SAT Math Calculator-Allowed practice tests.

## Quiz #111

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Question 1 |

**There are 5 pencil-cases on the desk. Each pencil-case contains at least 10 pencils, but not more than 14 pencils. Which of the following could be the total number of pencils in all five pencil-cases?**

35 | |

45 | |

65 | |

75 |

Question 1 Explanation:

The correct answer is (C). If all pencil-cases have 10 pencils (the minimum number of pencils every pencil-case can hold), then the total minimum number of pencils is 10 × 5 = 50 pencils.

Likewise, if all pencil-cases have 14 pencils (the maximum number of pencils every pencil-case can hold), then the total maximum number of pencils is 14 × 5 = 70 pencils.

Thus, the answer must be between 50 and 70. Out of all the possible answer choices, only (C), 65 pencils, matches these criteria.

Likewise, if all pencil-cases have 14 pencils (the maximum number of pencils every pencil-case can hold), then the total maximum number of pencils is 14 × 5 = 70 pencils.

Thus, the answer must be between 50 and 70. Out of all the possible answer choices, only (C), 65 pencils, matches these criteria.

Question 2 |

**A bag contains 3 black marbles, 6 red marbles and 9 white marbles. What is the probability of getting a red marble?**

$\dfrac{1}{7}$ | |

$\dfrac{6}{12}$ | |

$\dfrac{1}{2}$ | |

$\dfrac{1}{3}$ |

Question 2 Explanation:

The correct answer is (D). The probability of getting a red marble is:

$\frac{Number~of~red~marbles}{Total~number~of~marbles}$

The total number of red marbles: 6

The total number of marbles: 3 + 6 + 9 =18.

Simplify:

$\dfrac{6}{18} = \dfrac{1}{3}$

$\frac{Number~of~red~marbles}{Total~number~of~marbles}$

The total number of red marbles: 6

The total number of marbles: 3 + 6 + 9 =18.

Simplify:

$\dfrac{6}{18} = \dfrac{1}{3}$

Question 3 |

**The diameter of a circle is increased by 80%. By what percent is the area increased?**

80% | |

224% | |

180% | |

325% |

Question 3 Explanation:

The correct response is (B). This is a great question for plugging in your own values. Let’s say the value of the original diameter was 10. The original area would be:

area =

area = (5)

area= 25

If the diameter increased by 80%, the new diameter would be 18, and the new area would be:

area =

area= (9)

area = 81

The formula for percentage increase is: $\dfrac{New Number - Original Number}{Original Number} * 100$

The numerator is 81

$\dfrac{25π}{56π} * 100 = 224%$

area =

*πr*^{2}area = (5)

^{2}*π*area= 25

*π*.If the diameter increased by 80%, the new diameter would be 18, and the new area would be:

area =

*πr*^{2}area= (9)

^{2}*π*area = 81

*π*.The formula for percentage increase is: $\dfrac{New Number - Original Number}{Original Number} * 100$

The numerator is 81

*π*− 25*π*= 56*π*.$\dfrac{25π}{56π} * 100 = 224%$

Question 4 |

**A bird traveled 72 miles in 6 hours flying at constant speed. At this rate, how many miles did the bird travel in 5 hours?**

12 | |

30 | |

60 | |

14.4 |

Question 4 Explanation:

The correct answer is (C). First calculate the rate that the bird is traveling at:

$\frac{72~miles}{6~hour} = 12~miles~per~hour$

The distance traveled in 5 hours will be:

$12~miles~per~hour * 5~hours = 60~miles$

$\frac{72~miles}{6~hour} = 12~miles~per~hour$

The distance traveled in 5 hours will be:

$12~miles~per~hour * 5~hours = 60~miles$

Question 5 |

**Three kids own a total of 96 comic books. If one of the kids owns 16 of the comic books, what is the average (arithmetic mean) number of comic books owned by the other two kids?**

40 | |

42 | |

44 | |

46 |

Question 5 Explanation:

The correct answer is (A).

If one kid owns 16 of the comic books, then the number of comic books owned by the other two kids is: 96 − 16 = 80.

Therefore, the average is: 80 ÷ 2 = 40.

If one kid owns 16 of the comic books, then the number of comic books owned by the other two kids is: 96 − 16 = 80.

Therefore, the average is: 80 ÷ 2 = 40.

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## Quiz #112

Congratulations - you have completed *Quiz #112*.

You scored %%SCORE%% out of %%TOTAL%%.

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Question 1 |

**If David has twice as many nickels as Tom, and Tom has 15 more nickels than John, how many dollars does David have if John has 6 nickels?**

2.1 | |

21 | |

42 | |

14 |

Question 1 Explanation:

The correct answer is (A). If John has 6 nickels, then Tom has 15 + 6 = 21 nickels. Since the number of nickels David has is twice the nickels Tom has, then 21 × 2 = 42. A nickel is worth 5 cents, so 42 nickels will be worth 42 × 5 = 210 cents which is equivalent to 210 ÷ 100 (Every dollar worth 100 cents) = 2.1 dollars.

Question 2 |

**Franklin bought several kites at the store, and each kite cost 16 dollars. Richard purchased several different kites, and spent 20 dollars each. If the ratio of the number of kites Franklin purchased to the number of kites Richard purchased is 3 to 2, what is the average cost of a kite purchased by Richard and Franklin?**

16.8 | |

17.2 | |

17.6 | |

18.0 |

Question 2 Explanation:

The correct answer is (C). For every 5 kites purchased by these men, 3 of them are Franklin’s and 2 of them are Richard’s. We can set up an equation to find the total money spent:

3(16) + 2(20) = Total Money Spent for every 5 kites

88 = Total Money Spent for every 5 kites

To find the average per kite cost, we can simply divide 88 by 5:

88 ÷ 5 = 17.6

3(16) + 2(20) = Total Money Spent for every 5 kites

88 = Total Money Spent for every 5 kites

To find the average per kite cost, we can simply divide 88 by 5:

88 ÷ 5 = 17.6

Question 3 |

**Circle A is inside Circle B, and the two circles share the same center O. If the circumference of B is four times the circumference of A, and the radius of Circle A is three, what is the difference between Circle B’s diameter and Circle A’s diameter?**

6 | |

9 | |

12 | |

18 |

Question 3 Explanation:

The correct response is (D). Start by drawing the figure. If the radius of A is 3, then its diameter is 6. Its circumference is 2

The difference between the diameters is: 24 – 6 = 18

*πr*= 6*π*. B’s circumference is four times A’s circumference. B’s circumference = 24*π*= 2*πr*. B’s radius must be 12, and its diameter is 24.The difference between the diameters is: 24 – 6 = 18

Question 4 |

**On a number line with points**

*LMNOP*, the ratio of*LM*to*MN*is 1, and the ratio of*NO*to*OP*is 3/4. If the length of LP is 28 and the length of*MO*is 13, what is the ratio of*LO*to*MP*?7/18 | |

13/28 | |

15/28 | |

20/21 |

Question 4 Explanation:

The correct response is (D). Let’s draw the number line:

I chose variables

Let’s set up a few equations based on the given information:

2

Using substitution:

2(13 – 3

26 – 6

26 +

Now we can find

Now plug in these values to find the ratio:

= (2

= (14 + 6)/(7 + 14)

= 20/21

I chose variables

*x*and*y*to help express the ratios. Right now the ratio of*LO*to*MP*is (2*x*+ 3*y*)/(*x*+ 7*y*). If we could find the values of*x*and*y*we can determine the ratio.Let’s set up a few equations based on the given information:

*x*+ 3*y*= 13*x*= 13 – 3*y*2

*x*+ 7*y*= 28Using substitution:

2(13 – 3

*y*) + 7*y*= 2826 – 6

*y*+ 7*y*= 2826 +

*y*= 28*y*= 2Now we can find

*x*:*x*+ 3(2) = 13*x*+ 6 = 13*x*= 7Now plug in these values to find the ratio:

= (2

*x*+ 3*y*)/(*x*+ 7*y*)= (14 + 6)/(7 + 14)

= 20/21

Question 5 |

**The total cost of 4 equally priced pens is $6. If the cost per pen is increased by 25 cents, how much will 3 of these pens cost at the new rate?**

$5.25 | |

$5.00 | |

$5.50 | |

$8.25 |

Question 5 Explanation:

The correct answer is (A). First figure out the cost per pen:

Cost = 6 ÷ 4 = 1.5

After the 0.25 price increase:

Cost = 1.5 + 0.25 = 1.75

The cost of 3 pens will be 3 × 1.75 = 5.25

Cost = 6 ÷ 4 = 1.5

After the 0.25 price increase:

Cost = 1.5 + 0.25 = 1.75

The cost of 3 pens will be 3 × 1.75 = 5.25

Question 6 |

**A “Triangle” is any positive integer greater than 1 that has only three positive integer factors: itself, its square root, and 1. Which of the following is a Triangle?**

169 | |

100 | |

36 | |

81 |

Question 6 Explanation:

The correct answer is (A). The only factors of 169 are: 169, 13, and 1. The square root of 169 is 13. All the other options have more than 3 factors.

Question 7 |

**The table above, which describes the New York City workforce, is partially filled in. Based on this information, how many women in the New York City workforce are unemployed?**

500 | |

1,500 | |

2,500 | |

3,500 |

Question 7 Explanation:

The correct answer is (D). First we can compute the number of employed women by using the numbers in the first column:

40,000 − 22,000 = 18,000

Then we can compute the number of unemployed women by using the data in the second row:

21,500 − 18,000 = 3,500

40,000 − 22,000 = 18,000

Then we can compute the number of unemployed women by using the data in the second row:

21,500 − 18,000 = 3,500

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## Quiz #104

Congratulations - you have completed *Quiz #104*.

You scored %%SCORE%% out of %%TOTAL%%.

Your performance has been rated as %%RATING%%

Your answers are highlighted below.

Question 1 |

**The sum of two positive integers is 13. The difference between these numbers is 7. What is their product?**

12 | |

22 | |

30 | |

40 |

Question 1 Explanation:

The correct answer is (C). Use 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 = 7

The method of substitution can be used here. First solve for x in terms of y:

x = y + 7

Then substitute the right hand side for x in the other equation:

(y + 7) + y = 13

Now combine like terms and solve for y:

2y = 6, y = 3

Then substitute 3 for y in either equation and solve 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

x + y = 13

x − y = 7

The method of substitution can be used here. First solve for x in terms of y:

x = y + 7

Then substitute the right hand side for x in the other equation:

(y + 7) + y = 13

Now combine like terms and solve for y:

2y = 6, y = 3

Then substitute 3 for y in either equation and solve 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 2 |

**The table below 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 × 10**

^{24}?13% | |

33% | |

66% | |

100% |

Question 2 Explanation:

The correct answer is (B). First, analyze the table and note each planet’s mass:

Earth: 5.98 × 10

Mars: 6.37 × 10

Jupiter: 1.90 × 10

Only Jupiter is greater than 10 × 10

Earth: 5.98 × 10

^{24}kgMars: 6.37 × 10

^{23}kgJupiter: 1.90 × 10

^{27}kgOnly Jupiter is greater than 10 × 10

^{24}kg. One out of the three planets is 33.33%.Question 3 |

**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?**

$3$ | |

$3\sqrt{2}$ | |

$6$ | |

$6\sqrt{2}$ |

Question 3 Explanation:

The correct answer is (C). Since 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 4 |

**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

0 | |

2 | |

4 | |

6 |

Question 4 Explanation:

The correct answer is (A). Start by organizing the data numerically from least to greatest:
22, 32, 34, 44, 45, 58, 58, 67, 68, 72, 94

The 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.

The 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 5 |

**What is the total number of degrees in the interior angles of a regular hexagon?**

720 | |

810 | |

920 | |

1080 |

Question 5 Explanation:

The correct answer is (A). To 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 = (

Since we are told the figure is a hexagon, it has 6 sides, so

Sum of Interior Angles = (6 − 2) × 180 = 4 × 180 = 720

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 6 |

**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, how many cubic feet of water are draining out per minute?**

5 | |

15 | |

25 | |

35 |

Question 6 Explanation:

The correct answer is (C). Write an equation to model the situation:

$\frac{200 ft^3}{min}$

Substituting 200 for the number of minutes, the unknown rate can be found:

$\frac{200 ft^3}{minutes}$$ * $$200$ $minutes$ $-$ $\frac{x ft^3}{minutes}$ $= 35,000 ft^3$

$200(200-x) = 35,000$

$200 - x = 175$

$x = 25$ $\frac{ft^3}{min}$

$\frac{200 ft^3}{min}$

**(# of minutes)**$-$ $\frac{x ft^3}{min}$**(# of minutes)**$= 35,000 $$ft^3$Substituting 200 for the number of minutes, the unknown rate can be found:

$\frac{200 ft^3}{minutes}$$ * $$200$ $minutes$ $-$ $\frac{x ft^3}{minutes}$ $= 35,000 ft^3$

$200(200-x) = 35,000$

$200 - x = 175$

$x = 25$ $\frac{ft^3}{min}$

Question 7 |

**What is the sum of the areas of the three rectangles that are drawn below the graph of the line y = 2**

^{x}?6 | |

12 | |

14 | |

16 |

Question 7 Explanation:

The correct answer is (C). Start by plugging

The areas are:

2(1) = 2

4(1) = 4

8(1) = 8

The sum is 14.

*x*= 1,*x*= 2, and*x*= 3 into the equation*y*= 2^{x}to find the y-coordinates where the corner of each rectangle touches*y*= 2^{x}. Those y-values 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) = 8

The sum is 14.

Question 8 |

**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?**

15 in^{3} | |

25 in^{3} | |

45 in^{3} | |

60 in^{3} |

Question 8 Explanation:

The correct answer is (D). 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.

The volume of the cube is: $length * width * height = 5*5*5 = 125$

The volume of the sphere is:

$\frac{4}{3}$ $\pi$ $($$\frac{5}{2}$$)$$^3$ ≈ $65.44$

So the volume of the space between the cube is: $125 - 65.44 ≈ 59.55 ≈ 60$

The volume of the cube is: $length * width * height = 5*5*5 = 125$

The volume of the sphere is:

$\frac{4}{3}$ $\pi$ $($$\frac{5}{2}$$)$$^3$ ≈ $65.44$

So the volume of the space between the cube is: $125 - 65.44 ≈ 59.55 ≈ 60$

Question 9 |

**If AB is parallel to**

*DC*, and*AD*is parallel to*BC*, then what is the value of*b*−*a*?30 | |

50 | |

60 | |

70 |

Question 9 Explanation:

The correct answer is (B). If 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

a + b + 70 = 180

a + b = 110

(30) + b = 110; b = 80

Substitute the determined values and evaluate b − a: 80 − 30 = 50

Question 10 |

**The speed of a subway train is represented by the equation**

*z*=*s*^{2}+ 2*s*for all situations where 0 ≤*s*≤ 7, where*z*is the rate of speed in km per hour and*s*is the time in seconds from the moment the train starts moving. In km per hour, how much faster is the subway train moving after 7 seconds than it was moving after 3 seconds?4 | |

9 | |

15 | |

48 |

Question 10 Explanation:

The correct answer is (D). For 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
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## Quiz #105

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You scored %%SCORE%% out of %%TOTAL%%.

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Question 1 |

**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?**

$2 \pi x^3$ | |

$3 \pi x^3$ | |

$3$$\pi$$^2$$x^3$ | |

$4$$\pi$$^2$$x^3$ |

Question 1 Explanation:

The correct answer is (D). Begin by noting the relevant, necessary formulas:

Circumference: $C=2 \pi r$

Volume: $V = \pi r^2 h$

Note that radius is equal to

$C$ $=$ $2 \pi r$ $=$ $\frac{1}{2} h$, or $h$ $=$ $4 \pi r$

Substitute this expression for

$V$ $=$ $\pi x^2 (4 \pi x)$ $=$ $4$$\pi$$^2$$x^2$

Circumference: $C=2 \pi r$

Volume: $V = \pi r^2 h$

Note that radius is equal to

*x*and the circumference is equal to half the height, or:$C$ $=$ $2 \pi r$ $=$ $\frac{1}{2} h$, or $h$ $=$ $4 \pi r$

Substitute this expression for

*h*into the volume formula:$V$ $=$ $\pi x^2 (4 \pi x)$ $=$ $4$$\pi$$^2$$x^2$

Question 2 |

**What is the value of**

*a*+*b*?w − x − xy + z | |

360 − x + y + w + z | |

180 − y + z − w | |

w + x + y + z − 360 |

Question 2 Explanation:

The correct answer is (D). Recall 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

For the top triangle:

360 +

For the bottom triangle:

360 +

Adding the 2 equations together:

*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*+*x*For the bottom triangle:

*b*+ (180 −*y*) + (180 −*z*) = 180°360 +

*b*−*y*−*z*= 180*b*= −180 +*y*+*z*Adding the 2 equations together:

*a*+*b*= (−180 +*w*+*x*) + (−180 +*y*+*z*)*a*+*b*= −360 +*w*+*x*+*y*+*z*Question 3 |

**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)**

36 | |

38 | |

40 | |

42 |

Question 3 Explanation:

The correct answer is (A). In order to calculate the average speed of the trip, it is first necessary to solve for the total number of miles traveled and the total length of time traveled. 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:

$\frac{1 knot}{1.15 miles}$ $=$ $\frac{y knots}{x miles}$

where y = 20, 30, and 40:

$\frac{1 knot}{1.15 miles}$ $=$ $\frac{20 knots}{x miles}$

Solving for x:

$x$ $mph$ $=$ $20$

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:

$230$

$230$

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:

$=$ $\frac{1173 miles}{33 hours}$

$≈$ $35.55 \frac{miles}{hour}$

Use the provided unit conversion to set up a general expression relating knots to mph:

$\frac{1 knot}{1.15 miles}$ $=$ $\frac{y knots}{x miles}$

where y = 20, 30, and 40:

$\frac{1 knot}{1.15 miles}$ $=$ $\frac{20 knots}{x miles}$

Solving for x:

$x$ $mph$ $=$ $20$

**knots**$*$ $\frac{1.15 mph}{1 knot}$ $=$ $23$ $mph$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:

$230$

**miles**$=$ $\frac{23 miles}{hours}$ $*$**Time**$230$

**miles**$*$ $\frac{hours}{23 miles}$ $=$ $10$ $hours$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:

**Average speed**$=$ $\frac{230 + 345 + 598 miles}{10 + 10 + 13 hours}$$=$ $\frac{1173 miles}{33 hours}$

$≈$ $35.55 \frac{miles}{hour}$

**, which is approximately 36 mph**Question 4 |

**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?**

5 | |

10 | |

15 | |

20 |

Question 4 Explanation:

The correct answer is (B). Currently our ratio is

$\frac{(E + 50)}{(S + 25)}$ $=$ $\frac{3}{2}$

Substitute 5

$\frac{(E + 50)}{(5E + 25)}$ $=$ $\frac{3}{2}$

2(

2

100 = 15

25 = 15

$\frac{25}{13}$ =

1.9 =

Since there are 5 scientists for every engineer currently, then there are approximately 10 scientists.

*S/E*= 5/1, or 5*E*=*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:$\frac{(E + 50)}{(S + 25)}$ $=$ $\frac{3}{2}$

Substitute 5

*E*for*S*, and solve:$\frac{(E + 50)}{(5E + 25)}$ $=$ $\frac{3}{2}$

2(

*E*+ 50) = 3(5*E*+ 25)2

*E*+ 100 = 15*E*+ 75100 = 15

*E*+ 7525 = 15

*E*$\frac{25}{13}$ =

*E*1.9 =

*E*(or approximately 2)Since there are 5 scientists for every engineer currently, then there are approximately 10 scientists.

Question 5 |

**In the**

*xy*-coordinate plane, a circle with center (−4, 0) is tangent to the line*y*= −*x*. What is the circumference of the circle?$4\pi$ | |

$2\pi \sqrt{2}$ | |

$4\pi$ | |

$4\pi \sqrt{2}$ |

Question 5 Explanation:

The correct answer is (D). Start 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 45-45-90 triangle using the x-axis 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.

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 45-45-90 triangle using the x-axis 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.

**Circumference**$=$ $2 \pi r$ $=$ $2 \pi * 2 \sqrt{2}$ $=$ $4\pi \sqrt{2}$Question 6 |

**Under which conditions is the expression**$\frac{ab}{a-b}$

**always less than zero?**

$a$ $<$ $b$ $<$ $0$ | |

$0$ $<$ $b$ $<$ $a$ | |

$a$ $<$ $0$ $<$ $b$ | |

$b$ $<$ $a$ $<$ $0$ |

Question 6 Explanation:

The correct answer is (A). If the expression $\frac{ab}{a-b}$ $<$ $0$, then $ab$ $<$ $0$ or $a$ $-$ $b$ $<$ $0$ → $a$ $<$ $b.$

The only answer that guarantees a negative answer is $a$ $<$ $b$ $<$ $0.$

Consider $a$ $=$ $-2,$ $b$ $=$ $-1;$

$\frac{(-2)*(-1)}{-2-(-1)}$ $=$ $\frac{2}{-1}$ $=$ $-2.$

On the other hand, if $a$ $=$ $-1$ and $b$ $=$ $1$, then:

$\frac{(-1)*(1)}{-1-1}$ $=$ $\frac{-1}{-2}$ $=$ $\frac{1}{2}$

The only answer that guarantees a negative answer is $a$ $<$ $b$ $<$ $0.$

Consider $a$ $=$ $-2,$ $b$ $=$ $-1;$

$\frac{(-2)*(-1)}{-2-(-1)}$ $=$ $\frac{2}{-1}$ $=$ $-2.$

On the other hand, if $a$ $=$ $-1$ and $b$ $=$ $1$, then:

$\frac{(-1)*(1)}{-1-1}$ $=$ $\frac{-1}{-2}$ $=$ $\frac{1}{2}$

Question 7 |

**In a recent survey of two popular best-selling books, two-fifths 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 three-eighths 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?**

187 | |

352 | |

473 | |

626 |

Question 7 Explanation:

The correct answer is (C). Remember to find each category separately to keep the different numbers clear. The total number of people was 2,200.

$\frac{2}{5}$ 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 $\frac{3}{8}$ 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.

$\frac{2}{5}$ 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 $\frac{3}{8}$ 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 8 |

**If (**

*m*,ƒ(*m*)) represents a point on the graph ƒ(*m*) = 2*m*+ 1, which of the following could be a portion of the graph of the set of points (*m*,(ƒ(*m*))^{2})?Question 8 Explanation:

The correct answer is (C). Let’s re-write the “

Begin with the equation provided in the question, ƒ(

ƒ(

ƒ(

By factoring out a 4 from the expression, the vertex form of the quadratic can be found:

4(

The correct answer is (C). 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.

*m*” as an “*x*” to better understand how this function would look on an*xy*-coordinate plane.Begin with the equation provided in the question, ƒ(

*m*), and square it, ƒ(*m*)^{2}:ƒ(

*m*) = 2x + 1ƒ(

*m*)^{2}= (2*x*+ 1)^{2}= 4*x*^{2}+ 4*x*+ 1By factoring out a 4 from the expression, the vertex form of the quadratic can be found:

4(

*x*^{2}+*x*+ $\frac{1}{4}$) = 4(*x*+ $\frac{1}{2}$)^{2})The correct answer is (C). 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 9 |

**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?5 | |

8 | |

10 | |

13 |

Question 9 Explanation:

The correct answer is (C). 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:

$x:x\sqrt{3}:2x$

In this case, the smallest side length x is 5, so:

$5:5\sqrt{3}:2*5$

The hypotenuse is 2 * 5 = 10.

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:

$x:x\sqrt{3}:2x$

In this case, the smallest side length x is 5, so:

$5:5\sqrt{3}:2*5$

The hypotenuse is 2 * 5 = 10.

Question 10 |

**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?23 x^{23} | |

(2 ^{23})x | |

23 x^{24} | |

(2 ^{24})x − x |

Question 10 Explanation:

The correct answer is (D). To 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).
There are 10 questions to complete.

List |

## Quiz #106

Congratulations - you have completed *Quiz #106*.

You scored %%SCORE%% out of %%TOTAL%%.

Your performance has been rated as %%RATING%%

Your answers are highlighted below.

Question 1 |

**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?**

115° | |

120° | |

135° | |

145° |

Question 1 Explanation:

The correct answer is (C). Remember 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:

$\frac{3}{8} = \frac{x}{360}$

360(3) = 8

1080 = 8

135 =

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:

$\frac{3}{8} = \frac{x}{360}$

360(3) = 8

*x*1080 = 8

*x*135 =

*x*Question 2 |

**Line**

*x*can be described by the function ƒ(x) = 5*x*. Line*y*is parallel to Line x such that the shortest distance between Line*y*and Line*x*is 5, and the*y*-intercept of Line*y*is negative. What is a possible equation for line*y*?$f(x) = x - 5$ | |

$f(x) + 5\sqrt{2} = 5x$ | |

$f(x) = x - 5\sqrt{2}$ | |

$f(x) - 5 = 5x$ |

Question 2 Explanation:

The correct answer is (B). Start by drawing the lines:

The slope-intercept form of a line is

The distance between Line

The slope-intercept form of a line is

*y*=*mx*+*b*, where*m*is the slope and*b*is the*y*-intercept. 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*y*-axis 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*y*-intercept of Line*y*must be greater than 5. This eliminates choice (D).Question 3 |

$\dfrac{41}{12}$ | |

$\dfrac{41}{11}$ | |

$\dfrac{31}{9}$ | |

$\dfrac{31}{7}$ |

Question 3 Explanation:

The correct answer is (B). Use 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:

$\dfrac{2y + 3}{5y - 4} = \dfrac{5}{7}$

Cross multiply and combine like terms to solve for

7(2

14

41 = 11

$y = \frac{41}{11}$

$\dfrac{2y + 3}{5y - 4} = \dfrac{5}{7}$

Cross multiply and combine like terms to solve for

*y*:7(2

*y*+ 3) = 5(5*y*− 4)14

*y*+ 21 = 25*y*− 2041 = 11

*y*$y = \frac{41}{11}$

Question 4 |

**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?$\dfrac{v+e}{v+e+w}$ | |

$\dfrac{0.3v+0.4e}{v+e+w}$ | |

$\dfrac{3v+4e}{v+e+w}$ | |

$\dfrac{30v+40e}{v+e+w}$ |

Question 4 Explanation:

The correct answer is (D). The total number of gallons in the final mixture will be the sum of all the components:

$100 * \dfrac{0.3v + 0.4e}{v + e + w} = \dfrac{30v + 40e}{v + e + w}$

If you chose (B), remember that the question was asking for the percent, not the actual number in the mixture!

*v*+*e*+*w*. There are 0.3*v*gallons of nutrients from VitaDrink in the mixture, 0.4*e*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 0.3*v*+ 0.4*e*. To convert from a fraction to a percent, we simply multiply our value by 100:$100 * \dfrac{0.3v + 0.4e}{v + e + w} = \dfrac{30v + 40e}{v + e + w}$

If you chose (B), remember that the question was asking for the percent, not the actual number in the mixture!

Question 5 |

**On a coordinate plane, (**

*a*,*b*) and (*a*+ 5,*b*+*c*), and (13, 10) are three points on line*l*. If the*x*-intercept of line*l*is −7, what is the value of*c*?1.5 | |

2.0 | |

2.5 | |

3.0 |

Question 5 Explanation:

The correct answer is (C). Recall that all lines can be written in the form

Using this revised equation in conjunction with the given points, we can first solve for the slope of the line in terms of

$slope = \frac{y_2 - y_1}{x_2 - x_1} = \frac{(b+c)-b}{(a+5)-a} = \frac{c}{5}$

Substitute this value for the slope into the linear equation:

$b = \frac{c}{5} * a + k$

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 x-intercept 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):

$slope = \frac{10 - 0}{13-(-7)} = \frac{10}{20} = \frac{1}{2}$

Equate this value with the expression containing

$\frac{1}{2} = \frac{c}{5}$

$c = 5 * \frac{1}{2} = 2.5$

*y*=*mx*+*b*, where m is the slope of the line (rise/run), and*b*is the*y*-intercept 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*y*-intercept 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:$slope = \frac{y_2 - y_1}{x_2 - x_1} = \frac{(b+c)-b}{(a+5)-a} = \frac{c}{5}$

Substitute this value for the slope into the linear equation:

$b = \frac{c}{5} * a + k$

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 x-intercept 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):

$slope = \frac{10 - 0}{13-(-7)} = \frac{10}{20} = \frac{1}{2}$

Equate this value with the expression containing

*c*:$\frac{1}{2} = \frac{c}{5}$

$c = 5 * \frac{1}{2} = 2.5$

Question 6 |

**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?**

3.2% | |

8% | |

9.6% | |

18% |

Question 6 Explanation:

The correct answer is (C). The probability of getting one yellow marble is $\frac{4}{5}$.

The probability of getting one turquoise marble is $\frac{1}{5}$.

The probability of picking two turquoise marbles is $(\frac{1}{5})(\frac{1}{5})$ and picking one yellow marble is $(\frac{4}{5})$, and there are 3 ways of doing this:

Turquoise, Turquoise, Yellow

Turquoise, Yellow, Turquoise

Yellow, Turquoise, Turquoise

$(\frac{1}{5}) (\frac{1}{5}) (\frac{4}{5}) (3) = \frac{(1)(1)(4)(3)}{(5)(5)(5)}$

$\frac{12}{125}$ = 9.6%

The probability of getting one turquoise marble is $\frac{1}{5}$.

The probability of picking two turquoise marbles is $(\frac{1}{5})(\frac{1}{5})$ and picking one yellow marble is $(\frac{4}{5})$, and there are 3 ways of doing this:

Turquoise, Turquoise, Yellow

Turquoise, Yellow, Turquoise

Yellow, Turquoise, Turquoise

$(\frac{1}{5}) (\frac{1}{5}) (\frac{4}{5}) (3) = \frac{(1)(1)(4)(3)}{(5)(5)(5)}$

$\frac{12}{125}$ = 9.6%

There are 6 questions to complete.

List |