45 Random Questions

Hints will display for most wrong answers; explanations for most right answers. You can attempt a question multiple times; it will only be scored correct if you get it right the first time.

I used the official objectives and sample test to construct these questions, but cannot promise that they accurately reflect what’s on the real test. Some of the sample questions were more convoluted than I could bear to write. See terms of use. See the MTEL Practice Test main page to view questions on a particular topic or to download paper practice tests.

MTEL General Curriculum Mathematics Practice

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Question 1

Use the table below to answer the question that follows:

Gordon wants to buy three pounds of nuts. Each of the stores above ordinarily sells the nuts for $4.99 a pound, but is offering a discount this week. At which store can he buy the nuts for the least amount of money?

A	Store A Hint: This would save about $2.50. You can quickly see that D saves more.
B	Store B Hint: This saves 15% and C saves 25%.
C	Store C
D	Store D Hint: This is about 20% off, which is less of a discount than C.

Question 1 Explanation:

Topic: Understand the meanings and models of integers, fractions, decimals,percents, and mixed numbers and apply them to the solution of word problems (Objective 0017).

Question 2

Which of the following nets will not fold into a cube?

A	Hint: If you have trouble visualizing, cut them out and fold (during the test, you can tear paper to approximate).
B
C	Hint: If you have trouble visualizing, cut them out and fold (during the test, you can tear paper to approximate).
D	Hint: If you have trouble visualizing, cut them out and fold (during the test, you can tear paper to approximate).

Question 2 Explanation:

Topic: Match three-dimensional figures and their two-dimensional representations (e.g., nets, projections, perspective drawings) (Objective 0024).

Question 3

Given that 10 cm is approximately equal to 4 inches, which of the following expressions models a way to find out approximately how many inches are equivalent to 350 cm?

A	$ \large 350\times \left( \dfrac{10}{4} \right)$ Hint: The final result should be smaller than 350, and this answer is bigger.
B	$ \large 350\times \left( \dfrac{4}{10} \right)$ Hint: Dimensional analysis can help here: $350 \text{cm} \times \dfrac{4 \text{in}}{10 \text{cm}}$. The cm's cancel and the answer is in inches.
C	$ \large (10-4) \times 350 $ Hint: This answer doesn't make much sense. Try with a simpler example (e.g. 20 cm not 350 cm) to make sure that your logic makes sense.
D	$ \large (350-10) \times 4$ Hint: This answer doesn't make much sense. Try with a simpler example (e.g. 20 cm not 350 cm) to make sure that your logic makes sense.

Question 3 Explanation:

Topic: Applying fractions to word problems (Objective 0017) This problem is similar to one on the official sample test for that objective, but it might fit better into unit conversion and dimensional analysis (Objective 0023: Measurement)

Question 4

The expression $ \large{{8}^{3}}\cdot {{2}^{-10}}$ is equal to which of the following?

A	$ \large 2$ Hint: Write $8^3$ as a power of 2.
B	$ \large \dfrac{1}{2}$ Hint: $8^3 \cdot {2}^{-10}={(2^3)}^3 \cdot {2}^{-10}$ =$2^9 \cdot {2}^{-10} =2^{-1}$
C	$ \large 16$ Hint: Write $8^3$ as a power of 2.
D	$ \large \dfrac{1}{16}$ Hint: Write $8^3$ as a power of 2.

Question 4 Explanation:

Topic: Laws of Exponents (Objective 0019).

Question 5

The chairs in a large room can be arranged in rows of 18, 25, or 60 with no chairs left over. If C is the smallest possible number of chairs in the room, which of the following inequalities does C satisfy?

A	$ \large C\le 300$ Hint: Find the LCM.
B	$ \large 300 < C \le 500 $ Hint: Find the LCM.
C	$ \large 500 < C \le 700 $ Hint: Find the LCM.
D	$ \large C>700$ Hint: The LCM is 900, which is the smallest number of chairs.

Question 5 Explanation:

Topic: Apply LCM in "real-world" situations (according to standardized tests....) (Objective 0018).

Question 6

What is the mathematical name of the three-dimensional polyhedron depicted below?

A	Tetrahedron Hint: All the faces of a tetrahedron are triangles.
B	Triangular Prism Hint: A prism has two congruent, parallel bases, connected by parallelograms (since this is a right prism, the parallelograms are rectangles).
C	Triangular Pyramid Hint: A pyramid has one base, not two.
D	Trigon Hint: A trigon is a triangle (this is not a common term).

Question 6 Explanation:

Topic: Classify and analyze three-dimensional figures using attributes of faces, edges, and vertices (Objective 0024).

Question 7

A family went on a long car trip. Below is a graph of how far they had driven at each hour.

Which of the following is closest to their average speed driving on the trip?

A	$ \large d=20t$ Hint: Try plugging t=7 into the equation, and see how it matches the graph.
B	$ \large d=30t$ Hint: Try plugging t=7 into the equation, and see how it matches the graph.
C	$ \large d=40t$
D	$ \large d=50t$ Hint: Try plugging t=7 into the equation, and see how it matches the graph.

Question 7 Explanation:

Topic: Select the linear equation that best models a real-world situation (Objective 0022).

Question 8

In the triangle below, $\overline{AC}\cong \overline{AD}\cong \overline{DE}$ and $m\angle CAD=100{}^\circ $. What is $m\angle DAE$?

A	$ \large 20{}^\circ $ Hint: Angles ACD and ADC are congruent since they are base angles of an isosceles triangle. Since the angles of a triangle sum to 180, they sum to 80, and they are 40 deg each. Thus angle ADE is 140 deg, since it makes a straight line with angle ADC. Angles DAE and DEA are base angles of an isosceles triangle and thus congruent-- they sum to 40 deg, so are 20 deg each.
B	$ \large 25{}^\circ $ Hint: If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal.
C	$ \large 30{}^\circ $ Hint: If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal.
D	$ \large 40{}^\circ $ Hint: Make sure you're calculating the correct angle.

Question 8 Explanation:

Topic: Classify and analyze polygons using attributes of sides and angles, including real-world applications. (Objective 0024).

Question 9

What is the perimeter of a right triangle with legs of lengths x and 2x?

A	$ \large 6x$ Hint: Use the Pythagorean Theorem.
B	$ \large 3x+5{{x}^{2}}$ Hint: Don't forget to take square roots when you use the Pythagorean Theorem.
C	$ \large 3x+\sqrt{5}{{x}^{2}}$ Hint: $\sqrt {5 x^2}$ is not $\sqrt {5}x^2$.
D	$ \large 3x+\sqrt{5}{{x}^{{}}}$ Hint: To find the hypotenuse, h, use the Pythagorean Theorem: $x^2+(2x)^2=h^2.$ $5x^2=h^2,h=\sqrt{5}x$. The perimeter is this plus x plus 2x.

Question 9 Explanation:

Topic: Recognize and apply connections between algebra and geometry (e.g., the use of coordinate systems, the Pythagorean theorem) (Objective 0024).

Question 10

If x is an integer, which of the following must also be an integer?

A	$ \large \dfrac{x}{2}$ Hint: If x is odd, then $ \dfrac{x}{2} $ is not an integer, e.g. 3/2 = 1.5.
B	$ \large \dfrac{2}{x}$ Hint: Only an integer if x = -2, -1, 1, or 2.
C	$ \large-x$ Hint: -1 times any integer is still an integer.
D	$\large\sqrt{x}$ Hint: Usually not an integer, e.g. $ \sqrt{2} \approx 1.414 $.

Question 10 Explanation:

Topic: Integers (Objective 0016)

Question 11

Which of the lists below is in order from least to greatest value?

A	$ \large \dfrac{1}{2},\quad \dfrac{1}{3},\quad \dfrac{1}{4},\quad \dfrac{1}{5}$ Hint: This is ordered from greatest to least.
B	$ \large \dfrac{1}{3},\quad \dfrac{2}{7},\quad \dfrac{3}{8},\quad \dfrac{4}{11}$ Hint: 1/3 = 2/6 is bigger than 2/7.
C	$ \large \dfrac{1}{4},\quad \dfrac{2}{5},\quad \dfrac{2}{3},\quad \dfrac{4}{5}$ Hint: One way to look at this: 1/4 and 2/5 are both less than 1/2, and 2/3 and 4/5 are both greater than 1/2. 1/4 is 25% and 2/5 is 40%, so 2/5 is greater. The distance from 2/3 to 1 is 1/3 and from 4/5 to 1 is 1/5, and 1/5 is less than 1/3, so 4/5 is bigger.
D	$ \large \dfrac{7}{8},\quad \dfrac{6}{7},\quad \dfrac{5}{6},\quad \dfrac{4}{5}$ Hint: This is in order from greatest to least.

Question 11 Explanation:

Topic: Ordering Fractions (Objective 0017)

Question 12

Use the graph below to answer the question that follows.

Which of the following is a correct equation for the graph of the line depicted above?

A	$ \large y=-\dfrac{1}{2}x+2$ Hint: The slope is -1/2 and the y-intercept is 2. You can also try just plugging in points. For example, this is the only choice that gives y=1 when x=2.
B	$ \large 4x=2y$ Hint: This line goes through (0,0); the graph above does not.
C	$ \large y=x+2$ Hint: The line pictured has negative slope.
D	$ \large y=-x+2$ Hint: Try plugging x=4 into this equation and see if that point is on the graph above.

Question 12 Explanation:

Topic: Find a linear equation that represents a graph (Objective 0022).

Question 13

A publisher prints a series of books with covers made of identical material and using the same thickness of paper for each page. The covers of the book together are 0.4 cm thick, and 125 pieces of the paper used together are 1 cm thick.

The publisher uses a linear function to determine the total thickness, T(n) of a book made with n sheets of paper. What are the slope and intercept of T(n)?

A	Intercept = 0.4 cm, Slope = 125 cm/page Hint: This would mean that each page of the book was 125 cm thick.
B	Intercept =0.4 cm, Slope = $\dfrac{1}{125}$cm/page Hint: The intercept is how thick the book would be with no pages in it. The slope is how much 1 extra page adds to the thickness of the book.
C	Intercept = 125 cm, Slope = 0.4 cm Hint: This would mean that with no pages in the book, it would be 125 cm thick.
D	Intercept = $\dfrac{1}{125}$cm, Slope = 0.4 pages/cm Hint: This would mean that each new page of the book made it 0.4 cm thicker.

Question 13 Explanation:

Topic: Interpret the meaning of the slope and the intercepts of a linear equation that models a real-world situation (Objective 0022).

Question 14

Which of the following points is closest to $ \dfrac{34}{135} \times \dfrac{53}{86}$?

A	A Hint: $\frac{34}{135} \approx \frac{1}{4}$ and $ \frac{53}{86} \approx \frac {2}{3}$. $\frac {1}{4}$ of $\frac {2}{3}$ is small and closest to A.
B	B Hint: Estimate with simpler fractions.
C	C Hint: Estimate with simpler fractions.
D	D Hint: Estimate with simpler fractions.

Question 14 Explanation:

Topic: Understand meaning and models of operations on fractions (Objective 0019).

Question 15

A class is using base-ten block to represent numbers. A large cube represents 1000, a flat represents 100, a rod represents 10, and a little cube represents 1. Which of these is not a correct representation for 2,347?

A	23 flats, 4 rods, 7 little cubes Hint: Be sure you read the question carefully: 2300+40+7=2347
B	2 large cubes, 3 flats, 47 rods Hint: 2000+300+470 $ \neq$ 2347
C	2 large cubes, 34 rods, 7 little cubes Hint: Be sure you read the question carefully: 2000+340+7=2347
D	2 large cubes, 3 flats, 4 rods, 7 little cubes Hint: Be sure you read the question carefully: 2000+300+40+7=2347

Question 15 Explanation:

Topic: Place Value (Objective 0016)

Question 16

How many lines of reflective symmetry and how many centers of rotational symmetry does the parallelogram depicted below have?

A	4 lines of reflective symmetry, 1 center of rotational symmetry. Hint: Try cutting out a shape like this one from paper, and fold where you think the lines of reflective symmetry are (or put a mirror there). Do things line up as you thought they would?
B	2 lines of reflective symmetry, 1 center of rotational symmetry. Hint: Try cutting out a shape like this one from paper, and fold where you think the lines of reflective symmetry are (or put a mirror there). Do things line up as you thought they would?
C	0 lines of reflective symmetry, 1 center of rotational symmetry. Hint: The intersection of the diagonals is a center of rotational symmetry. There are no lines of reflective symmetry, although many people get confused about this fact (best to play with hands on examples to get a feel). Just fyi, the letter S also has rotational, but not reflective symmetry, and it's one that kids often write backwards.
D	2 lines of reflective symmetry, 0 centers of rotational symmetry. Hint: Try cutting out a shape like this one from paper. Trace onto another sheet of paper. See if there's a way to rotate the cut out shape (less than a complete turn) so that it fits within the outlines again.

Question 16 Explanation:

Topic: Analyze geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to concepts of symmetry (Objective 0024).

Question 17

The Venn Diagram below gives data on the number of seniors, athletes, and vegetarians in the student body at a college:

How many students at the college are seniors who are not vegetarians?

A	$ \large 137$ Hint: Doesn't include the senior athletes who are not vegetarians.
B	$ \large 167$
C	$ \large 197$ Hint: That's all seniors, including vegetarians.
D	$ \large 279$ Hint: Includes all athletes who are not vegetarians, some of whom are not seniors.

Question 17 Explanation:

Topic: Venn Diagrams (Objective 0025)

Question 18

There are six gumballs in a bag — two red and four green. Six children take turns picking a gumball out of the bag without looking. They do not return any gumballs to the bag. What is the probability that the first two children to pick from the bag pick the red gumballs?

A	$ \large \dfrac{1}{3}$ Hint: This is the probability that the first child picks a red gumball, but not that the first two children pick red gumballs.
B	$ \large \dfrac{1}{8}$ Hint: Are you adding things that you should be multiplying?
C	$ \large \dfrac{1}{9}$ Hint: This would be the probability if the gumballs were returned to the bag.
D	$ \large \dfrac{1}{15}$ Hint: The probability that the first child picks red is 2/6 = 1/3. Then there are 5 gumballs in the bag, one red, so the probability that the second child picks red is 1/5. Thus 1/5 of the time, after the first child picks red, the second does too, so the probability is 1/5 x 1/3 = 1/15.

Question 18 Explanation:

Topic: Calculate the probabilities of simple and compound events and of independent and dependent events (Objective 0026).

Question 19

Use the expression below to answer the question that follows.

$ \large 3\times {{10}^{4}}+2.2\times {{10}^{2}}$

Which of the following is closest to the expression above?

A	Five million Hint: Pay attention to the exponents. Adding 3 and 2 doesn't work because they have different place values.
B	Fifty thousand Hint: Pay attention to the exponents. Adding 3 and 2 doesn't work because they have different place values.
C	Three million Hint: Don't add the exponents.
D	Thirty thousand Hint: $ 3\times {{10}^{4}} = 30,000;$ the other term is much smaller and doesn't change the estimate.

Question 19 Explanation:

Topics: Place value, scientific notation, estimation (Objective 0016)

Question 20

Which of the numbers below is not equivalent to 4%?

A	$ \large \dfrac{1}{25}$ Hint: 1/25=4/100, so this is equal to 4% (be sure you read the question correctly).
B	$ \large \dfrac{4}{100}$ Hint: 4/100=4% (be sure you read the question correctly).
C	$ \large 0.4$ Hint: 0.4=40% so this is not equal to 4%
D	$ \large 0.04$ Hint: 0.04=4/100, so this is equal to 4% (be sure you read the question correctly).

Question 20 Explanation:

Converting between fractions, decimals, and percents (Objective 0017).

Question 21

The table below gives data from various years on how many young girls drank milk.

Based on the data given above, what was the probability that a randomly chosen girl in 1990 drank milk?

A	$ \large \dfrac{502}{1222}$ Hint: This is the probability that a randomly chosen girl who drinks milk was in the 1989-1991 food survey.
B	$ \large \dfrac{502}{2149}$ Hint: This is the probability that a randomly chosen girl from the whole survey drank milk and was also surveyed in 1989-1991.
C	$ \large \dfrac{502}{837}$
D	$ \large \dfrac{1222}{2149}$ Hint: This is the probability that a randomly chosen girl from any year of the survey drank milk.

Question 21 Explanation:

Topic: Recognize and apply the concept of conditional probability (Objective 0026).

Question 22

The histogram below shows the number of pairs of footware owned by a group of college students.

Which of the following statements can be inferred from the graph above?

A	The median number of pairs of footware owned is between 50 and 60 pairs. Hint: The same number of data points are less than the median as are greater than the median -- but on this histogram, clearly more than half the students own less than 50 pairs of shoes, so the median is less than 50.
B	The mode of the number of pairs of footware owned is 20. Hint: The mode is the most common number of pairs of footwear owned. We can't tell it from this histogram because each bar represents 10 different numbers-- perhaps 8 students each own each number from 10 to 19, but 40 students own exactly 6 pairs of shoes.... or perhaps not....
C	The mean number of pairs of footware owned is less than the median number of pairs of footware owned. Hint: This is a right skewed distribution, and so the mean is bigger than the median -- the few large values on the right pull up the mean, but have little effect on the median.
D	The median number of pairs of footware owned is between 10 and 20. Hint: There are approximately 230 students represented in this survey, and the 41st through 120th lowest values are between 10 and 20 -- thus the middle value is in that range.

Question 22 Explanation:

Topics: Analyze and interpret various graphic and data representations, and use measures of central tendency (e.g., mean, median, mode) and spread to describe and interpret real-world data (Objective 0025).

Question 23

Cell phone plan A charges $3 per month plus $0.10 per minute. Cell phone plan B charges $29.99 per month, with no fee for the first 400 minutes and then $0.20 for each additional minute.

Which equation can be used to solve for the number of minutes, m (with m>400) that a person would have to spend on the phone each month in order for the bills for plan A and plan B to be equal?

A	$ \large 3.10m=400+0.2m$ Hint: These are the numbers in the problem, but this equation doesn't make sense. If you don't know how to make an equation, try plugging in an easy number like m=500 minutes to see if each side equals what it should.
B	$ \large 3+0.1m=29.99+.20m$ Hint: Doesn't account for the 400 free minutes.
C	$ \large 3+0.1m=400+29.99+.20(m-400)$ Hint: Why would you add 400 minutes and $29.99? If you don't know how to make an equation, try plugging in an easy number like m=500 minutes to see if each side equals what it should.
D	$ \large 3+0.1m=29.99+.20(m-400)$ Hint: The left side is $3 plus $0.10 times the number of minutes. The right is $29.99 plus $0.20 times the number of minutes over 400.

Question 23 Explanation:

Identify variables and derive algebraic expressions that represent real-world situations (Objective 0020).

Question 24

The least common multiple of 60 and N is 1260. Which of the following could be the prime factorization of N?

A	$ \large2\cdot 5\cdot 7$ Hint: 1260 is divisible by 9 and 60 is not, so N must be divisible by 9 for 1260 to be the LCM.
B	$ \large{{2}^{3}}\cdot {{3}^{2}}\cdot 5 \cdot 7$ Hint: 1260 is not divisible by 8, so it isn't a multiple of this N.
C	$ \large3 \cdot 5 \cdot 7$ Hint: 1260 is divisible by 9 and 60 is not, so N must be divisible by 9 for 1260 to be the LCM.
D	$ \large{{3}^{2}}\cdot 5\cdot 7$ Hint: $1260=2^2 \cdot 3^2 \cdot 5 \cdot 7$ and $60=2^2 \cdot 3 \cdot 5$. In order for 1260 to be the LCM, N has to be a multiple of $3^2$ and of 7 (because 60 is not a multiple of either of these). N also cannot introduce a factor that would require the LCM to be larger (as in choice b).

Question 24 Explanation:

Topic: Least Common Multiple (Objective 0018)

Question 25

The expression $ \large {{7}^{-4}}\cdot {{8}^{-6}}$ is equal to which of the following?

A	$ \large \dfrac{8}{{{\left( 56 \right)}^{4}}}$ Hint: The bases are whole numbers, and the exponents are negative. How can the numerator be 8?
B	$ \large \dfrac{64}{{{\left( 56 \right)}^{4}}}$ Hint: The bases are whole numbers, and the exponents are negative. How can the numerator be 64?
C	$ \large \dfrac{1}{8\cdot {{\left( 56 \right)}^{4}}}$ Hint: $8^{-6}=8^{-4} \times 8^{-2}$
D	$ \large \dfrac{1}{64\cdot {{\left( 56 \right)}^{4}}}$

Question 25 Explanation:

Topics: Laws of exponents (Objective 0019).

Question 26

Which of the following values of x satisfies the inequality $ \large \left| {{(x+2)}^{3}} \right|<3?$

A	$ \large x=-3$ Hint: $ \left\| {{(-3+2)}^{3}} \right\|$=$ \left \| {(-1)}^3 \right \| $=$ \left \| -1 \right \|=1 $ .
B	$ \large x=0$ Hint: $ \left\| {{(0+2)}^{3}} \right\|$=$ \left \| {2}^3 \right \| $=$ \left \| 8 \right \| $ =$ 8$
C	$ \large x=-4$ Hint: $ \left\| {{(-4+2)}^{3}} \right\|$=$ \left \| {(-2)}^3 \right \| $=$ \left \| -8 \right \| $ =$ 8$
D	$ \large x=1$ Hint: $ \left\| {{(1+2)}^{3}} \right\|$=$ \left \| {3}^3 \right \| $=$ \left \| 27 \right \| $ = $27$

Question 26 Explanation:

Topics: Laws of exponents, order of operations, interpret absolute value (Objective 0019).

Question 27

A family on vacation drove the first 200 miles in 4 hours and the second 200 miles in 5 hours. Which expression below gives their average speed for the entire trip?

A	$ \large \dfrac{200+200}{4+5}$ Hint: Average speed is total distance divided by total time.
B	$ \large \left( \dfrac{200}{4}+\dfrac{200}{5} \right)\div 2$ Hint: This seems logical, but the problem is that it weights the first 4 hours and the second 5 hours equally, when each hour should get the same weight in computing the average speed.
C	$ \large \dfrac{200}{4}+\dfrac{200}{5} $ Hint: This would be an average of 90 miles per hour!
D	$ \large \dfrac{400}{4}+\dfrac{400}{5} $ Hint: This would be an average of 180 miles per hour! Even a family of race car drivers probably doesn't have that average speed on a vacation!

Question 27 Explanation:

Topic: Solve a variety of measurement problems (e.g., time, temperature, rates, average rates of change) in real-world situations (Objective 0023).

Question 28

Which of the graphs below represent functions?

II.

III.

IV.

A	I and IV only. Hint: There are vertical lines that go through 2 points in IV .
B	I and III only. Hint: Even though III is not continuous, it's still a function (assuming that vertical lines between the "steps" do not go through 2 points).
C	II and III only. Hint: Learn about the vertical line test.
D	I, II, and IV only. Hint: There are vertical lines that go through 2 points in II.

Question 28 Explanation:

Understand the definition of function and various representations of functions (e.g., input/output machines, tables, graphs, mapping diagrams, formulas). (Objective 0021).

Question 29

Use the problem below to answer the question that follows:

T shirts are on sale for 20% off. Tasha paid $8.73 for a shirt. What is the regular price of the shirt? There is no tax on clothing purchases under $175.

Let p represent the regular price of these t-shirt. Which of the following equations is correct?

A	$ \large 0.8p=\$8.73$ Hint: 80% of the regular price = $8.73.
B	$ \large \$8.73+0.2*\$8.73=p$ Hint: The 20% off was off of the ORIGINAL price, not off the $8.73 (a lot of people make this mistake). Plus this is the same equation as in choice c.
C	$ \large 1.2*\$8.73=p$ Hint: The 20% off was off of the ORIGINAL price, not off the $8.73 (a lot of people make this mistake). Plus this is the same equation as in choice b.
D	$ \large p-0.2*\$8.73=p$ Hint: Subtract p from both sides of this equation, and you have -.2 x 8.73 =0.

Question 29 Explanation:

Topics: Use algebra to solve word problems involving percents and identify variables, and derive algebraic expressions that represent real-world situations (Objective 0020).

Question 30

In January 2011, the national debt was about 14 trillion dollars and the US population was about 300 million people. Someone reading these figures estimated that the national debt was about $5,000 per person. Which of these statements best describes the reasonableness of this estimate?

A	It is too low by a factor of 10 Hint: 14 trillion $ \approx 15 \times {{10}^{12}} $ and 300 million $ \approx 3 \times {{10}^{8}}$, so the true answer is about $ 5 \times {{10}^{4}} $ or $50,000.
B	It is too low by a factor of 100
C	It is too high by a factor of 10
D	It is too high by a factor of 100

Question 30 Explanation:

Topics: Estimation, Scientific Notation in the real world (Objective 0016).

Question 31

An above-ground swimming pool is in the shape of a regular hexagonal prism, is one meter high, and holds 65 cubic meters of water. A second pool has a base that is also a regular hexagon, but with sides twice as long as the sides in the first pool. This second pool is also one meter high. How much water will the second pool hold?

A	$ \large 65\text{ }{{\text{m}}^{3}}$ Hint: A bigger pool would hold more water.
B	$ \large 65\cdot 2\text{ }{{\text{m}}^{3}}$ Hint: Try a simpler example, say doubling the sides of the base of a 1 x 1 x 1 cube.
C	$ \large 65\cdot 4\text{ }{{\text{m}}^{3}}$ Hint: If we think of the pool as filled with 1 x 1 x 1 cubes (and some fractions of cubes), then scaling to the larger pool changes each 1 x 1 x 1 cube to a 2 x 2 x 1 prism, or multiplies volume by 4.
D	$ \large 65\cdot 8\text{ }{{\text{m}}^{3}}$ Hint: Try a simpler example, say doubling the sides of the base of a 1 x 1 x 1 cube.

Question 31 Explanation:

Topic: Determine how the characteristics (e.g., area, volume) of geometric figures and shapes are affected by changes in their dimensions (Objective 0023).

Question 32

Here is a number trick:

1) Pick a whole number

2) Double your number.

3) Add 20 to the above result.

4) Multiply the above by 5

5) Subtract 100

6) Divide by 10

The result is always the number that you started with! Suppose you start by picking N. Which of the equations below best demonstrates that the result after Step 6 is also N?

A	$ \large N2+205-100\div 10=N$ Hint: Use parentheses or else order of operations is off.
B	$ \large \left( \left( 2N+20 \right)5-100 \right)\div 10=N$
C	$ \large \left( N+N+20 \right)*5-100\div 10=N$ Hint: With this answer you would subtract 10, instead of subtracting 100 and then dividing by 10.
D	$ \large \left( \left( \left( N\div 10 \right)-100 \right)5+20 \right)2=N$ Hint: This answer is quite backwards.

Question 32 Explanation:

Topic: Recognize and apply the concepts of variable, function, equality, and equation to express relationships algebraically (Objective 0020).

Question 33

Use the expression below to answer the question that follows.

$ \large \dfrac{\left( 4\times {{10}^{3}} \right)\times \left( 3\times {{10}^{4}} \right)}{6\times {{10}^{6}}}$

Which of the following is equivalent to the expression above?

A	2 Hint: $10^3 \times 10^4=10^7$, and note that if you're guessing when the answers are so closely related, you're generally better off guessing one of the middle numbers.
B	20 Hint: $ \dfrac{\left( 4\times {{10}^{3}} \right)\times \left( 3\times {{10}^{4}} \right)}{6\times {{10}^{6}}}=\dfrac {12 \times {{10}^{7}}}{6\times {{10}^{6}}}=$$2 \times {{10}^{1}}=20 $
C	200 Hint: $10^3 \times 10^4=10^7$
D	2000 Hint: $10^3 \times 10^4=10^7$, and note that if you're guessing when the answers are so closely related, you're generally better off guessing one of the middle numbers.

Question 33 Explanation:

Topics: Scientific notation, exponents, simplifying fractions (Objective 0016, although overlaps with other objectives too).

Question 34

Four children randomly line up, single file. What is the probability that they are in height order, with the shortest child in front? All of the children are different heights.

A	$ \large \dfrac{1}{4}$ Hint: Try a simpler question with 3 children -- call them big, medium, and small -- and list all the ways they could line up. Then see how to extend your logic to the problem with 4 children.
B	$ \large \dfrac{1}{256} $ Hint: Try a simpler question with 3 children -- call them big, medium, and small -- and list all the ways they could line up. Then see how to extend your logic to the problem with 4 children.
C	$ \large \dfrac{1}{16}$ Hint: Try a simpler question with 3 children -- call them big, medium, and small -- and list all the ways they could line up. Then see how to extend your logic to the problem with 4 children.
D	$ \large \dfrac{1}{24}$ Hint: The number of ways for the children to line up is $4!=4 \times 3 \times 2 \times 1 =24$ -- there are 4 choices for who is first in line, then 3 for who is second, etc. Only one of these lines has the children in the order specified.

Question 34 Explanation:

Topic: Apply knowledge of combinations and permutations to the computation of probabilities (Objective 0026).

Question 35

The histogram below shows the frequency of a class's scores on a 4 question quiz.

What was the mean score on the quiz?

A	$ \large 2.75$ Hint: There were 20 students who took the quiz. Total points earned: $2 \times 1+6 \times 2+ 7\times 3+5 \times 4=55$, and 55/20 = 2.75.
B	$ \large 2$ Hint: How many students are there total? Did you count them all?
C	$ \large 3$ Hint: How many students are there total? Did you count them all? Be sure you're finding the mean, not the median or the mode.
D	$ \large 2.5$ Hint: How many students are there total? Did you count them all? Don't just take the mean of 1, 2, 3, 4 -- you have to weight them properly.

Question 35 Explanation:

Topics: Analyze and interpret various graphic representations, and use measures of central tendency (e.g., mean, median, mode) and spread to describe and interpret real-world data (Objective 0025).

Question 36

Kendra is trying to decide which fraction is greater, $ \dfrac{4}{7}$ or $ \dfrac{5}{8}$. Which of the following answers shows the best reasoning?

A	$ \dfrac{4}{7}$ is $ \dfrac{3}{7}$away from 1, and $ \dfrac{5}{8}$ is $ \dfrac{3}{8}$away from 1. Since eighth‘s are smaller than seventh‘s, $ \dfrac{5}{8}$ is closer to 1, and is the greater of the two fractions.
B	$ 7-4=3$ and $ 8-5=3$, so the fractions are equal. Hint: Not how to compare fractions. By this logic, 1/2 and 3/4 are equal, but 1/2 and 2/4 are not.
C	$ 4\times 8=32$ and $ 7\times 5=35$. Since $ 32<35$ , $ \dfrac{5}{8}<\dfrac{4}{7}$ Hint: Starts out as something that works, but the conclusion is wrong. 4/7 = 32/56 and 5/8 = 35/56. The cross multiplication gives the numerators, and 35/56 is bigger.
D	$ 4<5$ and $ 7<8$, so $ \dfrac{4}{7}<\dfrac{5}{8}$ Hint: Conclusion is correct, logic is wrong. With this reasoning, 1/2 would be less than 2/100,000.

Question 36 Explanation:

Topics: Comparing fractions, and understanding the meaning of fractions (Objective 0017).

Question 37

There are 15 students for every teacher. Let t represent the number of teachers and let s represent the number of students. Which of the following equations is correct?

A	$ \large t=s+15$ Hint: When there are 2 teachers, how many students should there be? Do those values satisfy this equation?
B	$ \large s=t+15$ Hint: When there are 2 teachers, how many students should there be? Do those values satisfy this equation?
C	$ \large t=15s$ Hint: This is a really easy mistake to make, which comes from transcribing directly from English, "1 teachers equals 15 students." To see that it's wrong, plug in s=2; do you really need 30 teachers for 2 students? To avoid this mistake, insert the word "number," "Number of teachers equals 15 times number of students" is more clearly problematic.
D	$ \large s=15t$

Question 37 Explanation:

Topic: Select the linear equation that best models a real-world situation (Objective 0022).

Question 38

Which of the following inequalities describes all values of x with $\large \dfrac{x}{2}\le \dfrac{x}{3}$?

A	$ \large x < 0$ Hint: If x =0, then x/2 = x/3, so this answer can't be correct.
B	$ \large x \le 0$
C	$ \large x > 0$ Hint: If x =0, then x/2 = x/3, so this answer can't be correct.
D	$ \large x \ge 0$ Hint: Try plugging in x = 6.

Question 38 Explanation:

Topics: Inequalities, operations (Objective 0019) (not exactly sure how to classify, but this is like one of the problems on the official sample test).

Question 39

Solve for x: $\large 4-\dfrac{2}{3}x=2x$

A	$ \large x=3$ Hint: Try plugging x=3 into the equation.
B	$ \large x=-3$ Hint: Left side is positive, right side is negative when you plug this in for x.
C	$ \large x=\dfrac{3}{2}$ Hint: One way to solve: $4=\dfrac{2}{3}x+2x$ $=\dfrac{8}{3}x$.$x=\dfrac{3 \times 4}{8}=\dfrac{3}{2}$. Another way is to just plug x=3/2 into the equation and see that each side equals 3 -- on a multiple choice test, you almost never have to actually solve for x.
D	$ \large x=-\dfrac{3}{2}$ Hint: Left side is positive, right side is negative when you plug this in for x.

Question 39 Explanation:

Topic: Solve linear equations (Objective 0020).

Question 40

Which of the following is an irrational number?

A	$ \large \sqrt[3]{8}$ Hint: This answer is the cube root of 8. Since 2 x 2 x 2 =8, this is equal to 2, which is rational because 2 = 2/1.
B	$ \large \sqrt{8}$ Hint: It is not trivial to prove that this is irrational, but you can get this answer by eliminating the other choices.
C	$ \large \dfrac{1}{8}$ Hint: 1/8 is the RATIO of two integers, so it is rational.
D	$ \large -8$ Hint: Negative integers are also rational, -8 = -8/1, a ratio of integers.

Question 40 Explanation:

Topic: Identifying rational and irrational numbers (Objective 0016).

Question 41

Here is a student's work solving an equation:

$ x-4=-2x+6$

$ x-4+4=-2x+6+4$

$ x=-2x+10$

$ x-2x=10$

$ x=10$

Which of the following statements is true?

A	The student‘s solution is correct. Hint: Try plugging into the original solution.
B	The student did not correctly use properties of equality. Hint: After $ x=-2x+10$, the student subtracted 2x on the left and added 2x on the right.
C	The student did not correctly use the distributive property. Hint: Distributive property is $a(b+c)=ab+ac$.
D	The student did not correctly use the commutative property. Hint: Commutative property is $a+b=b+a$ or $ab=ba$.

Question 41 Explanation:

Topic: Justify algebraic manipulations by application of the properties of equality, the order of operations, the number properties, and the order properties (Objective 0020).

Question 42

Which of the following is closest to the height of a college student in centimeters?

A	1.6 cm Hint: This is more the height of a Lego toy college student -- less than an inch!
B	16 cm Hint: Less than knee high on most college students.
C	160 cm Hint: Remember, a meter stick (a little bigger than a yard stick) is 100 cm. Also good to know is that 4 inches is approximately 10 cm.
D	1600 cm Hint: This college student might be taller than some campus buildings!

Question 42 Explanation:

Topic: Estimate and calculate measurements using customary, metric, and nonstandard units of measurement (Objective 0023).

Question 43

Taxicab fares in Boston (Spring 2012) are $2.60 for the first $\dfrac{1}{7}$ of a mile or less and $0.40 for each $\dfrac{1}{7}$ of a mile after that.

Let d represent the distance a passenger travels in miles (with $d>\dfrac{1}{7}$). Which of the following expressions represents the total fare?

A	$ \large \$2.60+\$0.40d$ Hint: It's 40 cents for 1/7 of a mile, not per mile.
B	$ \large \$2.60+\$0.40\dfrac{d}{7}$ Hint: According to this equation, going 7 miles would cost $3; does that make sense?
C	$ \large \$2.20+\$2.80d$ Hint: You can think of the fare as $2.20 to enter the cab, and then $0.40 for each 1/7 of a mile, including the first 1/7 of a mile (or $2.80 per mile). Alternatively, you pay $2.60 for the first 1/7 of a mile, and then $2.80 per mile for d-1/7 miles. The total is 2.60+2.80(d-1/7) = 2.60+ 2.80d -.40 = 2.20+2.80d.
D	$ \large \$2.60+\$2.80d$ Hint: Don't count the first 1/7 of a mile twice.

Question 43 Explanation:

Topic: Identify variables and derive algebraic expressions that represent real-world situations (Objective 0020), and select the linear equation that best models a real-world situation (Objective 0022).

Question 44

Which of the following is equal to eleven billion four hundred thousand?

A	$ \large 11,400,000$ Hint: That's eleven million four hundred thousand.
B	$\large11,000,400,000$
C	$ \large11,000,000,400,000$ Hint: That's eleven trillion four hundred thousand (although with British conventions; this answer is correct, but in the US, it isn't).
D	$ \large 11,400,000,000$ Hint: That's eleven billion four hundred million

Question 44 Explanation:

Topic: Place Value (Objective 0016)

Question 45

Use the graph below to answer the question that follows:

The graph above best matches which of the following scenarios:

A	George left home at 10:00 and drove to work on a crooked path. He was stopped in traffic at 10:30 and 10:45. He drove 30 miles total. Hint: Just because he ended up 30 miles from home doesn't mean he drove 30 miles total.
B	George drove to work. On the way to work there is a little hill and a big hill. He slowed down for them. He made it to work at 11:15. Hint: The graph is not a picture of the roads.
C	George left home at 10:15. He drove 10 miles, then realized he‘d forgotten something at home. He turned back and got what he‘d forgotten. Then he drove in a straight line, at many different speeds, until he got to work around 11:15. Hint: A straight line on a distance versus time graph means constant speed.
D	George left home at 10:15. He drove 10 miles, then realized he‘d forgotten something at home. He turned back and got what he‘d forgotten. Then he drove at a constant speed until he got to work around 11:15.

Question 45 Explanation:

Topic: Use qualitative graphs to represent functional relationships in the real world (Objective 0021).

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If you found a mistake or have comments on a particular question, please contact me (please copy and paste at least part of the question into the form, as the numbers change depending on how quizzes are displayed). General comments can be left here.

A	\( \large 350\times \left( \dfrac{10}{4} \right)\) Hint: The final result should be smaller than 350, and this answer is bigger.
B	\( \large 350\times \left( \dfrac{4}{10} \right)\) Hint: Dimensional analysis can help here: \(350 \text{cm} \times \dfrac{4 \text{in}}{10 \text{cm}}\). The cm's cancel and the answer is in inches.
C	\( \large (10-4) \times 350 \) Hint: This answer doesn't make much sense. Try with a simpler example (e.g. 20 cm not 350 cm) to make sure that your logic makes sense.
D	\( \large (350-10) \times 4\) Hint: This answer doesn't make much sense. Try with a simpler example (e.g. 20 cm not 350 cm) to make sure that your logic makes sense.

A	\( \large 2\) Hint: Write \(8^3\) as a power of 2.
B	\( \large \dfrac{1}{2}\) Hint: \(8^3 \cdot {2}^{-10}={(2^3)}^3 \cdot {2}^{-10}\) =\(2^9 \cdot {2}^{-10} =2^{-1}\)
C	\( \large 16\) Hint: Write \(8^3\) as a power of 2.
D	\( \large \dfrac{1}{16}\) Hint: Write \(8^3\) as a power of 2.

A	\( \large C\le 300\) Hint: Find the LCM.
B	\( \large 300 < C \le 500 \) Hint: Find the LCM.
C	\( \large 500 < C \le 700 \) Hint: Find the LCM.
D	\( \large C>700\) Hint: The LCM is 900, which is the smallest number of chairs.

A	\( \large d=20t\) Hint: Try plugging t=7 into the equation, and see how it matches the graph.
B	\( \large d=30t\) Hint: Try plugging t=7 into the equation, and see how it matches the graph.
C	\( \large d=40t\)
D	\( \large d=50t\) Hint: Try plugging t=7 into the equation, and see how it matches the graph.

A	\( \large 20{}^\circ \) Hint: Angles ACD and ADC are congruent since they are base angles of an isosceles triangle. Since the angles of a triangle sum to 180, they sum to 80, and they are 40 deg each. Thus angle ADE is 140 deg, since it makes a straight line with angle ADC. Angles DAE and DEA are base angles of an isosceles triangle and thus congruent-- they sum to 40 deg, so are 20 deg each.
B	\( \large 25{}^\circ \) Hint: If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal.
C	\( \large 30{}^\circ \) Hint: If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal.
D	\( \large 40{}^\circ \) Hint: Make sure you're calculating the correct angle.

A	\( \large 6x\) Hint: Use the Pythagorean Theorem.
B	\( \large 3x+5{{x}^{2}}\) Hint: Don't forget to take square roots when you use the Pythagorean Theorem.
C	\( \large 3x+\sqrt{5}{{x}^{2}}\) Hint: \(\sqrt {5 x^2}\) is not \(\sqrt {5}x^2\).
D	\( \large 3x+\sqrt{5}{{x}^{{}}}\) Hint: To find the hypotenuse, h, use the Pythagorean Theorem: \(x^2+(2x)^2=h^2.\) \(5x^2=h^2,h=\sqrt{5}x\). The perimeter is this plus x plus 2x.

A	\( \large \dfrac{x}{2}\) Hint: If x is odd, then \( \dfrac{x}{2} \) is not an integer, e.g. 3/2 = 1.5.
B	\( \large \dfrac{2}{x}\) Hint: Only an integer if x = -2, -1, 1, or 2.
C	\( \large-x\) Hint: -1 times any integer is still an integer.
D	\(\large\sqrt{x}\) Hint: Usually not an integer, e.g. \( \sqrt{2} \approx 1.414 \).

A	\( \large \dfrac{1}{2},\quad \dfrac{1}{3},\quad \dfrac{1}{4},\quad \dfrac{1}{5}\) Hint: This is ordered from greatest to least.
B	\( \large \dfrac{1}{3},\quad \dfrac{2}{7},\quad \dfrac{3}{8},\quad \dfrac{4}{11}\) Hint: 1/3 = 2/6 is bigger than 2/7.
C	\( \large \dfrac{1}{4},\quad \dfrac{2}{5},\quad \dfrac{2}{3},\quad \dfrac{4}{5}\) Hint: One way to look at this: 1/4 and 2/5 are both less than 1/2, and 2/3 and 4/5 are both greater than 1/2. 1/4 is 25% and 2/5 is 40%, so 2/5 is greater. The distance from 2/3 to 1 is 1/3 and from 4/5 to 1 is 1/5, and 1/5 is less than 1/3, so 4/5 is bigger.
D	\( \large \dfrac{7}{8},\quad \dfrac{6}{7},\quad \dfrac{5}{6},\quad \dfrac{4}{5}\) Hint: This is in order from greatest to least.

A	\( \large y=-\dfrac{1}{2}x+2\) Hint: The slope is -1/2 and the y-intercept is 2. You can also try just plugging in points. For example, this is the only choice that gives y=1 when x=2.
B	\( \large 4x=2y\) Hint: This line goes through (0,0); the graph above does not.
C	\( \large y=x+2\) Hint: The line pictured has negative slope.
D	\( \large y=-x+2\) Hint: Try plugging x=4 into this equation and see if that point is on the graph above.

A	Intercept = 0.4 cm, Slope = 125 cm/page Hint: This would mean that each page of the book was 125 cm thick.
B	Intercept =0.4 cm, Slope = \(\dfrac{1}{125}\)cm/page Hint: The intercept is how thick the book would be with no pages in it. The slope is how much 1 extra page adds to the thickness of the book.
C	Intercept = 125 cm, Slope = 0.4 cm Hint: This would mean that with no pages in the book, it would be 125 cm thick.
D	Intercept = \(\dfrac{1}{125}\)cm, Slope = 0.4 pages/cm Hint: This would mean that each new page of the book made it 0.4 cm thicker.

A	A Hint: \(\frac{34}{135} \approx \frac{1}{4}\) and \( \frac{53}{86} \approx \frac {2}{3}\). \(\frac {1}{4}\) of \(\frac {2}{3}\) is small and closest to A.
B	B Hint: Estimate with simpler fractions.
C	C Hint: Estimate with simpler fractions.
D	D Hint: Estimate with simpler fractions.

A	\( \large 137\) Hint: Doesn't include the senior athletes who are not vegetarians.
B	\( \large 167\)
C	\( \large 197\) Hint: That's all seniors, including vegetarians.
D	\( \large 279\) Hint: Includes all athletes who are not vegetarians, some of whom are not seniors.

A	\( \large \dfrac{1}{3}\) Hint: This is the probability that the first child picks a red gumball, but not that the first two children pick red gumballs.
B	\( \large \dfrac{1}{8}\) Hint: Are you adding things that you should be multiplying?
C	\( \large \dfrac{1}{9}\) Hint: This would be the probability if the gumballs were returned to the bag.
D	\( \large \dfrac{1}{15}\) Hint: The probability that the first child picks red is 2/6 = 1/3. Then there are 5 gumballs in the bag, one red, so the probability that the second child picks red is 1/5. Thus 1/5 of the time, after the first child picks red, the second does too, so the probability is 1/5 x 1/3 = 1/15.

A	\( \large \dfrac{1}{25}\) Hint: 1/25=4/100, so this is equal to 4% (be sure you read the question correctly).
B	\( \large \dfrac{4}{100}\) Hint: 4/100=4% (be sure you read the question correctly).
C	\( \large 0.4\) Hint: 0.4=40% so this is not equal to 4%
D	\( \large 0.04\) Hint: 0.04=4/100, so this is equal to 4% (be sure you read the question correctly).

A	\( \large \dfrac{502}{1222}\) Hint: This is the probability that a randomly chosen girl who drinks milk was in the 1989-1991 food survey.
B	\( \large \dfrac{502}{2149}\) Hint: This is the probability that a randomly chosen girl from the whole survey drank milk and was also surveyed in 1989-1991.
C	\( \large \dfrac{502}{837}\)
D	\( \large \dfrac{1222}{2149}\) Hint: This is the probability that a randomly chosen girl from any year of the survey drank milk.

A	\( \large 3.10m=400+0.2m\) Hint: These are the numbers in the problem, but this equation doesn't make sense. If you don't know how to make an equation, try plugging in an easy number like m=500 minutes to see if each side equals what it should.
B	\( \large 3+0.1m=29.99+.20m\) Hint: Doesn't account for the 400 free minutes.
C	\( \large 3+0.1m=400+29.99+.20(m-400)\) Hint: Why would you add 400 minutes and $29.99? If you don't know how to make an equation, try plugging in an easy number like m=500 minutes to see if each side equals what it should.
D	\( \large 3+0.1m=29.99+.20(m-400)\) Hint: The left side is $3 plus $0.10 times the number of minutes. The right is $29.99 plus $0.20 times the number of minutes over 400.

A	\( \large2\cdot 5\cdot 7\) Hint: 1260 is divisible by 9 and 60 is not, so N must be divisible by 9 for 1260 to be the LCM.
B	\( \large{{2}^{3}}\cdot {{3}^{2}}\cdot 5 \cdot 7\) Hint: 1260 is not divisible by 8, so it isn't a multiple of this N.
C	\( \large3 \cdot 5 \cdot 7\) Hint: 1260 is divisible by 9 and 60 is not, so N must be divisible by 9 for 1260 to be the LCM.
D	\( \large{{3}^{2}}\cdot 5\cdot 7\) Hint: \(1260=2^2 \cdot 3^2 \cdot 5 \cdot 7\) and \(60=2^2 \cdot 3 \cdot 5\). In order for 1260 to be the LCM, N has to be a multiple of \(3^2\) and of 7 (because 60 is not a multiple of either of these). N also cannot introduce a factor that would require the LCM to be larger (as in choice b).

A	\( \large \dfrac{8}{{{\left( 56 \right)}^{4}}}\) Hint: The bases are whole numbers, and the exponents are negative. How can the numerator be 8?
B	\( \large \dfrac{64}{{{\left( 56 \right)}^{4}}}\) Hint: The bases are whole numbers, and the exponents are negative. How can the numerator be 64?
C	\( \large \dfrac{1}{8\cdot {{\left( 56 \right)}^{4}}}\) Hint: \(8^{-6}=8^{-4} \times 8^{-2}\)
D	\( \large \dfrac{1}{64\cdot {{\left( 56 \right)}^{4}}}\)

A	\( \large x=-3\) Hint: \( \left\| {{(-3+2)}^{3}} \right\|\)=\( \left \| {(-1)}^3 \right \| \)=\( \left \| -1 \right \|=1 \) .
B	\( \large x=0\) Hint: \( \left\| {{(0+2)}^{3}} \right\|\)=\( \left \| {2}^3 \right \| \)=\( \left \| 8 \right \| \) =\( 8\)
C	\( \large x=-4\) Hint: \( \left\| {{(-4+2)}^{3}} \right\|\)=\( \left \| {(-2)}^3 \right \| \)=\( \left \| -8 \right \| \) =\( 8\)
D	\( \large x=1\) Hint: \( \left\| {{(1+2)}^{3}} \right\|\)=\( \left \| {3}^3 \right \| \)=\( \left \| 27 \right \| \) = \(27\)

A	\( \large \dfrac{200+200}{4+5}\) Hint: Average speed is total distance divided by total time.
B	\( \large \left( \dfrac{200}{4}+\dfrac{200}{5} \right)\div 2\) Hint: This seems logical, but the problem is that it weights the first 4 hours and the second 5 hours equally, when each hour should get the same weight in computing the average speed.
C	\( \large \dfrac{200}{4}+\dfrac{200}{5} \) Hint: This would be an average of 90 miles per hour!
D	\( \large \dfrac{400}{4}+\dfrac{400}{5} \) Hint: This would be an average of 180 miles per hour! Even a family of race car drivers probably doesn't have that average speed on a vacation!

A	\( \large 0.8p=\$8.73\) Hint: 80% of the regular price = $8.73.
B	\( \large \$8.73+0.2*\$8.73=p\) Hint: The 20% off was off of the ORIGINAL price, not off the $8.73 (a lot of people make this mistake). Plus this is the same equation as in choice c.
C	\( \large 1.2*\$8.73=p\) Hint: The 20% off was off of the ORIGINAL price, not off the $8.73 (a lot of people make this mistake). Plus this is the same equation as in choice b.
D	\( \large p-0.2*\$8.73=p\) Hint: Subtract p from both sides of this equation, and you have -.2 x 8.73 =0.

A	\( \large 65\text{ }{{\text{m}}^{3}}\) Hint: A bigger pool would hold more water.
B	\( \large 65\cdot 2\text{ }{{\text{m}}^{3}}\) Hint: Try a simpler example, say doubling the sides of the base of a 1 x 1 x 1 cube.
C	\( \large 65\cdot 4\text{ }{{\text{m}}^{3}}\) Hint: If we think of the pool as filled with 1 x 1 x 1 cubes (and some fractions of cubes), then scaling to the larger pool changes each 1 x 1 x 1 cube to a 2 x 2 x 1 prism, or multiplies volume by 4.
D	\( \large 65\cdot 8\text{ }{{\text{m}}^{3}}\) Hint: Try a simpler example, say doubling the sides of the base of a 1 x 1 x 1 cube.

A	\( \large N2+205-100\div 10=N\) Hint: Use parentheses or else order of operations is off.
B	\( \large \left( \left( 2N+20 \right)5-100 \right)\div 10=N\)
C	\( \large \left( N+N+20 \right)*5-100\div 10=N\) Hint: With this answer you would subtract 10, instead of subtracting 100 and then dividing by 10.
D	\( \large \left( \left( \left( N\div 10 \right)-100 \right)5+20 \right)2=N\) Hint: This answer is quite backwards.

A	2 Hint: \(10^3 \times 10^4=10^7\), and note that if you're guessing when the answers are so closely related, you're generally better off guessing one of the middle numbers.
B	20 Hint: \( \dfrac{\left( 4\times {{10}^{3}} \right)\times \left( 3\times {{10}^{4}} \right)}{6\times {{10}^{6}}}=\dfrac {12 \times {{10}^{7}}}{6\times {{10}^{6}}}=\)\(2 \times {{10}^{1}}=20 \)
C	200 Hint: \(10^3 \times 10^4=10^7\)
D	2000 Hint: \(10^3 \times 10^4=10^7\), and note that if you're guessing when the answers are so closely related, you're generally better off guessing one of the middle numbers.

A	\( \large \dfrac{1}{4}\) Hint: Try a simpler question with 3 children -- call them big, medium, and small -- and list all the ways they could line up. Then see how to extend your logic to the problem with 4 children.
B	\( \large \dfrac{1}{256} \) Hint: Try a simpler question with 3 children -- call them big, medium, and small -- and list all the ways they could line up. Then see how to extend your logic to the problem with 4 children.
C	\( \large \dfrac{1}{16}\) Hint: Try a simpler question with 3 children -- call them big, medium, and small -- and list all the ways they could line up. Then see how to extend your logic to the problem with 4 children.
D	\( \large \dfrac{1}{24}\) Hint: The number of ways for the children to line up is \(4!=4 \times 3 \times 2 \times 1 =24\) -- there are 4 choices for who is first in line, then 3 for who is second, etc. Only one of these lines has the children in the order specified.

MTEL General Curriculum Mathematics Practice

Use the table below to answer the question that follows:

Gordon wants to buy three pounds of nuts. Each of the stores above ordinarily sells the nuts for $4.99 a pound, but is offering a discount this week. At which store can he buy the nuts for the least amount of money?

Store A

Store B

Store C

Store D

Which of the following nets will not fold into a cube?

Given that 10 cm is approximately equal to 4 inches, which of the following expressions models a way to find out approximately how many inches are equivalent to 350 cm?

The expression \( \large{{8}^{3}}\cdot {{2}^{-10}}\) is equal to which of the following?

The chairs in a large room can be arranged in rows of 18, 25, or 60 with no chairs left over. If C is the smallest possible number of chairs in the room, which of the following inequalities does C satisfy?

What is the mathematical name of the three-dimensional polyhedron depicted below?

Tetrahedron

Triangular Prism

Triangular Pyramid

Trigon

A family went on a long car trip. Below is a graph of how far they had driven at each hour.

Which of the following is closest to their average speed driving on the trip?

In the triangle below, \(\overline{AC}\cong \overline{AD}\cong \overline{DE}\) and \(m\angle CAD=100{}^\circ \). What is \(m\angle DAE\)?

What is the perimeter of a right triangle with legs of lengths x and 2x?

If x is an integer, which of the following must also be an integer?

Which of the lists below is in order from least to greatest value?

Use the graph below to answer the question that follows.

Which of the following is a correct equation for the graph of the line depicted above?

A publisher prints a series of books with covers made of identical material and using the same thickness of paper for each page. The covers of the book together are 0.4 cm thick, and 125 pieces of the paper used together are 1 cm thick.

The publisher uses a linear function to determine the total thickness, T(n) of a book made with n sheets of paper. What are the slope and intercept of T(n)?

Intercept = 0.4 cm, Slope = 125 cm/page

Intercept =0.4 cm, Slope = \(\dfrac{1}{125}\)cm/page

Intercept = 125 cm, Slope = 0.4 cm

Intercept = \(\dfrac{1}{125}\)cm, Slope = 0.4 pages/cm

Which of the following points is closest to \( \dfrac{34}{135} \times \dfrac{53}{86}\)?

A

B

C

D

A class is using base-ten block to represent numbers. A large cube represents 1000, a flat represents 100, a rod represents 10, and a little cube represents 1. Which of these is not a correct representation for 2,347?

23 flats, 4 rods, 7 little cubes

2 large cubes, 3 flats, 47 rods

2 large cubes, 34 rods, 7 little cubes

2 large cubes, 3 flats, 4 rods, 7 little cubes

How many lines of reflective symmetry and how many centers of rotational symmetry does the parallelogram depicted below have?

4 lines of reflective symmetry, 1 center of rotational symmetry.

2 lines of reflective symmetry, 1 center of rotational symmetry.

0 lines of reflective symmetry, 1 center of rotational symmetry.

2 lines of reflective symmetry, 0 centers of rotational symmetry.

The Venn Diagram below gives data on the number of seniors, athletes, and vegetarians in the student body at a college:

How many students at the college are seniors who are not vegetarians?

There are six gumballs in a bag — two red and four green. Six children take turns picking a gumball out of the bag without looking. They do not return any gumballs to the bag. What is the probability that the first two children to pick from the bag pick the red gumballs?

Use the expression below to answer the question that follows.

\( \large 3\times {{10}^{4}}+2.2\times {{10}^{2}}\)

Which of the following is closest to the expression above?

Five million

Fifty thousand

Three million

Thirty thousand

Which of the numbers below is not equivalent to 4%?

The table below gives data from various years on how many young girls drank milk.

Based on the data given above, what was the probability that a randomly chosen girl in 1990 drank milk?

The histogram below shows the number of pairs of footware owned by a group of college students.

Which of the following statements can be inferred from the graph above?

The median number of pairs of footware owned is between 50 and 60 pairs.

The mode of the number of pairs of footware owned is 20.

The mean number of pairs of footware owned is less than the median number of pairs of footware owned.

The median number of pairs of footware owned is between 10 and 20.

Cell phone plan A charges $3 per month plus $0.10 per minute. Cell phone plan B charges $29.99 per month, with no fee for the first 400 minutes and then $0.20 for each additional minute.

Which equation can be used to solve for the number of minutes, m (with m>400) that a person would have to spend on the phone each month in order for the bills for plan A and plan B to be equal?

The least common multiple of 60 and N is 1260. Which of the following could be the prime factorization of N?

The expression \( \large {{7}^{-4}}\cdot {{8}^{-6}}\) is equal to which of the following?

Which of the following values of x satisfies the inequality \( \large \left| {{(x+2)}^{3}} \right|<3?\)

A family on vacation drove the first 200 miles in 4 hours and the second 200 miles in 5 hours. Which expression below gives their average speed for the entire trip?

Which of the graphs below represent functions?

I and IV only.

I and III only.

II and III only.

I, II, and IV only.

Use the problem below to answer the question that follows:

T shirts are on sale for 20% off. Tasha paid $8.73 for a shirt. What is the regular price of the shirt? There is no tax on clothing purchases under $175.

Let p represent the regular price of these t-shirt. Which of the following equations is correct?

In January 2011, the national debt was about 14 trillion dollars and the US population was about 300 million people. Someone reading these figures estimated that the national debt was about $5,000 per person. Which of these statements best describes the reasonableness of this estimate?

A	\( \large 2.75\) Hint: There were 20 students who took the quiz. Total points earned: \(2 \times 1+6 \times 2+ 7\times 3+5 \times 4=55\), and 55/20 = 2.75.
B	\( \large 2\) Hint: How many students are there total? Did you count them all?
C	\( \large 3\) Hint: How many students are there total? Did you count them all? Be sure you're finding the mean, not the median or the mode.
D	\( \large 2.5\) Hint: How many students are there total? Did you count them all? Don't just take the mean of 1, 2, 3, 4 -- you have to weight them properly.

A	\( \dfrac{4}{7}\) is \( \dfrac{3}{7}\)away from 1, and \( \dfrac{5}{8}\) is \( \dfrac{3}{8}\)away from 1. Since eighth‘s are smaller than seventh‘s, \( \dfrac{5}{8}\) is closer to 1, and is the greater of the two fractions.
B	\( 7-4=3\) and \( 8-5=3\), so the fractions are equal. Hint: Not how to compare fractions. By this logic, 1/2 and 3/4 are equal, but 1/2 and 2/4 are not.
C	\( 4\times 8=32\) and \( 7\times 5=35\). Since \( 32<35\) , \( \dfrac{5}{8}<\dfrac{4}{7}\) Hint: Starts out as something that works, but the conclusion is wrong. 4/7 = 32/56 and 5/8 = 35/56. The cross multiplication gives the numerators, and 35/56 is bigger.
D	\( 4<5\) and \( 7<8\), so \( \dfrac{4}{7}<\dfrac{5}{8}\) Hint: Conclusion is correct, logic is wrong. With this reasoning, 1/2 would be less than 2/100,000.

A	\( \large t=s+15\) Hint: When there are 2 teachers, how many students should there be? Do those values satisfy this equation?
B	\( \large s=t+15\) Hint: When there are 2 teachers, how many students should there be? Do those values satisfy this equation?
C	\( \large t=15s\) Hint: This is a really easy mistake to make, which comes from transcribing directly from English, "1 teachers equals 15 students." To see that it's wrong, plug in s=2; do you really need 30 teachers for 2 students? To avoid this mistake, insert the word "number," "Number of teachers equals 15 times number of students" is more clearly problematic.
D	\( \large s=15t\)

A	\( \large x < 0\) Hint: If x =0, then x/2 = x/3, so this answer can't be correct.
B	\( \large x \le 0\)
C	\( \large x > 0\) Hint: If x =0, then x/2 = x/3, so this answer can't be correct.
D	\( \large x \ge 0\) Hint: Try plugging in x = 6.

A	\( \large x=3\) Hint: Try plugging x=3 into the equation.
B	\( \large x=-3\) Hint: Left side is positive, right side is negative when you plug this in for x.
C	\( \large x=\dfrac{3}{2}\) Hint: One way to solve: \(4=\dfrac{2}{3}x+2x\) \(=\dfrac{8}{3}x\).\(x=\dfrac{3 \times 4}{8}=\dfrac{3}{2}\). Another way is to just plug x=3/2 into the equation and see that each side equals 3 -- on a multiple choice test, you almost never have to actually solve for x.
D	\( \large x=-\dfrac{3}{2}\) Hint: Left side is positive, right side is negative when you plug this in for x.

A	\( \large \sqrt[3]{8}\) Hint: This answer is the cube root of 8. Since 2 x 2 x 2 =8, this is equal to 2, which is rational because 2 = 2/1.
B	\( \large \sqrt{8}\) Hint: It is not trivial to prove that this is irrational, but you can get this answer by eliminating the other choices.
C	\( \large \dfrac{1}{8}\) Hint: 1/8 is the RATIO of two integers, so it is rational.
D	\( \large -8\) Hint: Negative integers are also rational, -8 = -8/1, a ratio of integers.

A	\( \large \$2.60+\$0.40d\) Hint: It's 40 cents for 1/7 of a mile, not per mile.
B	\( \large \$2.60+\$0.40\dfrac{d}{7}\) Hint: According to this equation, going 7 miles would cost $3; does that make sense?
C	\( \large \$2.20+\$2.80d\) Hint: You can think of the fare as $2.20 to enter the cab, and then $0.40 for each 1/7 of a mile, including the first 1/7 of a mile (or $2.80 per mile). Alternatively, you pay $2.60 for the first 1/7 of a mile, and then $2.80 per mile for d-1/7 miles. The total is 2.60+2.80(d-1/7) = 2.60+ 2.80d -.40 = 2.20+2.80d.
D	\( \large \$2.60+\$2.80d\) Hint: Don't count the first 1/7 of a mile twice.

A	\( \large 11,400,000\) Hint: That's eleven million four hundred thousand.
B	\(\large11,000,400,000\)
C	\( \large11,000,000,400,000\) Hint: That's eleven trillion four hundred thousand (although with British conventions; this answer is correct, but in the US, it isn't).
D	\( \large 11,400,000,000\) Hint: That's eleven billion four hundred million

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