20 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

Please wait while the activity loads.
If this activity does not load, try refreshing your browser. Also, this page requires javascript. Please visit using a browser with javascript enabled.

If loading fails, click here to try again

Your answers are highlighted below.

Question 1

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 1 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 2

A solution requires 4 ml of saline for every 7 ml of medicine. How much saline would be required for 50 ml of medicine?

A	$ \large 28 \dfrac{4}{7}$ ml Hint: 49 ml of medicine requires 28 ml of saline. The extra ml of saline requires 4 ml saline/ 7 ml medicine = 4/7 ml saline per 1 ml medicine.
B	$ \large 28 \dfrac{1}{4}$ ml Hint: 49 ml of medicine requires 28 ml of saline. How much saline does the extra ml require?
C	$ \large 28 \dfrac{1}{7}$ ml Hint: 49 ml of medicine requires 28 ml of saline. How much saline does the extra ml require?
D	$ \large 87.5$ ml Hint: 49 ml of medicine requires 28 ml of saline. How much saline does the extra ml require?

Question 2 Explanation:

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

Question 3

A family has four children. What is the probability that two children are girls and two are boys? Assume the the probability of having a boy (or a girl) is 50%.

A	$ \large \dfrac{1}{2}$ Hint: How many different configurations are there from oldest to youngest, e.g. BGGG? How many of them have 2 boys and 2 girls?
B	$ \large \dfrac{1}{4}$ Hint: How many different configurations are there from oldest to youngest, e.g. BGGG? How many of them have 2 boys and 2 girls?
C	$ \large \dfrac{1}{5}$ Hint: Some configurations are more probable than others -- i.e. it's more likely to have two boys and two girls than all boys. Be sure you are weighting properly.
D	$ \large \dfrac{3}{8}$ Hint: There are two possibilities for each child, so there are $2 \times 2 \times 2 \times 2 =16$ different configurations, e.g. from oldest to youngest BBBG, BGGB, GBBB, etc. Of these configurations, there are 6 with two boys and two girls (this is the combination $_{4}C_{2}$ or "4 choose 2"): BBGG, BGBG, BGGB, GGBB, GBGB, and GBBG. Thus the probability is 6/16=3/8.

Question 3 Explanation:

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

Question 4

Which of the following is equivalent to

$ \large A-B+C\div D\times E$?

A	$ \large A-B-\dfrac{C}{DE} $ Hint: In the order of operations, multiplication and division have the same priority, so do them left to right; same with addition and subtraction.
B	$ \large A-B+\dfrac{CE}{D}$ Hint: In practice, you're better off using parentheses than writing an expression like the one in the question. The PEMDAS acronym that many people memorize is misleading. Multiplication and division have equal priority and are done left to right. They have higher priority than addition and subtraction. Addition and subtraction also have equal priority and are done left to right.
C	$ \large \dfrac{AE-BE+CE}{D}$ Hint: Use order of operations, don't just compute left to right.
D	$ \large A-B+\dfrac{C}{DE}$ Hint: In the order of operations, multiplication and division have the same priority, so do them left to right

Question 4 Explanation:

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

Question 5

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 5 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 6

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 6 Explanation:

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

Question 7

Exactly one of the numbers below is a prime number. Which one is it?

A	$ \large511 $ Hint: Divisible by 7.
B	$ \large517$ Hint: Divisible by 11.
C	$ \large519$ Hint: Divisible by 3.
D	$ \large521$

Question 7 Explanation:

Topics: Identify prime and composite numbers and demonstrate knowledge of divisibility rules (Objective 0018).

Question 8

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 8 Explanation:

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

Question 9

Below is a pictorial representation of $2\dfrac{1}{2}\div \dfrac{2}{3}$:

$Q45 frac div model$

Which of the following is the best description of how to find the quotient from the picture?

A	The quotient is $3\dfrac{3}{4}$. There are 3 whole blocks each representing $\dfrac{2}{3}$ and a partial block composed of 3 small rectangles. The 3 small rectangles represent $\dfrac{3}{4}$ of $\dfrac{2}{3}$.
B	The quotient is $3\dfrac{1}{2}$. There are 3 whole blocks each representing $\dfrac{2}{3}$ and a partial block composed of 3 small rectangles. The 3 small rectangles represent $\dfrac{3}{6}$ of a whole, or $\dfrac{1}{2}$. Hint: We are counting how many 2/3's are in 2 1/2: the unit becomes 2/3, not 1.
C	The quotient is $\dfrac{4}{15}$. There are four whole blocks separated into a total of 15 small rectangles. Hint: This explanation doesn't make much sense. Probably you are doing "invert and multiply," but inverting the wrong thing.
D	This picture cannot be used to find the quotient because it does not show how to separate $2\dfrac{1}{2}$ into equal sized groups. Hint: Study the measurement/quotative model of division. It's often very useful with fractions.

Question 9 Explanation:

Topic: Recognize and analyze pictorial representations of number operations. (Objective 0019).

Question 10

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 10 Explanation:

Topic: Solve linear equations (Objective 0020).

Question 11

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

$ \large \dfrac{2}{3}\times \dfrac{3}{4}=\dfrac{4\times 2}{3\times 3}=\dfrac{8}{9}$ $ \large \dfrac{2}{5}\times \dfrac{7}{7}=\dfrac{7\times 2}{5\times 7}=\dfrac{2}{5}$ $ \large \dfrac{7}{6}\times \dfrac{3}{4}=\dfrac{4\times 7}{6\times 3}=\dfrac{28}{18}=\dfrac{14}{9}$

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

A	It is not valid. It never produces the correct answer. Hint: In the middle example,the answer is correct.
B	It is not valid. It produces the correct answer in a few special cases, but it‘s still not a valid algorithm. Hint: Note that this algorithm gives a/b divided by c/d, not a/b x c/d, but some students confuse multiplication and cross-multiplication. If a=0 or if c/d =1, division and multiplication give the same answer.
C	It is valid if the rational numbers in the multiplication problem are in lowest terms. Hint: Lowest terms is irrelevant.
D	It is valid for all rational numbers. Hint: Can't be correct as the first and last examples have the wrong answers.

Question 11 Explanation:

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

Question 12

Use the table below to answer the question that follows:

Each number in the table above represents a value W that is determined by the values of x and y. For example, when x=3 and y=1, W=5. What is the value of W when x=9 and y=14? Assume that the patterns in the table continue as shown.

A	$ \large W=-5$ Hint: When y is even, W is even.
B	$ \large W=4$ Hint: Note that when x increases by 1, W increases by 2, and when y increases by 1, W decreases by 1. At x=y=0, W=0, so at x=9, y=14, W has increased by $9 \times 2$ and decreased by 14, or W=18-14=4.
C	$ \large W=6$ Hint: Try fixing x or y at 0, and start by finding W for x=0 y=14 or x=9, y=0.
D	$ \large W=32$ Hint: Try fixing x or y at 0, and start by finding W for x=0 y=14 or x=9, y=0.

Question 12 Explanation:

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

Question 13

What is the length of side $\overline{BD}$ in the triangle below, where $\angle DBA$ is a right angle?

A	$ \large 1$ Hint: Use the Pythagorean Theorem.
B	$ \large \sqrt{5}$ Hint: $2^2+e^2=3^2$ or $4+e^2=9;e^2=5; e=\sqrt{5}$.
C	$ \large \sqrt{13}$ Hint: e is not the hypotenuse.
D	$ \large 5$ Hint: Use the Pythagorean Theorem.

Question 13 Explanation:

Topic: Derive and use formulas for calculating the lengths, perimeters, areas, volumes, and surface areas of geometric shapes and figures (Objective 0023), and recognize and apply connections between algebra and geometry (e.g., the use of coordinate systems, the Pythagorean theorem) (Objective 0024).

Question 14

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 14 Explanation:

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

Question 15

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 15 Explanation:

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

Question 16

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 16 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 17

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 17 Explanation:

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

Question 18

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 18 Explanation:

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

Question 19

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 19 Explanation:

Topic: Laws of Exponents (Objective 0019).

Question 20

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 20 Explanation:

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

Once you are finished, click the button below. Any items you have not completed will be marked incorrect. Get Results

There are 20 questions to complete.

←

List

→

Return

Shaded items are complete.

1	2	3	4	5
6	7	8	9	10
11	12	13	14	15
16	17	18	19	20
End

Return

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	\( \large 28 \dfrac{4}{7}\) ml Hint: 49 ml of medicine requires 28 ml of saline. The extra ml of saline requires 4 ml saline/ 7 ml medicine = 4/7 ml saline per 1 ml medicine.
B	\( \large 28 \dfrac{1}{4}\) ml Hint: 49 ml of medicine requires 28 ml of saline. How much saline does the extra ml require?
C	\( \large 28 \dfrac{1}{7}\) ml Hint: 49 ml of medicine requires 28 ml of saline. How much saline does the extra ml require?
D	\( \large 87.5\) ml Hint: 49 ml of medicine requires 28 ml of saline. How much saline does the extra ml require?

A	\( \large \dfrac{1}{2}\) Hint: How many different configurations are there from oldest to youngest, e.g. BGGG? How many of them have 2 boys and 2 girls?
B	\( \large \dfrac{1}{4}\) Hint: How many different configurations are there from oldest to youngest, e.g. BGGG? How many of them have 2 boys and 2 girls?
C	\( \large \dfrac{1}{5}\) Hint: Some configurations are more probable than others -- i.e. it's more likely to have two boys and two girls than all boys. Be sure you are weighting properly.
D	\( \large \dfrac{3}{8}\) Hint: There are two possibilities for each child, so there are \(2 \times 2 \times 2 \times 2 =16\) different configurations, e.g. from oldest to youngest BBBG, BGGB, GBBB, etc. Of these configurations, there are 6 with two boys and two girls (this is the combination \(_{4}C_{2}\) or "4 choose 2"): BBGG, BGBG, BGGB, GGBB, GBGB, and GBBG. Thus the probability is 6/16=3/8.

A	\( \large A-B-\dfrac{C}{DE} \) Hint: In the order of operations, multiplication and division have the same priority, so do them left to right; same with addition and subtraction.
B	\( \large A-B+\dfrac{CE}{D}\) Hint: In practice, you're better off using parentheses than writing an expression like the one in the question. The PEMDAS acronym that many people memorize is misleading. Multiplication and division have equal priority and are done left to right. They have higher priority than addition and subtraction. Addition and subtraction also have equal priority and are done left to right.
C	\( \large \dfrac{AE-BE+CE}{D}\) Hint: Use order of operations, don't just compute left to right.
D	\( \large A-B+\dfrac{C}{DE}\) Hint: In the order of operations, multiplication and division have the same priority, so do them left to right

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	\( \large511 \) Hint: Divisible by 7.
B	\( \large517\) Hint: Divisible by 11.
C	\( \large519\) Hint: Divisible by 3.
D	\( \large521\)

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	The quotient is \(3\dfrac{3}{4}\). There are 3 whole blocks each representing \(\dfrac{2}{3}\) and a partial block composed of 3 small rectangles. The 3 small rectangles represent \(\dfrac{3}{4}\) of \(\dfrac{2}{3}\).
B	The quotient is \(3\dfrac{1}{2}\). There are 3 whole blocks each representing \(\dfrac{2}{3}\) and a partial block composed of 3 small rectangles. The 3 small rectangles represent \(\dfrac{3}{6}\) of a whole, or \(\dfrac{1}{2}\). Hint: We are counting how many 2/3's are in 2 1/2: the unit becomes 2/3, not 1.
C	The quotient is \(\dfrac{4}{15}\). There are four whole blocks separated into a total of 15 small rectangles. Hint: This explanation doesn't make much sense. Probably you are doing "invert and multiply," but inverting the wrong thing.
D	This picture cannot be used to find the quotient because it does not show how to separate \(2\dfrac{1}{2}\) into equal sized groups. Hint: Study the measurement/quotative model of division. It's often very useful with fractions.

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	\( \large W=-5\) Hint: When y is even, W is even.
B	\( \large W=4\) Hint: Note that when x increases by 1, W increases by 2, and when y increases by 1, W decreases by 1. At x=y=0, W=0, so at x=9, y=14, W has increased by \(9 \times 2\) and decreased by 14, or W=18-14=4.
C	\( \large W=6\) Hint: Try fixing x or y at 0, and start by finding W for x=0 y=14 or x=9, y=0.
D	\( \large W=32\) Hint: Try fixing x or y at 0, and start by finding W for x=0 y=14 or x=9, y=0.

A	\( \large 1\) Hint: Use the Pythagorean Theorem.
B	\( \large \sqrt{5}\) Hint: \(2^2+e^2=3^2\) or \(4+e^2=9;e^2=5; e=\sqrt{5}\).
C	\( \large \sqrt{13}\) Hint: e is not the hypotenuse.
D	\( \large 5\) Hint: Use the Pythagorean Theorem.

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	\( \large 6g\) 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	\( \large N2+205-100\div 10=N\) 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	\( \large \dfrac{1}{4}\) 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	\( \large 2\) 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	\( \large \dfrac{1}{7}\) 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

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 solution requires 4 ml of saline for every 7 ml of medicine. How much saline would be required for 50 ml of medicine?

A family has four children. What is the probability that two children are girls and two are boys? Assume the the probability of having a boy (or a girl) is 50%.

Which of the following is equivalent to

\( \large A-B+C\div D\times E\)?

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.

Exactly one of the numbers below is a prime number. Which one is it?

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?

Below is a pictorial representation of \(2\dfrac{1}{2}\div \dfrac{2}{3}\):

Which of the following is the best description of how to find the quotient from the picture?

The quotient is \(3\dfrac{3}{4}\). There are 3 whole blocks each representing \(\dfrac{2}{3}\) and a partial block composed of 3 small rectangles. The 3 small rectangles represent \(\dfrac{3}{4}\) of \(\dfrac{2}{3}\).

The quotient is \(3\dfrac{1}{2}\). There are 3 whole blocks each representing \(\dfrac{2}{3}\) and a partial block composed of 3 small rectangles. The 3 small rectangles represent \(\dfrac{3}{6}\) of a whole, or \(\dfrac{1}{2}\).

The quotient is \(\dfrac{4}{15}\). There are four whole blocks separated into a total of 15 small rectangles.

This picture cannot be used to find the quotient because it does not show how to separate \(2\dfrac{1}{2}\) into equal sized groups.

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

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

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

It is not valid. It never produces the correct answer.

It is not valid. It produces the correct answer in a few special cases, but it‘s still not a valid algorithm.

It is valid if the rational numbers in the multiplication problem are in lowest terms.

It is valid for all rational numbers.

Use the table below to answer the question that follows:

Each number in the table above represents a value W that is determined by the values of x and y. For example, when x=3 and y=1, W=5. What is the value of W when x=9 and y=14? Assume that the patterns in the table continue as shown.

What is the length of side \(\overline{BD}\) in the triangle below, where \(\angle DBA\) is a right angle?

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?

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

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?

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?

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

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

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.