ACT Practice Exam D06

​ACT Practice form D06

Multiple Choice

In the figure, C is on $$\overline{BD}$$ , $$\angle BAC$$ measures $$40^\circ$$ , and $$\angle ABC$$ measures $$110^\circ$$ . What is the measure of $$\angle ACD$$ ?

• 110°

• 120°

• 130°

• 140°

• 150°

Explanation Slide...

The exterior angle is equal to the sum of the 2 interior angles. 110 + 40 = 150

Multiple Choice

For what value of a is the equation $$\frac{1}{2}a + 10 = 6$$ true?

• -32

• -8

• -2

• 8

• 32

Explanation Slide...

To solve , subtract 10 from both sides. Multiply by 2. Thus, the correct value of a is -8.

Multiple Choice

What is the least common denominator of the fractions $$\frac{4}{15}$$ , $$\frac{1}{20}$$ , and $$\frac{3}{8}$$ ?

• 24

• 120

• 300

• 480

• 2,400

Explanation Slide...

To find the least common denominator (LCD), we determine the least common multiple (LCM) of the denominators 15, 20, and 8. The LCM is 120, making it the least common denominator. The LCM will be the SMALLEST number that can be divided by all the denominators without getting a decimal.

Multiple Choice

|5 - 3| - |1 - 6| = ?

• -7

• -3

• 3

• 7

• 15

Explanation Slide...

First, calculate the absolute values: |5 - 3| = 2 and |1 - 6| = 5. Then, substitute these values into the equation: 2 - 5 = -3. Thus, the correct answer is -3.

Multiple Choice

In the trapezoid, $$\overline{AB}$$ is parallel to $$\overline{DC}$$ . What is the measure of $$\angle C$$ ?

• 50°

• 95°

• 115°

• 130°

• 135°

Explanation Slide...

In a trapezoid with parallel sides, consecutive angles between the bases are supplementary. Therefore, if angle Dis 85°, then angle C is 180° - 85° = 95°. Thus, the measure of C is 95°.

Multiple Choice

Gao earns his regular pay of $12 per hour for up to 40 hours of work per week. For each hour over 40 hours of work per week, Gao is paid 1 1/2 times his regular pay. How much does Gao earn in a week in which he works 56 hours?

• $672

• $756

• $768

• $1,008

• $1,344

Explanation Slide...

Gao earns $12/hour for 40 hours: 40 x 12 = $480. For 16 overtime hours (56-40), he earns 1.5 x 12 = $18/hour: 16 x 18 = $288. Total: $480 + $288 = $768.

Multiple Choice

On the first day of school, Ms. Dubacek gave her third-grade students 6 new spelling words to learn. On each day of school after that, she gave the students 3 new spelling words. How many new spelling words had she given the students by the end of the 21st day of school?

• 60

• 63

• 66

• 69

• 72

Explanation Slide...

Ms. Dubacek gave 6 words on the first day and 3 words for the next 20 days. So, total words = 6 + (3 * 20) = 6 + 60 = 66. Thus, by the end of the 21st day, she had given 66 new spelling words.

Multiple Choice

What is the value of the expression $$\frac{8!}{(4!)^2}$$ ? (Note: 3! = 3(2)(1) and 6! = 6(5)(4)(3)(2)(1))

• 0

• $$\frac{1}{2}$$

• 1

• 70

• 420

Explanation Slide...

To find the factorial key: math, arrow right to PROB, scroll down to 4:! You will type in the number then the factorial,example: for 8! you will key in 8, math, arrow right, scroll down to 4, enter, then do it all over again for the next number.

Multiple Choice

Right triangle ΔABC and its side lengths given in inches are shown below. What is sin B?

• a/b

• a/c

• b/a

• b/c

• c/a

Explanation Slide...

In a right triangle, sin B is defined as the ratio of the length of the side opposite angle B (b) to the length of the hypotenuse (c). Therefore, sin B = b/c, making the correct answer b/c.

Multiple Choice

(6a² - 5ac² + 12c) - (4c - 3a² - 2ac²) is equivalent to:

• 2a² - 2ac² + 14c

• 3a² - 7ac² + 16c

• 9a² - 3ac² + 8c

• 3a⁶ - 7a²c⁴ + 16c²

• 9a⁶ - 3a²c⁴ + 8c²

Explanation Slide...

To simplify (6a² - 5ac² + 12c) - (4c - 3a² - 2ac²), distribute the negative sign and combine like terms: 6a² + 3a² - 5ac² + 2ac² + 12c - 4c = 9a² - 3ac² + 8c. Thus, the correct answer is 9a² - 3ac² + 8c.

Multiple Choice

Which of the following (x,y) pairs is the solution for the system of equations x + 2y = 2 and -x + y = 7?

• (-4,3)

• (-1,1.5)

• (1,0.5)

• (0,1)

• (2,0)

Explanation Slide...

To solve the system, substitute (x,y) pairs into both equations. Testing (-4,3): x + 2y = -4 + 6 = 2 and -x + y = 4 + 3 = 7. Both equations are satisfied, confirming (-4,3) is the correct solution.

Multiple Choice

Tim’s flight was originally scheduled to depart at 4:51 p.m., but it was delayed 563 minutes. What time did Tim’s flight eventually depart?

• 1:12 a.m.

• 1:28 a.m.

• 2:14 a.m.

• 10:14 p.m.

• 10:54 p.m.

Explanation Slide...

Tim's flight was delayed by 563 minutes, which is 9 hours and 23 minutes. Adding this to the original departure time of 4:51 p.m. results in a new departure time of 2:14 a.m.

Multiple Choice

A circle with the equation x² + y² = 144 is graphed in the standard (x,y) coordinate plane. At what points does the circle intersect the x-axis?

• (-6,0) and (6,0)

• (-12,0) and (12,0)

• (-24,0) and (24,0)

• (-72,0) and (72,0)

• (-144,0) and (144,0)

Explanation Slide...

The equation x² + y² = 144 represents a circle with a radius of 12. To find the x-intercepts, set y = 0: x² = 144, giving x = ±12. Thus, the circle intersects the x-axis at (-6,0) and (6,0).

Multiple Choice

Given that x² - 5x - 36 factors into 2 binomial factors with integer coefficients, which of the following binomials is 1 of those factors?

• x - 12

• x - 9

• x - 4

• x + 6

• x + 12

Explanation Slide...

To factor x² - 5x - 36, we look for two numbers that multiply to -36 and add to -5. The numbers -9 and 4 fit this, leading to the factors (x - 9)(x + 4). Thus, x - 9 is one of the factors.

Multiple Choice

Two vectors are shown in the standard (x,y) coordinate plane below. One of the following vectors in the standard (x,y) coordinate plane is the sum of these 2 vectors. Which one?

• A

• B

• C

• D

• E

Explanation Slide...

To find the sum of the two vectors, add their corresponding components. You can also add vectors graphically by drawing them "head-to-tail". 

Multiple Choice

A square and a rectangle have the same area. The length of the rectangle is 196 centimeters, and the width of the rectangle is 4 centimeters. What is the length, in centimeters, of a side of the square?

• 20

• 28

• 100

• 400

• 784

Explanation Slide...

The area of the rectangle is length × width = 196 cm × 4 cm = 784 cm². Since the square has the same area, its side length is √784 cm = 28 cm. Thus, the length of a side of the square is 28 cm.

Multiple Choice

T-shirts are on sale for D dollars each, including tax. Valentina has N dollars with which to purchase T-shirts. After she purchases the maximum number she can, Q T-shirts, she has R dollars left. For all possible choices of D and N, which of the following equations models a correct relationship between D, N, Q, and R, as defined?

• N = Q + R

• N = QD + RD

• N = QD + R

• N = Q + RD

• N = QR + D

Explanation Slide...

Valentina spends Q times D dollars on T-shirts, leaving her with R dollars. Thus, her total money N is the cost of T-shirts plus the remaining money: N = QD + R, making this the correct equation.

Multiple Choice

At a sandwich shop, customers can order either a meat or a vegetarian sandwich on either white or wheat bread. Out of a total of 50 customers, 20 ordered a sandwich on white bread, 28 ordered a meat sandwich, and 12 ordered a meat sandwich on white bread. How many customers ordered a vegetarian sandwich on wheat bread?

• 2

• 8

• 10

• 14

• 16

Explanation Slide...

Out of 50 customers, 20 ordered white bread and 12 ordered meat on white. Thus, 8 ordered vegetarian on white. With 28 total meat orders, 20 on white means 8 on wheat. Therefore, 50 - (8 + 8) = 14 ordered vegetarian on wheat.

Multiple Choice

A team of biologists tagged and released 90 deer in a forest. From the same forest 2 weeks later, the biologists collected a random sample of 30 deer, 5 of which were tagged. Let p be the proportion of deer in this forest that are tagged. What is p̂, the sample proportion, for this sample?

• 1/24

• 1/18

• 1/6

• 1/5

• 1/3

Explanation Slide...

The sample proportion p̂ is calculated as the number of tagged deer in the sample divided by the total number of deer in the sample. Here, p̂ = 5/30 = 1/6, which is the correct choice.

Multiple Choice

20. √2 + √8 + √18 = ?

• 2√7

• 6√2

• 12√2

• 14√2

• 14

Explanation Slide...

To simplify √2 + √8 + √18, we rewrite √8 as 2√2 and √18 as 3√2. Thus, √2 + 2√2 + 3√2 = 6√2. The correct answer is 6√2.

Multiple Choice

Which of the following inequalities is equivalent to 3 - 2x > 7 - x?

• x < -4/3

• x > -4/3

• x < -4

• x > -4

• x > 6

Explanation Slide...

To solve 3 - 2x > 7 - x, rearrange to get -2x + x > 7 - 3, simplifying to -x > 4. Dividing by -1 reverses the inequality, yielding x < -4. Thus, the correct choice is x < -4.

Multiple Choice

Let a be positive and b be negative. If it can be determined, in which quadrant of the standard (x,y) coordinate plane is the point (-a,b²) located?

• I

• II

• III

• IV

• Cannot be determined from the given information

Explanation Slide...

The point (-a, b^2) has x = -a (negative) and y = b^2 (positive, since b is negative). This places the point in the second quadrant, where x is negative and y is positive.

Multiple Choice

The mass of a certain type of bacteria grows exponentially, doubling every 20 minutes. What was the mass, in milligrams, of the bacteria exactly 2 hours after the mass first reached 10 milligrams?

• 70

• 200

• 320

• 640

• 4,000

Explanation Slide...

The bacteria doubles every 20 minutes. In 2 hours (120 minutes), there are 6 doubling periods. Starting from 10 mg, after 6 doublings: 10 mg * 2^6 = 10 mg * 64 = 640 mg. Thus, the mass is 640 mg.

Multiple Choice

One day will be randomly selected from the 7 days in a week. Then 1 month will be randomly selected from the 12 months in a year. What is the probability that the selected day will be Tuesday and the selected month will be January?

• $$\frac{1}{84}$$

• $$\frac{1}{42}$$

• $$\frac{1}{19}$$

• $$\frac{2}{19}$$

• $$\frac{19}{84}$$

Explanation Slide...

The probability of selecting a Tuesday is 1/7 and for January is 1/12. Thus, the combined probability is 1/7 times 1/12. Therefore, the correct answer is 1/84.

Multiple Choice

The average weight of Juan, Jim, and Malik is exactly 160 pounds. The average weight of Juan, Jim, Malik, and Harry is exactly 150 pounds. How many pounds does Harry weigh?

• 100

• 120

• 130

• 155

• 190

Explanation Slide...

The total weight of Juan, Jim, and Malik is 3 * 160 = 480 pounds. The total weight of all four is 4 * 150 = 600 pounds. Therefore, Harry's weight is 600 - 480 = 120 pounds.

Multiple Choice

What is the value of the expression (log₆(36)) (log₃(9)) ?

• 2

• 3

• 4

• 5

• 18

Explanation Slide...

To solve (log_6(36))(log_3(9)), we simplify: log_6(36) = 2 (since 6^2 = 36) and log_3(9) = 2 (since 3^2 = 9). Thus, (2)(2) = 4. The correct answer is 4.

Multiple Choice

Each of 2 identical number cubes, shown below, has a different integer, 1 through 6, on each face. Consider the sample space determined by rolling these number cubes and adding the 2 integers on the faces that land on top. What is the positive difference between the greatest sum and the least sum in this sample space?

• 5

• 10

• 11

• 12

• 34

Explanation Slide...

The least sum occurs with 1+1=2, and the greatest sum with 6+6=12. The positive difference between the greatest and least sums is 12-2=10.

Multiple Choice

What angle measure, in radians, is equal to 30° ?

• π/6

• π/5

• π/3

• π/2

• 5π/6

Explanation Slide...

To convert degrees to radians, use the formula: radians = degrees × (π/180). For 30°, it becomes 30 × (π/180) = π/6. Thus, the angle measure in radians equal to 30° is π/6.

Multiple Choice

For one school week, Hannah recorded the following temperatures, in degrees Fahrenheit, so she could investigate the difference between the high and low temperature each day. To the nearest degree, what was the mean of the differences in daily high and low temperatures for these 5 days?

• 28°

• 29°

• 30°

• 32°

• 34°

Explanation Slide...

To find the mean of the differences, calculate the daily differences between high and low temperatures, sum them, and divide by 5. The total difference is 170°, so the mean is 34° (170° ÷ 5). Thus, the correct answer is 34°.

Multiple Choice

A family’s budgeted items are expressed as a fraction of their weekly income in the chart below. What fractional part of their weekly income is left for unbudgeted items?

• 5/48

• 1/48

• 1/24

• 1/16

• 1/8

Explanation Slide...

To find the unbudgeted fraction, subtract the total budgeted fractions from 1.

Multiple Choice

What is the 322nd digit after the decimal point in the repeating decimal 0.1357?

• 0

• 1

• 3

• 5

• 7

Explanation Slide...

Divide the position of the digit we want to find by the length of the repeating block and find the remainder. The remainder will tell us the position of the digit within the repeating block.

Multiple Choice

You and a friend each have a can full of water. You start pouring the water from your can into an empty bucket at a constant rate of 4 ounces per second. While you are still pouring water, 3 seconds after you started, your friend starts pouring the water from her can into the same bucket at a constant rate of 2 ounces per second. How many seconds after you first started pouring the water into the bucket will it contain 24 ounces of water?

• 4

• 5

• 6

• 8

• 12

Explanation Slide...

Set up an equation representing the total amount of water in the bucket as a function of time, and solve for the time when the bucket contains 24 ounces. The total amount of water is the sum of the water from both people:4t + 2(t−3) Set this equal to 24 : 4𝑡 + 2(𝑡−3) = 24

Multiple Choice

A package of candy contains pieces each of which is 1 of 6 possible colors: brown, red, green, yellow, orange, and blue. In each package, $$\frac{1}{3}$$ of the pieces are brown and the remaining pieces have an even distribution of the other 5 colors. What is the probability that a piece drawn randomly from the package is red?

• $$\frac{1}{15}$$

• $$\frac{2}{15}$$

• $$\frac{1}{6}$$

• $$\frac{1}{5}$$

• $$\frac{2}{3}$$

Explanation Slide...

Calculate the proportion of non-brown candies: 1 - 1/3 = 2/3. Calculate the probability of drawing a red candy: 2/3 times 1/5 = 2/15.

Multiple Choice

Which of the following intervals is the range of the function $$f(x) = -(x - 3)^2 + 4$$ ?

• (-∞, 3]

• (-∞, 4]

• [3, 4]

• [3, ∞)

• [4, ∞)

Explanation Slide...

Graph the function then look ay the y values.

Multiple Choice

Anoki made a scale drawing of his rectangular classroom. The classroom is 7.5 meters by 9.0 meters. In his scale drawing, Anoki made the length of the shorter side of the classroom 9.0 centimeters. What is the length, in centimeters, of the longer side of the classroom in Anoki’s scale drawing?

• 7.5

• 10.5

• 10.8

• 15.0

• 16.5

Explanation Slide...

Set up a proportion and then solve.

Multiple Choice

In rhombus ABCD shown below, AC = 5 feet and BD = 6 feet. What is the area of ABCD, in square feet?

• 5.5

• 7.5

• 11

• 15

• 30

Explanation Slide...

Find the area of one triangle (A=1/2bh) then multiply by 2 because there are 2 triangles. The diagonals of a rhombus bisect each other. The base of the triangles = 5, the height of the triangles = 3. the area is 2 times 1/2 times 5 times 3 = 15An alternate formula: 1/2d1d2 = 1/2 times 5 times 6 = 15

Multiple Choice

One number is 25% of a second number, and the second number is 70% of a third number. The first number is what percent of the third number?

• A. 17.5%

• B. 42.5%

• C. 45%

• D. 87.5%

• E. 95%

Explanation Slide...

x = 25% of 70% of z; of means multiply; 0.25 times 0.7 = 0.175 = 17.5%

Multiple Choice

The monthly rent charged for a store at Center Street Mall is $2 per square foot of floor area. The floor plan of a store at Center Street Mall is shown in the figure below, with right angles as indicated and all distances given in feet. How much monthly rent is charged for this store?

• $1,656

• $1,872

• $6,624

• $7,380

• $7,488

Explanation Slide...

Know your formulas! The composite diagram is made from a rectangle and a trapezoid. Formulas: A=bh; A=1/2(b1+b2)h. Find the area of each figure, add them. Then multiply the total area by the cost to get the total cost.

Multiple Choice

Mrs. Neeson, a science teacher, told her students that 30.0% of their final semester grades will come from their homework averages, and the remaining 70.0% will come from their test averages. She also said that the final exam will count for 20.0% of the test average. What percent of the science final semester grade is the final exam grade?

• 6.0%

• 10.5%

• 14.0%

• 20.0%

• 28.6%

Explanation Slide...

Multiply the percentage of the test average by the percentage of the final exam within the test average. 0.70×0.20=0.14 = 14%

Multiple Choice

A rectangle with an area of 30 square inches has length and width, in inches, that are both integers. Which of the following CANNOT be the perimeter, in inches, of the rectangle?

• 22

• 26

• 34

• 60

• 62

Explanation Slide...

List all possible integer pairs of length and width for the rectangle, the factors of 30; then calculate the perimeter for each pair, and find the perimeter that is not in the options. The factors of 30 are(1,30), (2,15), (3,10), (5,6). To test the perimeter, add the pair of factors then multiply by 2. ex: 30+1 = 31 times 2 = 62. 62 is a possible perimeter and therefore not the answer.

Multiple Choice

A couple is deciding between 2 condos to purchase. Some information about each condo is given below. The couple assumes that the market value of either condo will increase exponentially at a rate of 4% per year. What is the positive difference, in dollars, of the 2 list prices?

• 2.01 × 10²

• 2.01 × 10³

• 2.01 × 10⁴

• 2.10 × 10⁴

• 2.10 × 10⁵

Explanation Slide...

To find the positive difference in list prices, we subtract. The answer choices are in scientific notation. To write a number in scientific notation, place the decimal point after the first non-zero digit, then multiply by a power of 10 where the exponent represents how many places the decimal point was moved

Multiple Choice

The couple assumes that the market value of either condo will increase exponentially at a rate of 4% per year. The couple will consider the price per square foot of each condo. Let x and y be the price per square foot, rounded to the nearest $1, of Condo X and Condo Y, respectively. One of the following comparisons is true. Which one?

• x is $3 greater than y.

• x is $6 less than y.

• x is $6 greater than y.

• x is $18 less than y.

• x is $18 greater than y.

Explanation Slide...

Divide the list price by the square footage to find the price per foot of condo Y. Then subtract the price per foot of condo X from condo Y to find the difference.

Multiple Choice

The couple assumes that the market value of either condo will increase exponentially at a rate of 4% per year. The annual property tax for Condo X is 2% of its assessed value. What is the assessed value of Condo X?

• $19,120

• $42,000

• $186,200

• $191,200

• $205,900

Explanation Slide...

To find the assessed value of Condo X, we set the property tax (2% of assessed value) equal to the expected increase in market value. 3,824 = 2%x; change the percent to a decimal then divide.

Multiple Choice

In the complex plane, consider the segment whose endpoints are the points corresponding to –6 + 3i and 2 – 7i. The midpoint of this segment corresponds to which of the following complex numbers?

• –4 – 4i

• –4 – 5i

• –2 – 2i

• –2 + 2i

• 4 + 5i

Explanation Slide...

To find the midpoint of the segment, average the real and imaginary parts of the endpoints: (-6 + 2)/2 = -2 and (3i - 7i)/2 = -2i. Thus, the midpoint is -2 - 2i

Multiple Choice

In the standard (x,y) coordinate plane, the 3 lines with equations y = $$\frac{3}{5}x - 3$$ , y = $$-\frac{2}{5}x + 2$$ , and x = 0 bound a triangular region. What is the area, in square coordinate units, of that triangular region? (A blank grid has been provided for your use.)

• 2.5

• 5.0

• 7.5

• 12.5

• 62.5

Explanation Slide...

To find the area of the triangle formed by the lines, we first find the vertices by solving the equations. The vertices are (0, -3), (0, 2), and (5, 0). The area is calculated as 0.5 * base * height = 0.5 * 5 * 5 = 12.5.

Multiple Choice

In square ABCD shown below, AC is a diagonal and the length of BC is 75 feet. Which of the following quantities is NOT a rational number?

• The perimeter of ABCD, in feet

• The area of ABCD, in square feet

• The length of AB, in feet

• The length of AC, in feet

• The measure of ∠CAD, in degrees

Explanation Slide...

The length of AC is calculated using the rules of 45-45-90 triangles; the hypotenuse is equal to the leg times the square root of 2. Because the length of AC involves a radical that is not a perfect square it is irrational.

Multiple Choice

The volume of a solid object is equal to the volume of water it displaces when completely submerged in water. A solid object will be placed in a rectangular tank that has a base of 35 cm by 30 cm and is filled with water to a uniform depth of 13 cm. When the object is completely submerged, the new depth of the water in the tank is 15 cm. What is the volume, in cubic centimeters, of the object?

• 135

• 525

• 780

• 1,212

• 2,100

Explanation Slide...

The volume of the object is equal to the change in water volume. The tank's base area is 35 cm x 30 cm = 1050 cm². The increase in water depth is 15 cm - 13 cm = 2 cm. Thus, the volume displaced is 1050 cm² x 2 cm = 2100 cm³.

Multiple Choice

The 2 circles graphed in the standard (x,y) coordinate plane are centered at the origin, O. In coordinate units, the radius of the smaller circle is 2, and the radius of the larger circle is 4. Points A(−4,0), B, and C(4,0) are on the larger circle. The measure of ∠BOC is 45°. What is the x-coordinate of B?

• F. $$\frac{4}{\sqrt{3}}$$

• G. $$\frac{4}{\sqrt{2}}$$

• H. 4

• J. $$4\sqrt{2}$$

• K. $$4\sqrt{3}$$

Explanation Slide...

Because the given angle is 45 degrees. The triangle is a 45-45-90. The hypotenuse is the radius which is 4. To find the length of the missing leg divide 4 by the square root of 2.

Multiple Choice

The 2 circles graphed in the standard (x,y) coordinate plane are centered at the origin, 0. In coordinate units, the radius of the smaller circle is 2, and the radius of the larger circle is 4. Points A(−4,0), B, and C(4,0) are on the larger circle. The measure of ∠BOC is 45°. A 3rd circle, not shown, is the image resulting from applying the 1st transformation listed below to the smaller circle and then applying the 2nd transformation listed below to the result of the 1st transformation. 1st: A dilation with center O and scale factor 2 2nd: A translation of 8 coordinate units to the right The 3rd circle has how many points in common with the larger circle?

• 0

• 1

• 2

• 4

• Infinitely many

Explanation Slide...

The smaller circle, after dilation and translation, has a radius of 4 and is centered at (8,0). It touches the larger circle at one point, making the answer 1.

Multiple Choice

What is the area, in square coordinate units, of the region that is outside the smaller circle and inside the larger circle?

• 4π

• 12π

• 20π

• 48π

• 80π

Explanation Slide...

The area of a circle is given by A = πr². If the larger circle has radius 4 (A = 16π) and the smaller circle has radius 2 (A = 4π), the area outside the smaller and inside the larger is 16π - 4π = 12π.

Multiple Choice

Which of the following is an equation of $$\overleftarrow{O}\overrightarrow{B}$$ ?

• $$y = -4x$$

• $$y = -x$$

• $$y = x$$

• $$y = 2x$$

• $$y = 4x$$

Multiple Choice

A sequence is given by s₁ = 4 and s_n+1 = 2s_n - 3 for n ≥ 1. What is s₅?

• 5

• 7

• 11

• 19

• 35

Explanation Slide...

Use the recursive formula to find subsequent terms until 𝑠5 is reached. s2 = 2(4) - 3 = 5; s3 = 2(5) - 3 = 7; s4 = 2(7) -3 = 11; s5 = 2(11) - 3 = 19

Multiple Choice

In the figure, the distances between 2 pairs of cities are shown, as well as the angle formed at Ewing, which has a measure of 127°. Which of the following values is closest to the distance, in miles, from Deerborn to Fergus? (Note: cos 127° ≈ -0.6; sin 127° ≈ 0.8)

• 100

• 140

• 160

• 180

• 200

Explanation Slide...

KNOW THE FORMULAS!! Law of cosines: c2 = a2+b2-2ab(cosC). Make sure the calculator is in degrees. Follow the order of operations - do all multiplication first, then add or subtract, lastly take the square root.

Multiple Choice

Which of the following expressions is equivalent to $$\frac{1}{x-a} - \frac{1}{x+a}$$ ?

• $$\frac{2a}{x^2-a^2}$$

• $$\frac{x^2-a^2}{x^2-a^2}$$

• $$\frac{-2x}{x^2-a^2}$$

• $$\frac{1}{x^2-a^2}$$

• $$\frac{-1}{2a}$$

Explanation Slide...

Multiply each fraction by the other denominator. Be sure to multiply both numerator and denominator. Then combine like terms in the numerator.

Multiple Choice

One of the following equations represents the ellipse shown below in the standard (x,y) coordinate plane. Which one?

• $$\frac{(x-2)^2}{3} + \frac{(y+3)^2}{5} = 1$$

• $$\frac{(x+2)^2}{3} + \frac{(y-3)^2}{5} = 1$$

• $$\frac{(x+2)^2}{5} + \frac{(y-3)^2}{3} = 1$$

• $$\frac{(x-2)^2}{9} + \frac{(y+3)^2}{25} = 1$$

• $$\frac{(x+2)^2}{9} + \frac{(y-3)^2}{25} = 1$$

Explanation Slide...

(h, k) is the center; notice that b and a are squared. Find the midpoint of (-1, -3) and (5, -3) to find the center. Remember the center and the equation will have opposite signs because of the minus signs.

Multiple Choice

One of the following equations is that of a parabola with x-intercepts -5 and $$\frac{3}{4}$$ in the standard (x,y) coordinate plane. Which equation?

• $$y = 3x^2 - 11x - 20$$

• $$y = 3x^2 + 11x - 20$$

• $$y = 4x^2 - 17x = 15$$

• $$y = 4x^2 + 17x - 15$$

• $$y = 15x^2 - 17x - 4$$

Explanation Slide...

Using the intercepts x=-5 and x=3/4, we can solve, work backwards, to get zero which gives us the factors of the parabola. The factors are (x+5) and (4x - 3). Now multiply the binomials using the foil method to get the equation.

Multiple Choice

There are 100 fractions in the following set. $$\frac{1}{4}, \frac{4}{7}, \frac{7}{10}, \frac{10}{13}, \ldots, \frac{292}{295}, \frac{295}{298}, \frac{298}{301}$$ Each fraction after the first is found by adding 3 to the preceding fraction’s numerator and denominator. What is the product of these 100 fractions?

• 1

• $$\frac{1}{3}$$

• $$\frac{1}{4}$$

• $$\frac{1}{100}$$

• $$\frac{1}{301}$$

Explanation Slide...

Notice that the numerator of each fraction cancels with the denominator of the previous fraction.

Multiple Choice

If $$2^x = 7$$ and $$2^y = 14$$ , then x - y = ?

• -14

• -7

• -1

• 1

• 49

Explanation Slide...

The exponential form 𝑏𝑦 = 𝑥 is equivalent to the logarithmic form log𝑏𝑥 = 𝑦; The logarithmic form of 2𝑥 = 7 is log27 = 𝑥. The logarithmic form of the equation 2𝑦 = 14 is log214 = 𝑦; there for x - y = log27 - log214. To type this in the calculator, key in alpha window then scroll down to logbase.

Multiple Choice

The table indicates the grade (10 or 11) and high school (North or South) of the 270 students enrolled in Algebra II in the Green City School District. Suppose 2 of these students will be chosen at random to represent the Algebra II classes at a local STEM event. Which of the following expressions gives the probability that both chosen students will be from the same grade and the same high school?

• 
  $$\frac{47(46)}{270(269)} + \frac{93(92)}{270(269)} + \frac{73(72)}{270(269)} + \frac{57(56)}{270(269)}$$

• 
  $$\frac{47}{270} + \frac{93}{270} + \frac{73}{270} + \frac{56}{270}$$

• 
  $$\frac{47(73)}{270(269)} + \frac{93(57)}{270(269)}$$

• $$\frac{47(93)}{270(269)} + \frac{73(57)}{270(269)}$$

• $$\frac{1}{4}\cdot\frac{1}{4}$$

Explanation Slide...

Add all the students to get the total number of students. The number of people selected / the total number of people gives the probability of the first person. Subtracting 1 from both the numerator and denominator will give you the probability of the 2nd person. Example; 10th grade north - (47/270)(46/269). Do this for each school and each grade.

Multiple Choice

A certain company has 120 employees, 85 of whom have business degrees. Of the employees with business degrees, 75 are certified public accountants (CPAs). There are 14 employees who are not CPAs and also do not hold a business degree. One employee of the company will be selected at random to be interviewed for a television news program. What is the probability that the selected employee will be a CPA? (Note: A business degree is NOT required to be a CPA.)

• $$\frac{75}{120}$$

• 
  $$\frac{85}{120}$$

• $$\frac{89}{120}$$

• 
  $$\frac{96}{120}$$

• $$\frac{99}{120}$$

Explanation Slide...

Basic probability is found by the desired outcome over the total.We know the total employees = 120; The desired outcome is the numbers of CPA's. 120 (employees) - 10(B.D. only) - 14 (no degree) = 96. The probability is 96/120