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

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 1 Explanation:

Topic: Laws of Exponents (Objective 0019).

Question 2

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 2 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 3

What fraction of the area of the picture below is shaded?

$q21 fraction area$

A	$ \large \dfrac{17}{24}$ Hint: You might try adding segments so each quadrant is divided into 6 pieces with equal area -- there will be 24 regions, not all the same shape, but all the same area, with 17 of them shaded (for the top left quarter, you could also first change the diagonal line to a horizontal or vertical line that divides the square in two equal pieces and shade one) .
B	$ \large \dfrac{3}{4}$ Hint: Be sure you're taking into account the different sizes of the pieces.
C	$ \large \dfrac{2}{3}$ Hint: The bottom half of the picture is 2/3 shaded, and the top half is more than 2/3 shaded, so this answer is too small.
D	$ \large \dfrac{17}{6} $ Hint: This answer is bigger than 1, so doesn't make any sense. Be sure you are using the whole picture, not one quadrant, as the unit.

Question 3 Explanation:

Topic: Models of Fractions (Objective 0017)

Question 4

A homeowner is planning to tile the kitchen floor with tiles that measure 6 inches by 8 inches. The kitchen floor is a rectangle that measures 10 ft by 12 ft, and there are no gaps between the tiles. How many tiles does the homeowner need?

A	30 Hint: The floor is 120 sq feet, and the tiles are smaller than 1 sq foot. Also, remember that 1 sq foot is 12 $\times$ 12=144 sq inches.
B	120 Hint: The floor is 120 sq feet, and the tiles are smaller than 1 sq foot.
C	300 Hint: Recheck your calculations.
D	360 Hint: One way to do this is to note that 6 inches = 1/2 foot and 8 inches = 2/3 foot, so the area of each tile is 1/2 $\times$ 2/3=1/3 sq foot, or each square foot of floor requires 3 tiles. The area of the floor is 120 square feet. Note that the tiles would fit evenly oriented in either direction, parallel to the walls.

Question 4 Explanation:

Topic: Estimate and calculate measurements, use unit conversions to solve measurement problems, solve measurement problems in real-world situations (Objective 0023).

Question 5

In which table below is y a function of x?

A	Hint: If x=3, y can have two different values, so it's not a function.
B	Hint: If x=3, y can have two different values, so it's not a function.
C	Hint: If x=1, y can have different values, so it's not a function.
D	Hint: Each value of x always corresponds to the same value of y.

Question 5 Explanation:

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

Question 6

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 6 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 7

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 7 Explanation:

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

Question 8

In March of 2012, 1 dollar was worth the same as 0.761 Euros, and 1 dollar was also worth the same as 83.03 Japanese Yen. Which of the expressions below gives the number of Yen that are worth 1 Euro?

A	$ \large {83}.0{3}\cdot 0.{761}$ Hint: This equation gives less than the number of yen per dollar, but 1 Euro is worth more than 1 dollar.
B	$ \large \dfrac{0.{761}}{{83}.0{3}}$ Hint: Number is way too small.
C	$ \large \dfrac{{83}.0{3}}{0.{761}}$ Hint: One strategy here is to use easier numbers, say 1 dollar = .5 Euros and 100 yen, then 1 Euro would be 200 Yen (change the numbers in the equations and see what works). Another is to use dimensional analysis: we want # yen per Euro, or yen/Euro = yen/dollar $\times$ dollar/Euro = $83.03 \times \dfrac {1}{0.761}$
D	$ \large \dfrac{1}{0.{761}}\cdot \dfrac{1}{{83}.0{3}}$ Hint: Number is way too small.

Question 8 Explanation:

Topic: Analyze the relationships among proportions, constant rates, and linear functions (Objective 0022).

Question 9

A cylindrical soup can has diameter 7 cm and height 11 cm. The can holds g grams of soup. How many grams of the same soup could a cylindrical can with diameter 14 cm and height 33 cm hold?

A	$ \large 6g$ Hint: You must scale in all three dimensions.
B	$ \large 12g$ Hint: Height is multiplied by 3, and diameter and radius are multiplied by 2. Since the radius is squared, final result is multiplied by $2^2\times 3=12$.
C	$ \large 18g$ Hint: Don't square the height scale factor.
D	$ \large 36g$ Hint: Don't square the height scale factor.

Question 9 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 10

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 10 Explanation:

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

Question 11

Use the four figures below to answer the question that follows:

How many of the figures pictured above have at least one line of reflective symmetry?

A	$ \large 1$
B	$ \large 2$ Hint: The ellipse has 2 lines of reflective symmetry (horizontal and vertical, through the center) and the triangle has 3. The other two figures have rotational symmetry, but not reflective symmetry.
C	$ \large 3$
D	$ \large 4$ Hint: All four have rotational symmetry, but not reflective symmetry.

Question 11 Explanation:

Topic: Analyze and apply geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to concepts of symmetry, similarity, and congruence; and use these concepts to solve problems (Objective 0024).

Question 12

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 12 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 13

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 13 Explanation:

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

Question 14

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 14 Explanation:

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

Question 15

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 15 Explanation:

Topic: Least Common Multiple (Objective 0018)

Question 16

What set of transformations will transform the leftmost image into the rightmost image?

A	A 90 degree clockwise rotation about (2,1) followed by a translation of two units to the right. Hint: Part of the figure would move below the x-axis with these transformations.
B	A translation 3 units up, followed by a reflection about the line y=x. Hint: See what happens to the point (5,1) under this set of transformations.
C	A 90 degree clockwise rotation about (5,1), followed by a translation of 2 units up.
D	A 90 degree clockwise rotation about (2,1) followed by a translation of 2 units to the right. Hint: See what happens to the point (3,3) under this set of transformations.

Question 16 Explanation:

Topic:Analyze and apply geometric transformations (e.g., translations, rotations, reflections, dilations) (Objective 0024).

Question 17

Use the samples of a student's work below to answer the question that follows:

This student divides fractions by first finding a common denominator, then dividing the numerators.

$ \large \dfrac{2}{3} \div \dfrac{3}{4} \longrightarrow \dfrac{8}{12} \div \dfrac{9}{12} \longrightarrow 8 \div 9 = \dfrac {8}{9}$ $ \large \dfrac{2}{5} \div \dfrac{7}{20} \longrightarrow \dfrac{8}{20} \div \dfrac{7}{20} \longrightarrow 8 \div 7 = \dfrac {8}{7}$ $ \large \dfrac{7}{6} \div \dfrac{3}{4} \longrightarrow \dfrac{14}{12} \div \dfrac{9}{12} \longrightarrow 14 \div 9 = \dfrac {14}{9}$

Which of the following best describes the mathematical validity of the algorithm the student is using?

A	It is not valid. Common denominators are for adding and subtracting fractions, not for dividing them. Hint: Don't be so rigid! Usually there's more than one way to do something in math.
B	It got the right answer in these three cases, but it isn‘t valid for all rational numbers. Hint: Did you try some other examples? What makes you say it's not valid?
C	It is valid if the rational numbers in the division problem are in lowest terms and the divisor is not zero. Hint: Lowest terms doesn't affect this problem at all.
D	It is valid for all rational numbers, as long as the divisor is not zero. Hint: When we have common denominators, the problem is in the form a/b divided by c/b, and the answer is a/c, as the student's algorithm predicts.

Question 17 Explanation:

Topic: Analyze Non-Standard Computational Algorithms (Objective 0019).

Question 18

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 18 Explanation:

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

Question 19

Which of the lists below contains only irrational numbers?

A	$ \large\pi , \quad \sqrt{6},\quad \sqrt{\dfrac{1}{2}}$
B	$ \large\pi , \quad \sqrt{9}, \quad \pi +1$ Hint: $ \sqrt{9}=3$
C	$ \large\dfrac{1}{3},\quad \dfrac{5}{4},\quad \dfrac{2}{9}$ Hint: These are all rational.
D	$ \large-3,\quad 14,\quad 0$ Hint: These are all rational.

Question 19 Explanation:

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

Question 20

Which of the following is equivalent to $ \dfrac{3}{4}-\dfrac{1}{8}+\dfrac{2}{8}\times \dfrac{1}{2}?$

A	$ \large \dfrac{7}{16}$ Hint: Multiplication comes before addition and subtraction in the order of operations.
B	$ \large \dfrac{1}{2}$ Hint: Addition and subtraction are of equal priority in the order of operations -- do them left to right.
C	$ \large \dfrac{3}{4}$ Hint: $ \dfrac{3}{4}-\dfrac{1}{8}+\dfrac{2}{8}\times \dfrac{1}{2}$=$ \dfrac{3}{4}-\dfrac{1}{8}+\dfrac{1}{8}$=$ \dfrac{3}{4}+-\dfrac{1}{8}+\dfrac{1}{8}$=$ \dfrac{3}{4}$
D	$ \large \dfrac{3}{16}$ Hint: Multiplication comes before addition and subtraction in the order of operations.

Question 20 Explanation:

Topic: Operations on Fractions, Order of Operations (Objective 0019).

Question 21

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 21 Explanation:

Topic: Place Value (Objective 0016)

Question 22

Which of the numbers below is a fraction equivalent to $ 0.\bar{6}$?

A	$ \large \dfrac{4}{6}$ Hint: $ 0.\bar{6}=\dfrac{2}{3}=\dfrac{4}{6}$
B	$ \large \dfrac{3}{5}$ Hint: This is equal to 0.6, without the repeating decimal. Answer is equivalent to choice c, which is another way to tell that it's wrong.
C	$ \large \dfrac{6}{10}$ Hint: This is equal to 0.6, without the repeating decimal. Answer is equivalent to choice b, which is another way to tell that it's wrong.
D	$ \large \dfrac{1}{6}$ Hint: This is less than a half, and $ 0.\bar{6}$ is greater than a half.

Question 22 Explanation:

Topic: Converting between fraction and decimal representations (Objective 0017)

Question 23

Use the expression below to answer the question that follows:

$ \large \dfrac{\left( 7,154 \right)\times \left( 896 \right)}{216}$

Which of the following is the best estimate of the expression above?

A	2,000 Hint: The answer is bigger than 7,000.
B	20,000 Hint: Estimate 896/216 first.
C	3,000 Hint: The answer is bigger than 7,000.
D	30,000 Hint: $ \dfrac{896}{216} \approx 4$ and $7154 \times 4$ is over 28,000, so this answer is closest.

Question 23 Explanation:

Topics: Estimation, simplifying fractions (Objective 0016, overlaps with other objectives).

Question 24

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 24 Explanation:

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

Question 25

On a map the distance from Boston to Detroit is 6 cm, and these two cities are 702 miles away from each other. Assuming the scale of the map is the same throughout, which answer below is closest to the distance between Boston and San Francisco on the map, given that they are 2,708 miles away from each other?

A	21 cm Hint: How many miles would correspond to 24 cm on the map? Try adjusting from there.
B	22 cm Hint: How many miles would correspond to 24 cm on the map? Try adjusting from there.
C	23 cm Hint: One way to solve this without a calculator is to note that 4 groups of 6 cm is 2808 miles, which is 100 miles too much. Then 100 miles would be about 1/7 th of 6 cm, or about 1 cm less than 24 cm.
D	24 cm Hint: 4 groups of 6 cm is over 2800 miles on the map, which is too much.

Question 25 Explanation:

Topic: Apply proportional thinking to estimate quantities in real world situations (Objective 0019).

Question 26

The table below gives the result of a survey at a college, asking students whether they were residents or commuters:

Based on the above data, what is the probability that a randomly chosen commuter student is a junior or a senior?

A	$ \large \dfrac{34}{43}$
B	$ \large \dfrac{34}{71}$ Hint: This is the probability that a randomly chosen junior or senior is a commuter student.
C	$ \large \dfrac{34}{147}$ Hint: This is the probability that a randomly chosen student is a junior or senior who is a commuter.
D	$ \large \dfrac{71}{147}$ Hint: This is the probability that a randomly chosen student is a junior or a senior.

Question 26 Explanation:

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

Question 27

Here is a method that a student used for subtraction:

Which of the following is correct?

A	The student used a method that worked for this problem and can be generalized to any subtraction problem. Hint: Note that this algorithm is taught as the "standard" algorithm in much of Europe (it's where the term "borrowing" came from -- you borrow on top and "pay back" on the bottom).
B	The student used a method that worked for this problem and that will work for any subtraction problem that only requires one regrouping; it will not work if more regrouping is required. Hint: Try some more examples.
C	The student used a method that worked for this problem and will work for all three-digit subtraction problems, but will not work for larger problems. Hint: Try some more examples.
D	The student used a method that does not work. The student made two mistakes that cancelled each other out and was lucky to get the right answer for this problem. Hint: Remember, there are many ways to do subtraction; there is no one "right" algorithm.

Question 27 Explanation:

Topic: Analyze and justify standard and non-standard computational techniques (Objective 0019).

Question 28

Use the graph below to answer the question that follows.

If the polygon shown above is reflected about the y axis and then rotated 90 degrees clockwise about the origin, which of the following graphs is the result?

A	Hint: Try following the point (1,4) to see where it goes after each transformation.
B
C	Hint: Make sure you're reflecting in the correct axis.
D	Hint: Make sure you're rotating the correct direction.

Question 28 Explanation:

Question 29

Each individual cube that makes up the rectangular solid depicted below has 6 inch sides. What is the surface area of the solid in square feet?

A	$ \large 11\text{ f}{{\text{t}}^{2}}$ Hint: Check your units and make sure you're using feet and inches consistently.
B	$ \large 16.5\text{ f}{{\text{t}}^{2}}$ Hint: Each square has surface area $\dfrac{1}{2} \times \dfrac {1}{2}=\dfrac {1}{4}$ sq feet. There are 9 squares on the top and bottom, and 12 on each of 4 sides, for a total of 66 squares. 66 squares $\times \dfrac {1}{4}$ sq feet/square =16.5 sq feet.
C	$ \large 66\text{ f}{{\text{t}}^{2}}$ Hint: The area of each square is not 1.
D	$ \large 2376\text{ f}{{\text{t}}^{2}}$ Hint: Read the question more carefully -- the answer is supposed to be in sq feet, not sq inches.

Question 29 Explanation:

Topics: Use unit conversions to solve measurement problems, and derive and use formulas for calculating surface areas of geometric shapes and figures (Objective 0023).

Question 30

Which of the following is the equation of a linear function?

A	$ \large y={{x}^{2}}+2x+7$ Hint: This is a quadratic function.
B	$ \large y={{2}^{x}}$ Hint: This is an exponential function.
C	$ \large y=\dfrac{15}{x}$ Hint: This is an inverse function.
D	$ \large y=x+(x+4)$ Hint: This is a linear function, y=2x+4, it's graph is a straight line with slope 2 and y-intercept 4.

Question 30 Explanation:

Topic: Distinguish between linear and nonlinear functions (Objective 0022).

Question 31

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 31 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 32

The equation $ \large F=\frac{9}{5}C+32$ is used to convert a temperature measured in Celsius to the equivalent Farentheit temperature.

A patient's temperature increased by 1.5° Celcius. By how many degrees Fahrenheit did her temperature increase?

A	1.5° Hint: Celsius and Fahrenheit don't increase at the same rate.
B	1.8° Hint: That's how much the Fahrenheit temp increases when the Celsius temp goes up by 1 degree.
C	2.7° Hint: Each degree increase in Celsius corresponds to a $\dfrac{9}{5}=1.8$ degree increase in Fahrenheit. Thus the increase is 1.8+0.9=2.7.
D	Not enough information. Hint: A linear equation has constant slope, which means that every increase of the same amount in one variable, gives a constant increase in the other variable. It doesn't matter what temperature the patient started out at.

Question 32 Explanation:

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

Question 33

The prime factorization of n can be written as n=pqr, where p, q, and r are distinct prime numbers. How many factors does n have, including 1 and itself?

A	$ \large3$ Hint: 1, p, q, r, and pqr are already 5, so this isn't enough. You might try plugging in p=2, q=3, and r=5 to help with this problem.
B	$ \large5$ Hint: Don't forget pq, etc. You might try plugging in p=2, q=3, and r=5 to help with this problem.
C	$ \large6$ Hint: You might try plugging in p=2, q=3, and r=5 to help with this problem.
D	$ \large8$ Hint: 1, p, q, r, pq, pr, qr, pqr.

Question 33 Explanation:

Topic: Recognize uses of prime factorization of a number (Objective 0018).

Question 34

M is a multiple of 26. Which of the following cannot be true?

A	M is odd. Hint: All multiples of 26 are also multiples of 2, so they must be even.
B	M is a multiple of 3. Hint: 3 x 26 is a multiple of both 3 and 26.
C	M is 26. Hint: 1 x 26 is a multiple of 26.
D	M is 0. Hint: 0 x 26 is a multiple of 26.

Question 34 Explanation:

Topic: Characteristics of composite numbers (Objective 0018).

Question 35

The first histogram shows the average life expectancies for women in different countries in Africa in 1998; the second histogram gives similar data for Europe:

How much bigger is the range of the data for Africa than the range of the data for Europe?

A	0 years Hint: Range is the maximum life expectancy minus the minimum life expectancy.
B	12 years Hint: Are you subtracting frequencies? Range is about values of the data, not frequency.
C	18 years Hint: It's a little hard to read the graph, but it doesn't matter if you're consistent. It looks like the range for Africa is 80-38= 42 years and for Europe is 88-64 = 24; 42-24=18.
D	42 years Hint: Read the question more carefully.

Question 35 Explanation:

Topic: Compare different data sets (Objective 0025).

Question 36

If two fair coins are flipped, what is the probability that one will come up heads and the other tails?

A	$ \large \dfrac{1}{4}$ Hint: Think of the coins as a penny and a dime, and list all possibilities.
B	$ \large \dfrac{1}{3} $ Hint: This is a very common misconception. There are three possible outcomes -- both heads, both tails, and one of each -- but they are not equally likely. Think of the coins as a penny and a dime, and list all possibilities.
C	$ \large \dfrac{1}{2}$ Hint: The possibilities are HH, HT, TH, TT, and all are equally likely. Two of the four have one of each coin, so the probability is 2/4=1/2.
D	$ \large \dfrac{3}{4}$ Hint: Think of the coins as a penny and a dime, and list all possibilities.

Question 36 Explanation:

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

Question 37

Below are front, side, and top views of a three-dimensional solid.

Which of the following could be the solid shown above?

A	A sphere Hint: All views would be circles.
B	A cylinder
C	A cone Hint: Two views would be triangles, not rectangles.
D	A pyramid Hint: How would one view be a circle?

Question 37 Explanation:

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

Question 38

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 38 Explanation:

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

Question 39

The chart below gives percentiles for the number of sit-ups that boys of various ages can do in 60 seconds (source , June 24, 2011)

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

A	95% of 12 year old boys can do 56 sit-ups in 60 seconds. Hint: The 95th percentile means that 95% of scores are less than or equal to 56, and 5% are greater than or equal to 56.
B	At most 25% of 7 year old boys can do 19 or more sit-ups in 60 seconds. Hint: The 25th percentile means that 25% of scores are less than or equal to 19, and 75% are greater than or equal to 19.
C	Half of all 13 year old boys can do less than 41 sit-ups in 60 seconds and half can do more than 41 sit-ups in 60 seconds. Hint: Close, but not quite. There's no accounting for boys who can do exactly 41 sit ups. Look at these data: 10, 20, 41, 41, 41, 41, 50, 60, 90. The median is 41, but more than half can do 41 or more.
D	At least 75% of 16 year old boys can only do 51 or fewer sit-ups in 60 seconds. Hint: The "at least" is necessary due to duplicates. Suppose the data were 10, 20, 51, 51. The 75th percentile is 51, but 100% of the boys can only do 51 or fewer situps.

Question 39 Explanation:

Topic: Analyze and interpret various graphic and nongraphic data representations (e.g., frequency distributions, percentiles) (Objective 0025).

Question 40

Below are four inputs and outputs for a function machine representing the function A:

Which of the following equations could also represent A for the values shown?

A	$ \large A(n)=n+4$ Hint: For a question like this, you don't have to find the equation yourself, you can just try plugging the function machine inputs into the equation, and see if any values come out wrong. With this equation n= -1 would output 3, not 0 as the machine does.
B	$ \large A(n)=n+2$ Hint: For a question like this, you don't have to find the equation yourself, you can just try plugging the function machine inputs into the equation, and see if any values come out wrong. With this equation n= 2 would output 4, not 6 as the machine does.
C	$ \large A(n)=2n+2$ Hint: Simply plug in each of the four function machine input values, and see that the equation produces the correct output, e.g. A(2)=6, A(-1)=0, etc.
D	$ \large A(n)=2\left( n+2 \right)$ Hint: For a question like this, you don't have to find the equation yourself, you can just try plugging the function machine inputs into the equation, and see if any values come out wrong. With this equation n= 2 would output 8, not 6 as the machine does.

Question 40 Explanation:

Topics: Understand various representations of functions, and translate among different representations of functional relationships (Objective 0021).

Question 41

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 41 Explanation:

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

Question 42

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 42 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 43

The "houses" below are made of toothpicks and gum drops.

How many toothpicks are there in a row of 53 houses?

A	212 Hint: Can the number of toothpicks be even?
B	213 Hint: One way to see this is that every new "house" adds 4 toothpicks to the leftmost vertical toothpick -- so the total number is 1 plus 4 times the number of "houses." There are many other ways to look at the problem too.
C	217 Hint: Try your strategy with a smaller number of "houses" so you can count and find your mistake.
D	265 Hint: Remember that the "houses" overlap some walls.

Question 43 Explanation:

Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic). (Objective 0021).

Question 44

What is the probability that two randomly selected people were born on the same day of the week? Assume that all days are equally probable.

A	$ \large \dfrac{1}{7}$ Hint: It doesn't matter what day the first person was born on. The probability that the second person will match is 1/7 (just designate one person the first and the other the second). Another way to look at it is that if you list the sample space of all possible pairs, e.g. (Wed, Sun), there are 49 such pairs, and 7 of them are repeats of the same day, and 7/49=1/7.
B	$ \large \dfrac{1}{14}$ Hint: What would be the sample space here? Ie, how would you list 14 things that you pick one from?
C	$ \large \dfrac{1}{42}$ Hint: If you wrote the seven days of the week on pieces of paper and put the papers in a jar, this would be the probability that the first person picked Sunday and the second picked Monday from the jar -- not the same situation.
D	$ \large \dfrac{1}{49}$ Hint: This is the probability that they are both born on a particular day, e.g. Sunday.

Question 44 Explanation:

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

Question 45

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 45 Explanation:

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

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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	\( \large 0.8p=\$8.73\) 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	\( \large \dfrac{17}{24}\) Hint: You might try adding segments so each quadrant is divided into 6 pieces with equal area -- there will be 24 regions, not all the same shape, but all the same area, with 17 of them shaded (for the top left quarter, you could also first change the diagonal line to a horizontal or vertical line that divides the square in two equal pieces and shade one) .
B	\( \large \dfrac{3}{4}\) Hint: Be sure you're taking into account the different sizes of the pieces.
C	\( \large \dfrac{2}{3}\) Hint: The bottom half of the picture is 2/3 shaded, and the top half is more than 2/3 shaded, so this answer is too small.
D	\( \large \dfrac{17}{6} \) Hint: This answer is bigger than 1, so doesn't make any sense. Be sure you are using the whole picture, not one quadrant, as the unit.

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	\( \large t=s+15\) 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	\( \large {83}.0{3}\cdot 0.{761}\) Hint: This equation gives less than the number of yen per dollar, but 1 Euro is worth more than 1 dollar.
B	\( \large \dfrac{0.{761}}{{83}.0{3}}\) Hint: Number is way too small.
C	\( \large \dfrac{{83}.0{3}}{0.{761}}\) Hint: One strategy here is to use easier numbers, say 1 dollar = .5 Euros and 100 yen, then 1 Euro would be 200 Yen (change the numbers in the equations and see what works). Another is to use dimensional analysis: we want # yen per Euro, or yen/Euro = yen/dollar \(\times\) dollar/Euro = \(83.03 \times \dfrac {1}{0.761}\)
D	\( \large \dfrac{1}{0.{761}}\cdot \dfrac{1}{{83}.0{3}}\) Hint: Number is way too small.

A	\( \large 6g\) Hint: You must scale in all three dimensions.
B	\( \large 12g\) Hint: Height is multiplied by 3, and diameter and radius are multiplied by 2. Since the radius is squared, final result is multiplied by \(2^2\times 3=12\).
C	\( \large 18g\) Hint: Don't square the height scale factor.
D	\( \large 36g\) Hint: Don't square the height scale factor.

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	\( \large 1\)
B	\( \large 2\) Hint: The ellipse has 2 lines of reflective symmetry (horizontal and vertical, through the center) and the triangle has 3. The other two figures have rotational symmetry, but not reflective symmetry.
C	\( \large 3\)
D	\( \large 4\) Hint: All four have rotational symmetry, but not reflective symmetry.

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.

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	\( \large2\cdot 5\cdot 7\) 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	\( \large\pi , \quad \sqrt{6},\quad \sqrt{\dfrac{1}{2}}\)
B	\( \large\pi , \quad \sqrt{9}, \quad \pi +1\) Hint: \( \sqrt{9}=3\)
C	\( \large\dfrac{1}{3},\quad \dfrac{5}{4},\quad \dfrac{2}{9}\) Hint: These are all rational.
D	\( \large-3,\quad 14,\quad 0\) Hint: These are all rational.

A	\( \large \dfrac{7}{16}\) Hint: Multiplication comes before addition and subtraction in the order of operations.
B	\( \large \dfrac{1}{2}\) Hint: Addition and subtraction are of equal priority in the order of operations -- do them left to right.
C	\( \large \dfrac{3}{4}\) Hint: \( \dfrac{3}{4}-\dfrac{1}{8}+\dfrac{2}{8}\times \dfrac{1}{2}\)=\( \dfrac{3}{4}-\dfrac{1}{8}+\dfrac{1}{8}\)=\( \dfrac{3}{4}+-\dfrac{1}{8}+\dfrac{1}{8}\)=\( \dfrac{3}{4}\)
D	\( \large \dfrac{3}{16}\) Hint: Multiplication comes before addition and subtraction in the order of operations.

A	\( \large \dfrac{4}{6}\) Hint: \( 0.\bar{6}=\dfrac{2}{3}=\dfrac{4}{6}\)
B	\( \large \dfrac{3}{5}\) Hint: This is equal to 0.6, without the repeating decimal. Answer is equivalent to choice c, which is another way to tell that it's wrong.
C	\( \large \dfrac{6}{10}\) Hint: This is equal to 0.6, without the repeating decimal. Answer is equivalent to choice b, which is another way to tell that it's wrong.
D	\( \large \dfrac{1}{6}\) Hint: This is less than a half, and \( 0.\bar{6}\) is greater than a half.

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	\( \large \dfrac{34}{43}\)
B	\( \large \dfrac{34}{71}\) Hint: This is the probability that a randomly chosen junior or senior is a commuter student.
C	\( \large \dfrac{34}{147}\) Hint: This is the probability that a randomly chosen student is a junior or senior who is a commuter.
D	\( \large \dfrac{71}{147}\) Hint: This is the probability that a randomly chosen student is a junior or a senior.

A	\( \large 11\text{ f}{{\text{t}}^{2}}\) Hint: Check your units and make sure you're using feet and inches consistently.
B	\( \large 16.5\text{ f}{{\text{t}}^{2}}\) Hint: Each square has surface area \(\dfrac{1}{2} \times \dfrac {1}{2}=\dfrac {1}{4}\) sq feet. There are 9 squares on the top and bottom, and 12 on each of 4 sides, for a total of 66 squares. 66 squares \(\times \dfrac {1}{4}\) sq feet/square =16.5 sq feet.
C	\( \large 66\text{ f}{{\text{t}}^{2}}\) Hint: The area of each square is not 1.
D	\( \large 2376\text{ f}{{\text{t}}^{2}}\) Hint: Read the question more carefully -- the answer is supposed to be in sq feet, not sq inches.

A	\( \large y={{x}^{2}}+2x+7\) Hint: This is a quadratic function.
B	\( \large y={{2}^{x}}\) Hint: This is an exponential function.
C	\( \large y=\dfrac{15}{x}\) Hint: This is an inverse function.
D	\( \large y=x+(x+4)\) Hint: This is a linear function, y=2x+4, it's graph is a straight line with slope 2 and y-intercept 4.

A	\( \large3\) Hint: 1, p, q, r, and pqr are already 5, so this isn't enough. You might try plugging in p=2, q=3, and r=5 to help with this problem.
B	\( \large5\) Hint: Don't forget pq, etc. You might try plugging in p=2, q=3, and r=5 to help with this problem.
C	\( \large6\) Hint: You might try plugging in p=2, q=3, and r=5 to help with this problem.
D	\( \large8\) Hint: 1, p, q, r, pq, pr, qr, pqr.

A	\( \large \dfrac{1}{4}\) Hint: Think of the coins as a penny and a dime, and list all possibilities.
B	\( \large \dfrac{1}{3} \) Hint: This is a very common misconception. There are three possible outcomes -- both heads, both tails, and one of each -- but they are not equally likely. Think of the coins as a penny and a dime, and list all possibilities.
C	\( \large \dfrac{1}{2}\) Hint: The possibilities are HH, HT, TH, TT, and all are equally likely. Two of the four have one of each coin, so the probability is 2/4=1/2.
D	\( \large \dfrac{3}{4}\) Hint: Think of the coins as a penny and a dime, and list all possibilities.

A	\( \large A(n)=n+4\) Hint: For a question like this, you don't have to find the equation yourself, you can just try plugging the function machine inputs into the equation, and see if any values come out wrong. With this equation n= -1 would output 3, not 0 as the machine does.
B	\( \large A(n)=n+2\) Hint: For a question like this, you don't have to find the equation yourself, you can just try plugging the function machine inputs into the equation, and see if any values come out wrong. With this equation n= 2 would output 4, not 6 as the machine does.
C	\( \large A(n)=2n+2\) Hint: Simply plug in each of the four function machine input values, and see that the equation produces the correct output, e.g. A(2)=6, A(-1)=0, etc.
D	\( \large A(n)=2\left( n+2 \right)\) Hint: For a question like this, you don't have to find the equation yourself, you can just try plugging the function machine inputs into the equation, and see if any values come out wrong. With this equation n= 2 would output 8, not 6 as the machine does.

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	\( \large x < 0\) 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	\( \large \dfrac{1}{7}\) Hint: It doesn't matter what day the first person was born on. The probability that the second person will match is 1/7 (just designate one person the first and the other the second). Another way to look at it is that if you list the sample space of all possible pairs, e.g. (Wed, Sun), there are 49 such pairs, and 7 of them are repeats of the same day, and 7/49=1/7.
B	\( \large \dfrac{1}{14}\) Hint: What would be the sample space here? Ie, how would you list 14 things that you pick one from?
C	\( \large \dfrac{1}{42}\) Hint: If you wrote the seven days of the week on pieces of paper and put the papers in a jar, this would be the probability that the first person picked Sunday and the second picked Monday from the jar -- not the same situation.
D	\( \large \dfrac{1}{49}\) Hint: This is the probability that they are both born on a particular day, e.g. Sunday.

MTEL General Curriculum Mathematics Practice

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

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?

What fraction of the area of the picture below is shaded?

A homeowner is planning to tile the kitchen floor with tiles that measure 6 inches by 8 inches. The kitchen floor is a rectangle that measures 10 ft by 12 ft, and there are no gaps between the tiles. How many tiles does the homeowner need?

30

120

300

360

In which table below is y a function of x?

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?

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?

In March of 2012, 1 dollar was worth the same as 0.761 Euros, and 1 dollar was also worth the same as 83.03 Japanese Yen. Which of the expressions below gives the number of Yen that are worth 1 Euro?

A cylindrical soup can has diameter 7 cm and height 11 cm. The can holds g grams of soup. How many grams of the same soup could a cylindrical can with diameter 14 cm and height 33 cm hold?

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?

Use the four figures below to answer the question that follows:

How many of the figures pictured above have at least one line of reflective symmetry?

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?

The student‘s solution is correct.

The student did not correctly use properties of equality.

The student did not correctly use the distributive property.

The student did not correctly use the commutative property.

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.

Which of the following is an irrational number?

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

What set of transformations will transform the leftmost image into the rightmost image?

A 90 degree clockwise rotation about (2,1) followed by a translation of two units to the right.

A translation 3 units up, followed by a reflection about the line y=x.

A 90 degree clockwise rotation about (5,1), followed by a translation of 2 units up.

A 90 degree clockwise rotation about (2,1) followed by a translation of 2 units to the right.

Use the samples of a student's work below to answer the question that follows:

This student divides fractions by first finding a common denominator, then dividing the numerators.

Which of the following best describes the mathematical validity of the algorithm the student is using?

It is not valid. Common denominators are for adding and subtracting fractions, not for dividing them.

It got the right answer in these three cases, but it isn‘t valid for all rational numbers.

It is valid if the rational numbers in the division problem are in lowest terms and the divisor is not zero.

It is valid for all rational numbers, as long as the divisor is not zero.

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

1.6 cm

16 cm

160 cm

1600 cm

Which of the lists below contains only irrational numbers?

Which of the following is equivalent to \( \dfrac{3}{4}-\dfrac{1}{8}+\dfrac{2}{8}\times \dfrac{1}{2}?\)

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

Which of the numbers below is a fraction equivalent to \( 0.\bar{6}\)?

Use the expression below to answer the question that follows:

\( \large \dfrac{\left( 7,154 \right)\times \left( 896 \right)}{216}\)

Which of the following is the best estimate of the expression above?

2,000

20,000

3,000

30,000

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

21 cm

22 cm

23 cm

24 cm

The table below gives the result of a survey at a college, asking students whether they were residents or commuters:

Based on the above data, what is the probability that a randomly chosen commuter student is a junior or a senior?

Here is a method that a student used for subtraction:

Which of the following is correct?

The student used a method that worked for this problem and can be generalized to any subtraction problem.

The student used a method that worked for this problem and that will work for any subtraction problem that only requires one regrouping; it will not work if more regrouping is required.

The student used a method that worked for this problem and will work for all three-digit subtraction problems, but will not work for larger problems.

The student used a method that does not work. The student made two mistakes that cancelled each other out and was lucky to get the right answer for this problem.

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: \( \left\| {{(1+2)}^{3}} \right\|\)=\( \left \| {3}^3 \right \| \)=\( \left \| 27 \right \| \) = \(27\)

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