This is our free SAT Math practice test for section 4 of the new SAT. This is the portion of the math test that allows the usage of a calculator. The calculator is meant to save you time, which will allow you to focus on complex modeling and reasoning. Be aware that not all of the problems will require the use of a calculator. The SAT math test is designed to measure fluency, conceptual understanding, and applications. This section will include 30 multiple choice questions and 8 gridin questions.
Directions for Questions 130: Solve each problem and then select the best answer from the choices provided. Directions for Questions 3138: Solve each problem and then type in the correct answer. Use the forward slash symbol ( / ) for fractions.
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Question 1 of 38
1. Question
A teacher gave his students 5 vocabulary words to memorize. Each day thereafter, he gave his students an additional 3 vocabulary words to memorize. After 25 days, how many vocabulary words had his students memorized?
CorrectTo solve this problem, translate the question into a mathematical expression based on the key context words used. On the 1st day, the students are given 5 vocabulary words. Then they are given 3 more vocabulary words each day for the next 24 days.
The question can be restated mathematically as:
Total number of vocabulary words = initial number of vocabulary words + number of vocabulary words per day x number of days; or:
Notice that days cancel out in the expression and the right side of the equation yields a total number of words. This is a useful method of confirming that the equation was properly set up.Plugging in values and evaluating:
Total number of words = 5 + (3 * 24) = 77 words.IncorrectTo solve this problem, translate the question into a mathematical expression based on the key context words used. On the 1st day, the students are given 5 vocabulary words. Then they are given 3 more vocabulary words each day for the next 24 days.
The question can be restated mathematically as:
Total number of vocabulary words = initial number of vocabulary words + number of vocabulary words per day x number of days; or:
Notice that days cancel out in the expression and the right side of the equation yields a total number of words. This is a useful method of confirming that the equation was properly set up.Plugging in values and evaluating:
Total number of words = 5 + (3 * 24) = 77 words. 
Question 2 of 38
2. Question
The table above shows the mass, radius, axis period, radius of orbit, and period of revolution of the Sun and the planets in our solar system. Based on the table, if Earth, Mars, or Jupiter was chosen at random, what is the probability that the chosen planet’s mass would be greater than 10 x 10^{24} ?CorrectFirst, analyze the table and note each planet’s mass:
Earth: 5.98 x 10^{24} kg
Mars: 6.37 x 10^{23} kg
Jupiter: 1.90 x 10^{27} kgOnly Jupiter is greater than 10 x 10^{24} kg. One out of the three planets is 33.33%.
IncorrectFirst, analyze the table and note each planet’s mass:
Earth: 5.98 x 10^{24} kg
Mars: 6.37 x 10^{23} kg
Jupiter: 1.90 x 10^{27} kgOnly Jupiter is greater than 10 x 10^{24} kg. One out of the three planets is 33.33%.

Question 3 of 38
3. Question
The sum of two positive integers is 13. The difference between these numbers is 7. What is their product?
CorrectUse the given information to create a system of equations in two variables and then solve with the method of combination or substitution. Translating the problem statement into algebra:
x + y = 13
x − y = 7The method of substitution can be used here: solve for x in terms of y, x = y + 7, then substituting the right hand side for x in the other equation, (y + 7) + y = 13, then combining like terms and solving for y, 2y = 6, y = 3, then substituting 3 for y in either equation and solving for x, x + 3 = 13; x = 10.
However, you should notice that the method of combination reduces the number of steps necessary to solve the system. Add the 2 equations together and solve for x:
x + x = 13 + 7; 2x = 20; x = 10
The value of y can then be found. Once the variable values are found, their product can be computed:
x * y = 10 * 3 = 30
IncorrectUse the given information to create a system of equations in two variables and then solve with the method of combination or substitution. Translating the problem statement into algebra:
x + y = 13
x − y = 7The method of substitution can be used here: solve for x in terms of y, x = y + 7, then substituting the right hand side for x in the other equation, (y + 7) + y = 13, then combining like terms and solving for y, 2y = 6, y = 3, then substituting 3 for y in either equation and solving for x, x + 3 = 13; x = 10.
However, you should notice that the method of combination reduces the number of steps necessary to solve the system. Add the 2 equations together and solve for x:
x + x = 13 + 7; 2x = 20; x = 10
The value of y can then be found. Once the variable values are found, their product can be computed:
x * y = 10 * 3 = 30

Question 4 of 38
4. Question
In the given figure, the measure of angle OAC is 60 degrees, and the center of the circle is 0. If the circle has a radius of 6, what is the length of segment DB?CorrectSince O is the center, AO = DO = CO = BO = 6. This means we can divide these pieces into isosceles triangles. If triangle AOC is isosceles, and angle OAC = 60°, then angle ACO = 60°. This leaves 180 − 120 = 60 degrees for the third angle AOC. Triangle AOC = equilateral, and angle DOB is vertical with angle AOC and will also equal 60 degrees. Therefore, triangle DOB will also be equilateral, and the length of DB = 6, choice (C).
IncorrectSince O is the center, AO = DO = CO = BO = 6. This means we can divide these pieces into isosceles triangles. If triangle AOC is isosceles, and angle OAC = 60°, then angle ACO = 60°. This leaves 180 − 120 = 60 degrees for the third angle AOC. Triangle AOC = equilateral, and angle DOB is vertical with angle AOC and will also equal 60 degrees. Therefore, triangle DOB will also be equilateral, and the length of DB = 6, choice (C).

Question 5 of 38
5. Question
Which of the following lists every positive nonprime factor of 20?
CorrectThe positive factors of 20 are: 1, 2, 4, 5, 10, and 20. Of these, two are prime: 2 and 5. Recall that a prime number must have 2 distinct factors including itself; for this reason, 1 is not a prime a number, because it has only 1 factor. Therefore, the nonprime factors are 1, 4, 10, and 20.
IncorrectThe positive factors of 20 are: 1, 2, 4, 5, 10, and 20. Of these, two are prime: 2 and 5. Recall that a prime number must have 2 distinct factors including itself; for this reason, 1 is not a prime a number, because it has only 1 factor. Therefore, the nonprime factors are 1, 4, 10, and 20.

Question 6 of 38
6. Question
What is the difference between the median and the mode in the following set of data?
72, 44, 58, 32, 34, 68, 94, 22, 67, 45, 58
CorrectStart by organizing the data numerically from least to greatest:
22, 32, 34, 44, 45, 58, 58, 67, 68, 72, 94The mode, or data value that occurs most often, is 58.
The median, or data value in the middle of the data set, is also 58.
The difference between these values is 0.
IncorrectStart by organizing the data numerically from least to greatest:
22, 32, 34, 44, 45, 58, 58, 67, 68, 72, 94The mode, or data value that occurs most often, is 58.
The median, or data value in the middle of the data set, is also 58.
The difference between these values is 0.

Question 7 of 38
7. Question
What is the total number of degrees in the interior angles of a regular hexagon?
CorrectTo find the total number of degrees in the interior angles of any polygon, all we need to know is the number of sides. Each time we add a side (triangle to quadrilateral, quadrilateral to pentagon, etc), we add another 180° to the total. This can be expressed with the formula:
Sum of Interior Angles = (n − 2) * 180, where n is the number of sides.
Since we are told the figure is a hexagon, it has 6 sides, so n = 6:
Sum of Interior Angles = (6 − 2) * 180
= 4 * 180
= 720IncorrectTo find the total number of degrees in the interior angles of any polygon, all we need to know is the number of sides. Each time we add a side (triangle to quadrilateral, quadrilateral to pentagon, etc), we add another 180° to the total. This can be expressed with the formula:
Sum of Interior Angles = (n − 2) * 180, where n is the number of sides.
Since we are told the figure is a hexagon, it has 6 sides, so n = 6:
Sum of Interior Angles = (6 − 2) * 180
= 4 * 180
= 720 
Question 8 of 38
8. Question
A pool that holds 35,000 cubic feet of water is being filled by a pump at a rate of 200 cubic feet per minute. At the same time, water is draining out through an open valve accidentally left open. If the pool is full in 200 minutes, at what rate is the water draining out?
CorrectWrite an equation to model the situation:
IncorrectWrite an equation to model the situation:

Question 9 of 38
9. Question
What is the sum of the areas of the three rectangles that are drawn below the graph of the line y = 2^{x} ?CorrectStart by plugging in x = 1, x = 2, and x = 3 into the equation y = 2^{x} to find the ycoordinates where the corner of each rectangle touches y = 2^{x}. Those yvalues will be the height of each rectangle. Since we know each rectangle is 1 space apart, we can then find each area.
The areas are:
2(1) = 2
4(1) = 4
8(1) = 8The sum is 14.
IncorrectStart by plugging in x = 1, x = 2, and x = 3 into the equation y = 2^{x} to find the ycoordinates where the corner of each rectangle touches y = 2^{x}. Those yvalues will be the height of each rectangle. Since we know each rectangle is 1 space apart, we can then find each area.
The areas are:
2(1) = 2
4(1) = 4
8(1) = 8The sum is 14.

Question 10 of 38
10. Question
A perfect sphere with a diameter of 5 meters is inscribed in a cube. Which of the following best approximates the volume of the space between the sphere and the cube?
Correct
It may help you to draw a picture to better visualize the sphere and the cube. The volume of the space between the figures is the total volume of the cube minus the volume of the sphere.
Incorrect
It may help you to draw a picture to better visualize the sphere and the cube. The volume of the space between the figures is the total volume of the cube minus the volume of the sphere.

Question 11 of 38
11. Question
What is the value of a + b?CorrectRecall that angles forming a straight line and the 3 interior angles of a triangle each sum to 180°. Using these relationships and substituting expressions for the unlabeled angles enables us to then solve the equations in terms of a and b. We can then combine these equations to solve for a + b.
For the top triangle:
a + (180 − w) + (180 − x) = 180°
360 + a − w − x = 180
a = −180 + w + xFor the bottom triangle:
b + (180 − y) + (180 − z) = 180°
360 + b − y − z = 180
b = −180 + y + zAdding the 2 equations together:
a + b = (−180 + w + x) + (−180 + y + z)
a + b = −360 + w + x + y + zIncorrectRecall that angles forming a straight line and the 3 interior angles of a triangle each sum to 180°. Using these relationships and substituting expressions for the unlabeled angles enables us to then solve the equations in terms of a and b. We can then combine these equations to solve for a + b.
For the top triangle:
a + (180 − w) + (180 − x) = 180°
360 + a − w − x = 180
a = −180 + w + xFor the bottom triangle:
b + (180 − y) + (180 − z) = 180°
360 + b − y − z = 180
b = −180 + y + zAdding the 2 equations together:
a + b = (−180 + w + x) + (−180 + y + z)
a + b = −360 + w + x + y + z 
Question 12 of 38
12. Question
A passenger ship left Southampton, England for the Moroccan coast. The ship travelled the first 230 miles at an average speed of 20 knots, then increased its speed for the next 345 miles to 30 knots. It travelled the remaining 598 miles at an average speed of 40 knots. What was the ship’s approximate average speed in miles per hour? (1 knot = 1.15 miles per hour)
CorrectIn order to calculate the average speed of the trip, it is first necessary to solve for the total number of miles travelled and the total length of time travelled. Because the question asks for the average speed in miles per hour (and not knots), it is also necessary to convert all knots into miles per hour (mph).
Use the provided unit conversion to set up a general expression relating knots to mph:
Notice that knots cancel from the numerator and denominator leaving units of mph, which is what we are looking for; this verifies that we are on the right track toward the solution. Repeat this calculation for the other knot values to find: 30 knots = 34.5 mph; 40 knots = 46 mph.Using the calculated mph values, we can now solve for the length of time each portion of the trip took. Recall that a total distance travelled is equal to a rate, or speed of travel, multiplied with a length of time; or: Distance = Rate * Time. Dimensional analysis is again useful to verify that the equation results in the appropriate units.
Use the distances and corresponding rates to solve for each time:
Repeat this calculation for the other times: 345 miles takes 10 hours, and 598 miles takes 13 hours.Use the calculated values to divide the total distance travelled by the total length of time travelled to determine the overall average rate of travel:
IncorrectIn order to calculate the average speed of the trip, it is first necessary to solve for the total number of miles travelled and the total length of time travelled. Because the question asks for the average speed in miles per hour (and not knots), it is also necessary to convert all knots into miles per hour (mph).
Use the provided unit conversion to set up a general expression relating knots to mph:
Notice that knots cancel from the numerator and denominator leaving units of mph, which is what we are looking for; this verifies that we are on the right track toward the solution. Repeat this calculation for the other knot values to find: 30 knots = 34.5 mph; 40 knots = 46 mph.Using the calculated mph values, we can now solve for the length of time each portion of the trip took. Recall that a total distance travelled is equal to a rate, or speed of travel, multiplied with a length of time; or: Distance = Rate * Time. Dimensional analysis is again useful to verify that the equation results in the appropriate units.
Use the distances and corresponding rates to solve for each time:
Repeat this calculation for the other times: 345 miles takes 10 hours, and 598 miles takes 13 hours.Use the calculated values to divide the total distance travelled by the total length of time travelled to determine the overall average rate of travel:

Question 13 of 38
13. Question
What is the value of y, if lines a, b, and c are parallel?CorrectUse the given information to relate the values on the left with the values on the right. Remember that if two lines are divided proportionally, the corresponding segments are in proportion and the two lines and either pair of corresponding segments are in proportion. Set up the proportion and solve for the unknown:
IncorrectUse the given information to relate the values on the left with the values on the right. Remember that if two lines are divided proportionally, the corresponding segments are in proportion and the two lines and either pair of corresponding segments are in proportion. Set up the proportion and solve for the unknown:

Question 14 of 38
14. Question
If AB is parallel to DC, and AD is parallel to BC, what is the value of b − a?CorrectIf AB is parallel to DC, then BD is a transversal. Alternate interior angles between two parallel lines cut by a transversal are equal. Angle BDC = Angle ABD, so a = 30.
The sum of the interior angles of a triangle is 180, so an equation can be written:a + b + 70 = 180
a + b = 110
(30) + b = 110; b = 80
Substitute the determined values and evaluate b − a: 80 − 30 = 50
IncorrectIf AB is parallel to DC, then BD is a transversal. Alternate interior angles between two parallel lines cut by a transversal are equal. Angle BDC = Angle ABD, so a = 30.
The sum of the interior angles of a triangle is 180, so an equation can be written:a + b + 70 = 180
a + b = 110
(30) + b = 110; b = 80
Substitute the determined values and evaluate b − a: 80 − 30 = 50

Question 15 of 38
15. Question
The circumference of a right circular cylinder is half its height. The radius of the cylinder is x. What is the volume of the cylinder in terms of x?
CorrectBegin by noting the relevant, necessary formulas:
IncorrectBegin by noting the relevant, necessary formulas:

Question 16 of 38
16. Question
The speed of a subway train is represented by the equation z = s^{2} + 2s for all situations where 0 ≤ s ≤ 7, where z is the rate of speed in kilometers per hour and s is the time in seconds from the moment the train starts moving. In kilometers per hour, how much faster is the subway train moving after 7 seconds than it was moving after 3 seconds?
CorrectFor this word problem, we’re asked for the difference between the train’s speed after 7 seconds and the train’s speed after 3 seconds. First evaluate the function at s = 7 and from this value, then evaluate the function at s = 3 and find the difference of the two:
z(7) = (7)^{2} + 2(7) = 63
z(3) = (3)^{2} + 2(3) = 15
z(7) − z(3) = 63 − 15 = 48IncorrectFor this word problem, we’re asked for the difference between the train’s speed after 7 seconds and the train’s speed after 3 seconds. First evaluate the function at s = 7 and from this value, then evaluate the function at s = 3 and find the difference of the two:
z(7) = (7)^{2} + 2(7) = 63
z(3) = (3)^{2} + 2(3) = 15
z(7) − z(3) = 63 − 15 = 48 
Question 17 of 38
17. Question
At a certain lab, the ratio of scientists to engineers is 5:1. If 75 new team members are hired in the ratio of two engineers per scientist, the new ratio of scientists to engineers would be approximately 2:3. Approximately how many scientists currently work at the lab?
CorrectCurrently our ratio is S/E = 5/1, or 5E = S. Out of the 75 hires, for every 3 hires: two will be engineers and one will be a scientist, so that will add 50 engineers and 25 scientists. We’re told that adding these people to the mix creates a new ratio:
Since there are 5 scientists for every engineer currently, then there are approximately 10 scientists.IncorrectCurrently our ratio is S/E = 5/1, or 5E = S. Out of the 75 hires, for every 3 hires: two will be engineers and one will be a scientist, so that will add 50 engineers and 25 scientists. We’re told that adding these people to the mix creates a new ratio:
Since there are 5 scientists for every engineer currently, then there are approximately 10 scientists. 
Question 18 of 38
18. Question
In the xycoordinate plane, a circle with center (−4, 0) is tangent to the line y = −x. What is the circumference of the circle?
CorrectStart by drawing a diagram to better visualize the problem.
The line y = −x makes a 45 degree angle with each axis in the second quadrant. Connect the center of the circle to the point of tangency on y = −x. The radius of a circle is perpendicular to its point of tangency.We can draw a 454590 triangle using the xaxis and y = −x, and use our knowledge of right triangle ratios to find the radius (or hypotenuse of the triangle) is 2√2. Recall that 45°, 45°, 90° right triangles share a side: side: hypotenuse ratio of x: x: x√2.
IncorrectStart by drawing a diagram to better visualize the problem.
The line y = −x makes a 45 degree angle with each axis in the second quadrant. Connect the center of the circle to the point of tangency on y = −x. The radius of a circle is perpendicular to its point of tangency.We can draw a 454590 triangle using the xaxis and y = −x, and use our knowledge of right triangle ratios to find the radius (or hypotenuse of the triangle) is 2√2. Recall that 45°, 45°, 90° right triangles share a side: side: hypotenuse ratio of x: x: x√2.

Question 19 of 38
19. Question
CorrectIncorrect 
Question 20 of 38
20. Question
Line x can be described by the function ƒ(x) = 5x. Line y is parallel to Line x such that the shortest distance between Line y and Line x is 5, and the yintercept of Line y is negative. What is a possible equation for line y?
CorrectStart by drawing the lines:
The slopeintercept form of a line is y = mx + b, where m is the slope and b is the yintercept. Parallel lines have the same slope, so Line y must also have a slope of 5; therefore, we can eliminate choices (A) and (C).The distance between Line x and Line y is 5. If we drew a perpendicular line from the origin to Line y, we can form a right triangle with the yaxis as the hypotenuse and the distance between the lines as one of the legs. Since the hypotenuse is longer than either of the sides in a triangle, the yintercept of Line y must be greater than 5. This eliminates choice (D).
IncorrectStart by drawing the lines:
The slopeintercept form of a line is y = mx + b, where m is the slope and b is the yintercept. Parallel lines have the same slope, so Line y must also have a slope of 5; therefore, we can eliminate choices (A) and (C).The distance between Line x and Line y is 5. If we drew a perpendicular line from the origin to Line y, we can form a right triangle with the yaxis as the hypotenuse and the distance between the lines as one of the legs. Since the hypotenuse is longer than either of the sides in a triangle, the yintercept of Line y must be greater than 5. This eliminates choice (D).

Question 21 of 38
21. Question
In a recent survey of two popular bestselling books, twofifths of the 2,200 polled said they did not enjoy the second book, but did enjoy the first book. Of those, 40% were adults over 18. If threeeighths of those surveyed were adults over 18, how many adults over 18 did not report that they enjoyed the first book but not the second book?
CorrectRemember to find each category separately to keep the different numbers clear. The total number of people was 2,200.
^{2}/_{5} of 2,200 = 880, so 880 of those surveyed said they did not enjoy the second book, but enjoyed the first book. Of these 880 people, 40% were adults over 18, so in this group there were 880 * 0.4 = 352 people.
It is also stated that ^{3}/_{8} of the 2,200 surveyed, or 825 people who are adults over 18. To find the number of adults who did not report that they enjoyed the first book but not the second, subtract the portion who did report they enjoyed the first book but not the second from the total number of adults:
825 − 352 = 473 adults.
IncorrectRemember to find each category separately to keep the different numbers clear. The total number of people was 2,200.
^{2}/_{5} of 2,200 = 880, so 880 of those surveyed said they did not enjoy the second book, but enjoyed the first book. Of these 880 people, 40% were adults over 18, so in this group there were 880 * 0.4 = 352 people.
It is also stated that ^{3}/_{8} of the 2,200 surveyed, or 825 people who are adults over 18. To find the number of adults who did not report that they enjoyed the first book but not the second, subtract the portion who did report they enjoyed the first book but not the second from the total number of adults:
825 − 352 = 473 adults.

Question 22 of 38
22. Question
At the Monterey Bay Aquarium, the ratio of the number of stingrays to starfish is 7 to 56. If 6 more starfish were added to the exhibits, the new ratio would be 7 to 58. How many total stingrays and starfish will be in the Monterey Bay Aquarium’s exhibits after the starfish are added?
CorrectBegin by representing the ratio of stingrays to starfish as an equation. Let R equal the number of stingrays, and F equal the number of starfish:
Or:
56R = 7FAfter the 6 starfish are added, the new proportion is:
Or:
58R = 7(F + 6)
58R = 7F + 42We now have a system of equations that can be solved through the method of combination:
58R = 7F + 42
−56R = 7F
2R = 42; R = 21Because the question asks for the total number of stingrays and starfish after the addition of the starfish, plug the found number of stingrays into the second equation to solve for the number of starfish:
58(21) = 7F + 42
1218 − 42 = 7F; F = 168Combine the number of stingrays with the number of starfish to find the total number:
21 + 168 = 189IncorrectBegin by representing the ratio of stingrays to starfish as an equation. Let R equal the number of stingrays, and F equal the number of starfish:
Or:
56R = 7FAfter the 6 starfish are added, the new proportion is:
Or:
58R = 7(F + 6)
58R = 7F + 42We now have a system of equations that can be solved through the method of combination:
58R = 7F + 42
−56R = 7F
2R = 42; R = 21Because the question asks for the total number of stingrays and starfish after the addition of the starfish, plug the found number of stingrays into the second equation to solve for the number of starfish:
58(21) = 7F + 42
1218 − 42 = 7F; F = 168Combine the number of stingrays with the number of starfish to find the total number:
21 + 168 = 189 
Question 23 of 38
23. Question
If (m,ƒ(m)) represents a point on the graph ƒ(m) = 2m + 1, which of the following could be a portion of the graph of the set of points (m,(ƒ(m))^{2})?
CorrectLet’s rewrite the “m” as an “x” to better understand how this function would look on an xycoordinate plane.
Begin with the equation provided in the question, ƒ(m), and square it, ƒ(m)^{2}:
ƒ(m) = 2x + 1
ƒ(m)^{2} = (2x + 1)^{2} = 4x^{2} + 4x + 1By factoring out a 4 from the expression, the vertex form of the quadratic can be found:
This function translates graphically into a parabola with a vertex at (−½, 0) that is vertically stretched and opens upwards. Only answer choice (C) shows an appropriate possibility.IncorrectLet’s rewrite the “m” as an “x” to better understand how this function would look on an xycoordinate plane.
Begin with the equation provided in the question, ƒ(m), and square it, ƒ(m)^{2}:
ƒ(m) = 2x + 1
ƒ(m)^{2} = (2x + 1)^{2} = 4x^{2} + 4x + 1By factoring out a 4 from the expression, the vertex form of the quadratic can be found:
This function translates graphically into a parabola with a vertex at (−½, 0) that is vertically stretched and opens upwards. Only answer choice (C) shows an appropriate possibility. 
Question 24 of 38
24. Question
Parallelogram QRST has an area of 120 and its longest side (QT) is 24. The angle opposite the vertical is 30°, and the vertical is from R to point U, which lies along QT. What is the length of the hypotenuse of the triangle formed from segments RU, QU, and QR, rounded to the nearest whole number?
CorrectThis is a tough word problem to visualize, so start by drawing the figure:
Since the area is 120 and the base is 24, we know from the question stem that the height (RU) is 5.Given that the angle opposite the vertical is 30°, a 30°,60°,90° triangle should be seen. Recall that the ratio of the side lengths of a 30, 60, 90 triangle is:
In this case, the smallest side length x is 5, so:
The hypotenuse is 2 * 5 = 10.IncorrectThis is a tough word problem to visualize, so start by drawing the figure:
Since the area is 120 and the base is 24, we know from the question stem that the height (RU) is 5.Given that the angle opposite the vertical is 30°, a 30°,60°,90° triangle should be seen. Recall that the ratio of the side lengths of a 30, 60, 90 triangle is:
In this case, the smallest side length x is 5, so:
The hypotenuse is 2 * 5 = 10. 
Question 25 of 38
25. Question
The initial number of elements in Set A is x, where x > 0. If the number of elements in Set A doubles every hour, which of the following represents the increase in the number of elements in Set A after exactly one day?
CorrectTo find the INCREASE in the elements, we need to subtract the original number from the final total number. Since the original amount is going to be multiplied by 2 (or doubled) 24 times, we can express this as the exponent: 2^{24}. The final total increase will be 2^{24}x (the final total) − x, (the original number).
IncorrectTo find the INCREASE in the elements, we need to subtract the original number from the final total number. Since the original amount is going to be multiplied by 2 (or doubled) 24 times, we can express this as the exponent: 2^{24}. The final total increase will be 2^{24}x (the final total) − x, (the original number).

Question 26 of 38
26. Question
A birthday cake with a height of 4 inches is cut into two pieces such that each piece is of a different size. If the ratio of the volume of the larger slice to the volume of the smaller slice is 5 to 3, what is the degree measure of the cut made into the cake?
CorrectRemember that the ratio between the volumes of the two pieces will be the same as the ratio of the areas of their bases, and also that the ratio between the interior angle of a sector of a circle and 360 degrees is the same as the ratio between the area of a sector and the area of an entire circle.
Since the ratio of the larger slice to the smaller slice is 5 to 3, the ratio of the area of the smaller slice to the area of the entire cake must be 3 to 8. This ratio is the same as the ratio of the interior angle of the sector representing the smaller slice to 360 degrees. We can therefore set up a proportion:
IncorrectRemember that the ratio between the volumes of the two pieces will be the same as the ratio of the areas of their bases, and also that the ratio between the interior angle of a sector of a circle and 360 degrees is the same as the ratio between the area of a sector and the area of an entire circle.
Since the ratio of the larger slice to the smaller slice is 5 to 3, the ratio of the area of the smaller slice to the area of the entire cake must be 3 to 8. This ratio is the same as the ratio of the interior angle of the sector representing the smaller slice to 360 degrees. We can therefore set up a proportion:

Question 27 of 38
27. Question
An individual invested $25,000 for 1 year at the rate of 15% annually, compounded quarterly. What percent of the individual’s return after nine months was profit?
CorrectLet’s calculate this in stages; first calculate the interest after the first quarter, then calculate the interest after the second quarter, then the third quarter, and finally subtract the investment from the return to find the total profit then divide this by the total return to find what percentage the profit represented.
The interest on the first quarter was 25,000 * 0.15 = $3,750. The new principal would be 25,000 + 3,750 = $28,750.
The interest on the second quarter was 28,750 * 0.15 = $4,312.50. The new principal would be 28,750 + 4,312.50 = $33,062.50.
The interest on the third quarter was 33,062.50 * 0.15 = $4,959.38 (since the math starts to get complicated here, we could also round to $5,000). The final principal after 9 months would be 33,062.50 + 4,959.38 = $38,021.88. The total profit made after 9 months was $38,021.88 − $25,000 = $13,021.88.
IncorrectLet’s calculate this in stages; first calculate the interest after the first quarter, then calculate the interest after the second quarter, then the third quarter, and finally subtract the investment from the return to find the total profit then divide this by the total return to find what percentage the profit represented.
The interest on the first quarter was 25,000 * 0.15 = $3,750. The new principal would be 25,000 + 3,750 = $28,750.
The interest on the second quarter was 28,750 * 0.15 = $4,312.50. The new principal would be 28,750 + 4,312.50 = $33,062.50.
The interest on the third quarter was 33,062.50 * 0.15 = $4,959.38 (since the math starts to get complicated here, we could also round to $5,000). The final principal after 9 months would be 33,062.50 + 4,959.38 = $38,021.88. The total profit made after 9 months was $38,021.88 − $25,000 = $13,021.88.

Question 28 of 38
28. Question
VitaDrink contains 30 percent concentrated nutrients by volume. EnergyPlus contains 40 percent concentrated nutrients by volume. Which of the following represents the percent of concentrated nutrients by volume in a mixture of v gallons of VitaDrink, e gallons of EnergyPlus, and w gallons of water?
CorrectThe total number of gallons in the final mixture will be the sum of all the components: v + e + w. There are 0.3v gallons of nutrients from VitaDrink in the mixture, 0.4e gallons of nutrients from EnergyPlus in the mixture, and no nutrients from the water. The total number of gallons of nutrients in the new mixture will be 0.3v + 0.4e. To convert from a fraction to a percent, we simply multiply our value by 100:
If you chose (B), remember that the question was asking for the percent, not the actual number in the mixture!IncorrectThe total number of gallons in the final mixture will be the sum of all the components: v + e + w. There are 0.3v gallons of nutrients from VitaDrink in the mixture, 0.4e gallons of nutrients from EnergyPlus in the mixture, and no nutrients from the water. The total number of gallons of nutrients in the new mixture will be 0.3v + 0.4e. To convert from a fraction to a percent, we simply multiply our value by 100:
If you chose (B), remember that the question was asking for the percent, not the actual number in the mixture! 
Question 29 of 38
29. Question
On a coordinate plane, (a, b) and (a + 5, b + c), and (13, 10) are three points on line l. If the xintercept of line l is −7, what is the value of c?
CorrectRecall that all lines can be written in the form y = mx + b, where m is the slope of the line (rise/run), and b is the yintercept of the line. Given that the coordinate plane uses the variables a and b for x and y, we can rewrite the line equation as: b = ma + k, where the variable k is used to replace the original variable b for the yintercept to avoid confusion.
Using this revised equation in conjunction with the given points, we can first solve for the slope of the line in terms of c. Given that the slope of a line is the change in the y variable divided by the change in the x variable, calculate the line’s slope:
Notice that the question only asks for the value of c, which is a part of the slope of the line. If we use the given information to determine the actual value of the slope, we can equate the expression containing c with the actual value and solve for c. Given that the line has an xintercept of −7, we can deduce that (−7, 0) is a point on the line. Calculate the slope of the line using this point and the point (13,10):
IncorrectRecall that all lines can be written in the form y = mx + b, where m is the slope of the line (rise/run), and b is the yintercept of the line. Given that the coordinate plane uses the variables a and b for x and y, we can rewrite the line equation as: b = ma + k, where the variable k is used to replace the original variable b for the yintercept to avoid confusion.
Using this revised equation in conjunction with the given points, we can first solve for the slope of the line in terms of c. Given that the slope of a line is the change in the y variable divided by the change in the x variable, calculate the line’s slope:
Notice that the question only asks for the value of c, which is a part of the slope of the line. If we use the given information to determine the actual value of the slope, we can equate the expression containing c with the actual value and solve for c. Given that the line has an xintercept of −7, we can deduce that (−7, 0) is a point on the line. Calculate the slope of the line using this point and the point (13,10):

Question 30 of 38
30. Question
A bag contains 80% yellow marbles and 20% turquoise marbles. What is the probability, approximately, of obtaining exactly two turquoise marbles out of three randomly selected marbles?
CorrectIncorrect 
Question 31 of 38
31. Question
The sum of the two digits for positive twodigit integer x is 5. If x is prime and x < 25, what is the value of x?
CorrectThe correct answer is 23.
The possible values for the digits are 0 and 5, 1 and 4, or 2 and 3, since those are the only pairs of digits that sum to 5. Therefore, the possible numbers are: 50, 14, 41, 23, or 32. We cannot use 05, since we don’t place a 0 first in the tens digit. Of these, only 14 and 23 are less than 25. The only prime number option is 23.
IncorrectThe correct answer is 23.
The possible values for the digits are 0 and 5, 1 and 4, or 2 and 3, since those are the only pairs of digits that sum to 5. Therefore, the possible numbers are: 50, 14, 41, 23, or 32. We cannot use 05, since we don’t place a 0 first in the tens digit. Of these, only 14 and 23 are less than 25. The only prime number option is 23.

Question 32 of 38
32. Question
If the remainder is 3 when a positive twodigit integer, z, is divided into 15, what is the remainder when z is divided by 11?
CorrectThe correct answer is 1.
Start by considering what a “remainder” is. We know that the remainder, 3, is leftover after the twodigit number, z, is divided into 15. It is given that z is a twodigit integer, from which we can deduce that z is the number 12, because 12 is the only twodigit number that has a remainder of 3 when divided into 15:
15 ÷ 12 = 1 remainder 3
Dividing z by 11 yields a remainder of 1:
12 ÷ 11 = 1 remainder 1
IncorrectThe correct answer is 1.
Start by considering what a “remainder” is. We know that the remainder, 3, is leftover after the twodigit number, z, is divided into 15. It is given that z is a twodigit integer, from which we can deduce that z is the number 12, because 12 is the only twodigit number that has a remainder of 3 when divided into 15:
15 ÷ 12 = 1 remainder 3
Dividing z by 11 yields a remainder of 1:
12 ÷ 11 = 1 remainder 1

Question 33 of 38
33. Question
A circle is drawn on the coordinate plane such that two points lie on the circumference of the circle. What is the maximum possible distance between these two points if the area of the circle is 64π?
CorrectThe correct answer is 16.
The definition of the diameter is the maximum straight line distance passing from side to side through the center of a circle. The maximum “possible” distance between two points on a circumference will be equivalent to the diameter. If the area of a circle = πr^{2} = 64π then the radius, r = 8, and the diameter, which is twice the radius, is 16.
IncorrectThe correct answer is 16.
The definition of the diameter is the maximum straight line distance passing from side to side through the center of a circle. The maximum “possible” distance between two points on a circumference will be equivalent to the diameter. If the area of a circle = πr^{2} = 64π then the radius, r = 8, and the diameter, which is twice the radius, is 16.

Question 34 of 38
34. Question
A certain circle has a circumference of 12π. If line segment AC is tangent to the circle at point B, AB = BC, and AC = 16, what is the distance between the center of Circle X and point C?
CorrectThe correct answer is 10.
For this question, and any questions regarding geometric shapes that do not include a picture, start by drawing the figure. If the circumference of the circle is 2πr = 12π, then the radius of the circle is 6. We also know that AC = 16, AB + BC = AC and AB = BC, so BC must equal 8.
Labeling the figure with this information, and knowing that a line tangent to a circle forms a right angle with the radius, it can be seen that we have a right triangle with side lengths of 6 and 8, and an unknown hypotenuse for which we are solving.
It should be known that this right triangle is a multiple of a Pythagorean triple of 3,4,5; so the hypotenuse is a length of 10. However, if this is not recognized, the Pythagorean Theorem can be used to solve for the hypotenuse:a^{2} + b^{2} = c^{2}
6^{2} +8^{2} = c^{2}
36 + 64 = c^{2}
100 = c^{2}
c=10IncorrectThe correct answer is 10.
For this question, and any questions regarding geometric shapes that do not include a picture, start by drawing the figure. If the circumference of the circle is 2πr = 12π, then the radius of the circle is 6. We also know that AC = 16, AB + BC = AC and AB = BC, so BC must equal 8.
Labeling the figure with this information, and knowing that a line tangent to a circle forms a right angle with the radius, it can be seen that we have a right triangle with side lengths of 6 and 8, and an unknown hypotenuse for which we are solving.
It should be known that this right triangle is a multiple of a Pythagorean triple of 3,4,5; so the hypotenuse is a length of 10. However, if this is not recognized, the Pythagorean Theorem can be used to solve for the hypotenuse:a^{2} + b^{2} = c^{2}
6^{2} +8^{2} = c^{2}
36 + 64 = c^{2}
100 = c^{2}
c=10 
Question 35 of 38
35. Question
Among a group of pet owners, 80 people own dogs, 41 people own cats, and 54 people own fish. Ten people in the group own dogs and cats. Nineteen own dogs and fish, and a dozen own cats and fish. If seven people own all three types of pets, how many of them own dogs only?
CorrectThe correct answer is 44.
For this problem, it is best to construct, label, and use a Venn diagram to determine the number of people who own only dogs:
The number of dog owners, cat owners, and fish owners are given; as are the number of dog and cat owners, dog and fish owners, and dog and cat and fish owners. The labeled Venn diagram shows the relevant information. To solve for the number of dog owners only, subtract the sum of the owners that have one of the other animals:80 − (10 + 7 + 19) = 44
IncorrectThe correct answer is 44.
For this problem, it is best to construct, label, and use a Venn diagram to determine the number of people who own only dogs:
The number of dog owners, cat owners, and fish owners are given; as are the number of dog and cat owners, dog and fish owners, and dog and cat and fish owners. The labeled Venn diagram shows the relevant information. To solve for the number of dog owners only, subtract the sum of the owners that have one of the other animals:80 − (10 + 7 + 19) = 44

Question 36 of 38
36. Question
What is the area of triangle ABC that lies on the coordinate plane with vertices at (2,4), (5,8), and (8,2)?
CorrectThe correct answer is 15.
Start by drawing the figure on the coordinate plane. You’ll notice that none of the sides of the triangle are parallel with either the x or yaxis, making it slightly more challenging to find the base and the height. Find the area of the rectangle which encloses the triangle, then subtract out the three triangles outside our triangle’s area:
Area of triangle ABC = Area of square DEFC − Area of triangle ACD − Area of triangle AEB − Area of triangle BFC
IncorrectThe correct answer is 15.
Start by drawing the figure on the coordinate plane. You’ll notice that none of the sides of the triangle are parallel with either the x or yaxis, making it slightly more challenging to find the base and the height. Find the area of the rectangle which encloses the triangle, then subtract out the three triangles outside our triangle’s area:
Area of triangle ABC = Area of square DEFC − Area of triangle ACD − Area of triangle AEB − Area of triangle BFC

Question 37 of 38
37. Question
If Angle BAC = 70 and Angle ABC = 50, how much greater than a is the value of b?CorrectThe correct answer is 70.
Begin by labeling the figure so that the known and unknown values are clearly visible. From the labeled figure, generate equations including a common unknown, the variables a and b, and the fact that the interior angles of a triangle sum to 180°:
Using the larger triangle first, angle A + angle B + angle C = 180°; substituting known values:
70 + 50 + ∠BCD + a = 180
And from the smaller triangle on the left:
∠BCD + b + 50 = 180
∠BCD + b = 130
∠BCD = 130 − bNow that we have solved for ∠BCD, we can substitute the expression back into the first equation and relate angle b and angle a:
70 + 50 + 130 − b + a = 180
250 − b + a = 180
−b + a = −70
70 = b − aAngle a is 70 less than angle b. Notice that it was not necessary to individually solve for either of the angles, and instead we were able to use a third unknown value to relate the two.
IncorrectThe correct answer is 70.
Begin by labeling the figure so that the known and unknown values are clearly visible. From the labeled figure, generate equations including a common unknown, the variables a and b, and the fact that the interior angles of a triangle sum to 180°:
Using the larger triangle first, angle A + angle B + angle C = 180°; substituting known values:
70 + 50 + ∠BCD + a = 180
And from the smaller triangle on the left:
∠BCD + b + 50 = 180
∠BCD + b = 130
∠BCD = 130 − bNow that we have solved for ∠BCD, we can substitute the expression back into the first equation and relate angle b and angle a:
70 + 50 + 130 − b + a = 180
250 − b + a = 180
−b + a = −70
70 = b − aAngle a is 70 less than angle b. Notice that it was not necessary to individually solve for either of the angles, and instead we were able to use a third unknown value to relate the two.

Question 38 of 38
38. Question
A gym teacher is dividing a group of students into different teams of at least four or more students for various sports events at a track meet. There is no limit to the number of teams that can enter the competition, and students can be members of multiple teams. If the number of possible teams is less than 180, what is the greatest number of students that can make up a team?
CorrectThe correct answer is 8.
This is a great question to use the technique of picking potential answer choices and testing their validity. The question is looking for the greatest number of students that could make up a team if there are less than 180 possible teams and students can be on multiple teams. Let’s pick a value and try it out!
For example, if the greatest number was “5,” our team options would be teams of 4 and 5.
To find the total number of ways to choose 4 from 5, we can use the Combination formula:
The total number of possible teams would be 5 + 1 = 6. Since this is much less than 180, there’s probably a number greater than 5 that could be the “greatest possible” number of teams.Let’s try, 8. Our team options would be 4, 5, 6, 7, and 8:
8C4 = 70
8C5 = 56
8C6 = 28
8C7 = 8
8C8 = 1The total number of possible teams would be 56 + 70 + 28 + 8 + 1 = 163.
163 is still less than 180, but close enough to 180 to indicate that the nextgreatest integer, 9, would probably give us a total number of possible teams greater than 180. We can conclude that the answer is 8.
IncorrectThe correct answer is 8.
This is a great question to use the technique of picking potential answer choices and testing their validity. The question is looking for the greatest number of students that could make up a team if there are less than 180 possible teams and students can be on multiple teams. Let’s pick a value and try it out!
For example, if the greatest number was “5,” our team options would be teams of 4 and 5.
To find the total number of ways to choose 4 from 5, we can use the Combination formula:
The total number of possible teams would be 5 + 1 = 6. Since this is much less than 180, there’s probably a number greater than 5 that could be the “greatest possible” number of teams.Let’s try, 8. Our team options would be 4, 5, 6, 7, and 8:
8C4 = 70
8C5 = 56
8C6 = 28
8C7 = 8
8C8 = 1The total number of possible teams would be 56 + 70 + 28 + 8 + 1 = 163.
163 is still less than 180, but close enough to 180 to indicate that the nextgreatest integer, 9, would probably give us a total number of possible teams greater than 180. We can conclude that the answer is 8.