An EST I Math
Practice Booklet

Question 01

⚑

An amount of $14,000 is divided among three sisters, Layla, Yasmine, and Aya, in the ratio 1 : 2 : 4, respectively. How much did Yasmine receive?

A

$2,000

B

$4,000

C

$7,000

D

$8,000

Solution Total parts = 1+2+4 = 7. Yasmine's share = (2/7) × $14,000 = $4,000.

Question 02

⚑

6, 7, 7, 6, 1, 5, 9, 9, 7, 10

What is the median of the set of data above?

A

5.0

B

6.5

C

7.0

D

7.5

Solution Sorted: 1, 5, 6, 6, 7, 7, 7, 9, 9, 10. With 10 values the median is the average of the 5^th and 6^th: (7 + 7) / 2 = 7.0.

Question 03

⚑

Questions 3 – 5 refer to the table below. A car-rental company leased a total of 54 cars to an embassy for a one-year period.

Brand	Cars rented	Revenue (USD)
Hyundai	14	134,400
Toyota	8	85,440
Jeep	19	239,400
Honda	13	117,000

What is the monthly rental cost paid by the embassy for each Hyundai car?

A

$750

B

$760

C

$775

D

$800

Solution Per Hyundai per year: 134,400 ÷ 14 = $9,600. Per month: 9,600 ÷ 12 = $800.

Question 04

⚑

At the end of the year, the company paid 11% of its total revenue to the government as taxes. What was the total amount paid as taxes for the year?

A

$14,784.00

B

$37,052.40

C

$50,770.60

D

$63,386.40

Solution Total revenue = 134,400 + 85,440 + 239,400 + 117,000 = $576,240. Tax = 0.11 × 576,240 = $63,386.40.

Question 05

⚑

The company offered the embassy a 20% discount on all Jeep rentals for the renewed second year. What will be the new monthly rental cost per Jeep?

A

$800

B

$840

C

$850

D

$900

Solution Per Jeep per month original: 239,400 ÷ (19 × 12) = $1,050. After 20% off: 1,050 × 0.80 = $840.

Question 06

⚑

The probability that Mohammad successfully scores a penalty in a football match is 0.3. If he takes 3 penalty shots in a single game, what is the probability that he scores all three?

A

0.027

B

0.343

C

0.540

D

0.90

Solution Independent events: P(all 3) = 0.3 × 0.3 × 0.3 = 0.027.

Question 07

⚑

How many possible arrangements can be formed using all the letters in the word OFFICE?

A

180

B

360

C

720

D

1,440

Solution 6 letters with the letter F repeated twice: 6! / 2! = 720 / 2 = 360.

Question 08

⚑

What is the equation of the line passing through A(−9, 3) and B(1, 4)?

A

y = −x + 5

B

y = 0.1x − 3.9

C

y = 0.1x + 3.9

D

y = x + 3

Solution Slope = (4 − 3) / (1 − (−9)) = 1/10 = 0.1. Through B(1, 4): y − 4 = 0.1(x − 1) ⇒ y = 0.1x + 3.9.

Question 09

⚑

Given f(x) = −3(x + 4) − 4x² and g(x) = 2x² − x + 1, the value of (f/g)(1) is:

A

−9.5

B

−4.0

C

4.5

D

9.5

Solution f(1) = −3(5) − 4 = −19. g(1) = 2 − 1 + 1 = 2. f(1)/g(1) = −9.5.

Question 10

⚑

Questions 10 and 11 refer to the table below — number of locals and expatriate students enrolled at a high school.

	Locals	Expats
Boys	101	60
Girls	125	34

If a student is selected at random, what is the probability that the student is a boy?

A

0.316

B

0.467

C

0.503

D

0.519

Solution Boys = 101 + 60 = 161. Total = 320. P(boy) = 161/320 ≈ 0.503.

Question 11

⚑

If a student is selected at random, what is the probability that the student is a boy given that the student is local?

A

0.265

B

0.316

C

0.447

D

0.638

Solution P(boy | local) = local boys / all locals = 101/(101 + 125) = 101/226 ≈ 0.447.

Question 12

⚑

The price of a mobile phone increased by 40%, resulting in a final price of $308. What was the original price of the phone before the increase?

A

$220

B

$250

C

$280

D

$288

Solution Original × 1.40 = 308 ⇒ Original = 308 / 1.40 = $220.

Question 13

⚑

Which of the following is not a factor of P(x) = 3x⁴ + 5x³ − 64x² − 164x − 80?

A

x − 5

B

x − 2

C

x + 4

D

3x + 2

Solution Test P(2) = 3(16) + 5(8) − 64(4) − 164(2) − 80 = 48 + 40 − 256 − 328 − 80 = −576 ≠ 0. So (x − 2) is not a factor (the others all give 0).

Question 14

⚑

Jalal invested $4,500 for 2 years at an annual simple interest rate of 2%. How much interest did he earn at the end of the 2 years?

A

$90.0

B

$180.0

C

$181.8

D

$300.2

Solution I = P · r · t = 4,500 × 0.02 × 2 = $180.

Question 15

⚑

7k,   k + 10,   3k + 2,   4k − 4

The average of the set of data above is 5. What is the value of k?

A

0.6

B

0.8

C

1.2

D

1.4

Solution (7k + k + 10 + 3k + 2 + 4k − 4) / 4 = 5 ⇒ 15k + 8 = 20 ⇒ 15k = 12 ⇒ k = 0.8.

Question 16

⚑

Figure not drawn to scale.

What is the approximate perimeter of the shape shown in the figure above?

A

10.19 cm

B

12.0 cm

C

14.73 cm

D

16.19 cm

Solution AB = AC = 6 cm (radii). Arc CB = (40/360) × 2π × 6 = (1/9)(12π) ≈ 4.19 cm. Perimeter ≈ 6 + 6 + 4.19 ≈ 16.19 cm.

Question 17

⚑

A book contains 400 pages, of which 4 are defective. If 230 copies of the book were distributed each week over the past 5 weeks, how many non-defective pages were distributed in total?

A

4,600

B

91,080

C

182,160

D

455,400

Solution Books = 230 × 5 = 1,150. Non-defective pages each = 400 − 4 = 396. Total = 1,150 × 396 = 455,400.

Question 18

⚑

Which of the following is a simplified form of √(125a²) ⁄ √(5y³) ?

A

|a|·√(5y) / y²

B

5|a|·√y / y²

C

5|a|·√(5y) / y²

D

5|a|·√(5y)

Solution Combine radicals: √(125a² / 5y³) = √(25a²/y³) = 5|a| / √(y³) = 5|a| / (y√y). Rationalize: 5|a|·√y / y². Answer = B.

Question 19

⚑

Which of the following is a solution of |3x − 5| + 1 = 2?

A

1/3

B

1

C

2

D

7/3

Solution |3x − 5| = 1 gives 3x − 5 = ±1, so x = 2 or x = 4/3. Among the options, x = 2.

Question 20

⚑

Which of the following is the equation of the axis of symmetry of the graph of f(x) = 4x² − 20x + 7?

A

y = −18

B

x = −2.5

C

x = 1.5

D

x = 2.5

Solution x = −b/(2a) = 20/8 = 2.5.

Question 21

⚑

{ 3x − 1 ≥ −10
  x + 4 < −6

Which of the following represents the solution set of the system of inequalities above?

A

B

C

D

Empty: the system has no solution.

Solution First inequality: x ≥ −3. Second: x < −10. There is no x that is both ≥ −3 and < −10. Empty solution set.

Question 22

⚑

What is the y-coordinate of the midpoint of FG given F(2, −5) and G(0, 11)?

A

3

B

6

C

8

D

16

Solution y-midpoint = (−5 + 11)/2 = 3.

Question 23

⚑

Which of the following has infinitely many solutions?

A

4x − 1 = 9

B

|3x + 7| < 4

C

2x − 6 = 3(x + 7)

D

|−4x + 1| > −5

Solution An absolute value is always ≥ 0, which is always > −5, so option D is true for every real x — infinitely many solutions.

Question 24

⚑

Questions 24 and 25 refer to the function f(x) = 3ax³ + 4bx² + x + x³, where a and b are real numbers.

In order to make the equation a quadratic one, a must be equal to:

A

−1

B

−1/3

C

0

D

3

Solution Combine x³ terms: (3a + 1)x³. For a quadratic, this coefficient must be 0 ⇒ a = −1/3.

Question 25

⚑

If a = 1, and the coefficient of the x² term equals the coefficient of the x³ term, what is the value of b?

A

−1

B

1/2

C

1

D

3/2

Solution With a = 1, the x³ coefficient is 3(1) + 1 = 4 and the x² coefficient is 4b. Set 4b = 4 ⇒ b = 1.

Question 26

⚑

x	y
2	5
5	a
7	15

In the table above, x and y represent a linear relationship. What is the value of 2a?

A

5.5

B

13.0

C

17.5

D

22.0

Solution Slope from (2,5) to (7,15) is 10/5 = 2, line is y = 2x + 1. At x = 5, a = 11. So 2a = 22.0.

Question 27

⚑

If 1.2 m³ of a pool can be filled by 9 gallons of water, how many gallons are needed to fill a 26 m³ pool?

A

195

B

202

C

223

D

234

Solution Proportion: 1.2 / 9 = 26 / x ⇒ x = (26 × 9) / 1.2 = 195 gallons.

Question 28

⚑

The volume of a sphere of radius 12 cm is kπ. The value of k is:

A

192

B

576

C

2,304

D

6,912

Solution V = (4/3)π·r³ = (4/3)·12³·π = (4/3)·1728·π = 2304π. So k = 2,304.

Question 29

⚑

What is the absolute value of the negative root of p(x) = x³ − x² − 14x + 24?

A

2

B

3

C

4

D

8

Solution p(2) = 0, so factor: p(x) = (x − 2)(x² + x − 12) = (x − 2)(x + 4)(x − 3). Roots are 2, 3, −4. The negative root is −4 ⇒ |−4| = 4.

Question 30

⚑

3x − 4 ≤ 1

The greatest positive integer in the solution set of the inequality above is:

A

1

B

2

C

3

D

5

Solution 3x ≤ 5 ⇒ x ≤ 5/3 ≈ 1.67. Greatest positive integer ≤ 1.67 is 1.

Question 31

⚑

The number of players in a basketball academy increased from 20 to 34 in one month. If m% is the percent increase, then m is equal to:

A

30

B

45

C

50

D

70

Solution Increase = 14. % increase = (14/20) × 100 = 70%.

Question 32

⚑

If f(x) = 2x + 1 and g(x) = −3x + 1, then what is the value of (f + g)(1)?

A

1

B

3

C

6

D

7

Solution f(1) + g(1) = 3 + (−2) = 1.

Question 33

⚑

If 3x − 7 = 20, what is the value of 2x?

A

4.5

B

9.0

C

12.5

D

18.0

Solution 3x = 27 ⇒ x = 9 ⇒ 2x = 18.0.

Question 34

⚑

{ ax + 3y = 24
  x + 5y = 33

For what value of a in the system above is y = 6?

A

−2

B

−1

C

2

D

3

Solution y = 6 in the second eq. gives x + 30 = 33 ⇒ x = 3. Sub into first: 3a + 18 = 24 ⇒ a = 2.

Question 35

⚑

What is the slope of the line passing through A(−3, 4) and B(1, −2)?

A

−1.5

B

0.5

C

1.0

D

1.5

Solution Slope = (−2 − 4) / (1 − (−3)) = −6 / 4 = −1.5.

Question 36

⚑

Rounding 0.05674 kg to the nearest gram is equivalent to:

A

6 g

B

56 g

C

57 g

D

567 g

Solution 0.05674 kg = 56.74 g ≈ 57 g.

Question 37

⚑

3x² − 3x + 1 = 0

What is the discriminant of the equation above?

A

3

B

−3

C

−15

D

−21

Solution Δ = b² − 4ac = (−3)² − 4(3)(1) = 9 − 12 = −3.

Question 38

⚑

If 25^(3x − 1) = 125ˣ, then x =

A

1/3

B

0

C

2/3

D

1

Solution Rewrite both sides with base 5: 5^(2(3x−1)) = 5^(3x) ⇒ 6x − 2 = 3x ⇒ 3x = 2 ⇒ x = 2/3.

Question 39

⚑

Which of the following is the equation of the line perpendicular to the line graphed above and passing through (1, 0)?

A

y = −2x + 2

B

y = −2x + 1

C

y = (1/2)x − 1/2

D

y = 2x − 2

Solution Graphed line slope ≈ 1/2 (rises 2 from x = −4 to x = 0). Perpendicular slope = −2. Through (1, 0): y = −2(x − 1) = −2x + 2.

Question 40

⚑

The coordinates of the highest point on the graph of f(x) = −2x² − 4x + 1 are:

A

(2, −15)

B

(1, −5)

C

(−1, 3)

D

(−2, 1)

Solution Vertex at x = −b/(2a) = 4/(−4) = −1. f(−1) = −2 + 4 + 1 = 3. Vertex = (−1, 3).

Question 41

⚑

What is the value of |2x − 3y| + 4y for x = −1 and y = −2?

A

−8

B

−4

C

0

D

4

Solution |2(−1) − 3(−2)| + 4(−2) = |−2 + 6| − 8 = 4 − 8 = −4.

Question 42

⚑

Questions 42 and 43 refer to the graph of functions f (V-shape) and g (parabola opening downward) shown below.

What is the value of f(−6) + g(−1)?

A

1

B

3

C

4

D

8

Solution Read directly off the graph. The V-shape f has its vertex at (−4, 0) with arm slope ±½, so f(−6) = ½·|−6 − (−4)| = 1. The downward parabola g has vertex (−2, 4) with roots at −4 and 0, so g(−1) = 3. Sum = 1 + 3 = 4.

Question 43

⚑

Which of the following represents the solution set of g(x) > 3?

A

[−3, −1]

B

(−∞, −3] ∪ [−1, +∞)

C

(−∞, −1] ∪ [−1, +∞)

D

(−3, −1)

Solution The downward parabola g exceeds 3 between its two intersection points with y = 3, namely x = −3 and x = −1, exclusive. So the solution is (−3, −1).

Question 44

⚑

Given i = √(−1),   (2i + 5)(i − 1) =

A

−7 + 3i

B

−7 − 3i

C

−3 + 3i

D

3 + 3i

Solution (2i + 5)(i − 1) = 2i² − 2i + 5i − 5 = −2 + 3i − 5 = −7 + 3i.

Question 45

⚑

Solving 2x + 3yx + 1 = 4x² for y will give:

A

y = (4x² − 2x − 1) / 3x

B

y = 4x² − 2x − 1

C

y = 4x² − 5x − 1

D

y = (4x² + 2x + 1) / 3x

Solution Isolate 3yx: 3yx = 4x² − 2x − 1 ⇒ y = (4x² − 2x − 1) / 3x.

Question 46

⚑

If x = √3 − 1, what is the value of 3x² − x + 6√3?

A

13 − √3

B

13

C

13 + √3

D

15

Solution x² = (√3 − 1)² = 4 − 2√3, so 3x² = 12 − 6√3. Then 3x² − x + 6√3 = (12 − 6√3) − (√3 − 1) + 6√3 = 13 − √3.

Question 47

⚑

Figure not drawn to scale.

The area of △ABC in the figure above is:

A

131.5 cm²

B

144.5 cm²

C

150.0 cm²

D

300.0 cm²

Solution AD = 9, AC = 15 ⇒ DC = √(15² − 9²) = √144 = 12. BC = 20, DC = 12 ⇒ BD = √(400 − 144) = 16. AB = 9 + 16 = 25. Area = ½ · AB · DC = ½ · 25 · 12 = 150 cm².

Question 48

⚑

The distance between A(1, −1) and B(3, −2) is √k, where k is a real number. What is the value of k?

A

2

B

3

C

5

D

7

Solution d² = (3 − 1)² + (−2 + 1)² = 4 + 1 = 5. So k = 5.

Question 49

⚑

If P(x) = 3x³ − 2x + m is divisible by (x + 1), then the value of m is:

A

1

B

3

C

6

D

9

Solution By the Factor Theorem, P(−1) = 0: 3(−1)³ − 2(−1) + m = −3 + 2 + m = m − 1 = 0 ⇒ m = 1.

Question 50

⚑

What is the product of the positive integers in the solution set of 3x + 1 < 10?

A

1

B

2

C

3

D

6

Solution 3x < 9 ⇒ x < 3. Positive integers satisfying x < 3 are 1 and 2. Product = 2.