The correct answer choice is (A).
Question 9 :
The sales manager of a company awarded a total of $3000 in bonuses to the most productive salespeople. The bonuses were awarded in amounts of $250 or $750. If at least one $250 bonus and at least one $750 bonus were awarded, what is one possible number of $250 bonuses awarded?
Answer :
Let x be the number of $250 and y be the number of $750.
Since x and y represent number of bonuses, they can be only positive integers.
Since there were at least one $250 bonus and at least one $750 bonus awarded,
x ≥ 1 and y ≥ 1
Given : Total bonus awarded is $3000.
250x + 750y = 3000
250(x + 3y) = 3000
Divide both sides by 250.
x + 3y = 12
If x = 1,
1 + 3y = 12
3y = 11
y = 3.666.... (not an integer)
If x = 2,
2 + 3y = 12
3y = 10
y = 3.333.... (not an integer)
If x = 3,
3 + 3y = 12
3y = 9
y = 3.... (an integer)
So, x = 3 can be accepted.
One possible number of $250 bonuses awarded is 3.
Question 10 :
In the figure above, AE || CD and segment AD intersects segment CE at B. What is the length of segment CE?
Answer :
Let x be the length of CB.
Then,
CE = CB + BE
CE = x + 8
Given : AE || CD.
Segments AD and EC are transversals.
∠A and ∠D are alternate interior angles,
m∠A = m∠D
∠E and ∠C are alternate interior angles,
m∠E = m∠C
m∠A and m∠E of ΔAEB are equal to m∠D and m∠C ΔDCB respectively.
By AA Similarity Postulate, ΔAEB and ΔDCB are similar triangles.
Then, the corresponding sides are proportional.
CE = x + 8
Substitute x = 4.
CE = 4 + 8
CE = 12
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