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

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

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

Question 1 Explanation:

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

Question 2

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 2 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 3

Some children explored the diagonals in 2 x 2 squares on pages of a calendar (where all four squares have numbers in them). They conjectured that the sum of the diagonals is always equal; in the example below, 8+16=9+15.

Which of the equations below could best be used to explain why the children's conjecture is correct?

A	$ \large 8x+16x=9x+15x$ Hint: What would x represent in this case? Make sure you can describe in words what x represents.
B	$ \large x+(x+2)=(x+1)+(x+1)$ Hint: What would x represent in this case? Make sure you can describe in words what x represents.
C	$ \large x+(x+8)=(x+1)+(x+7)$ Hint: x is the number in the top left square, x+8 is one below and to the right, x+1 is to the right of x, and x+7 is below x.
D	$ \large x+8+16=x+9+15$ Hint: What would x represent in this case? Make sure you can describe in words what x represents.

Question 3 Explanation:

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

Question 4

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 4 Explanation:

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

Question 5

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 5 Explanation:

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

Question 6

Below is a portion of a number line.

Point A is one-quarter of the distance from 0.26 to 0.28. What number is represented by point A?

A	$ \large0.26$ Hint: Please reread the question.
B	$ \large0.2625$ Hint: This is one-quarter of the distance between 0.26 and 0.27, which is not what the question asked.
C	$ \large0.265$
D	$ \large0.27$ Hint: Please read the question more carefully. This answer would be correct if Point A were halfway between the tick marks, but it's not.

Question 6 Explanation:

Topic: Using number lines (Objective 0017)

Question 7

I. $\large \dfrac{1}{2}+\dfrac{1}{3}$	II. $ \large .400000$	III. $\large\dfrac{1}{5}+\dfrac{1}{5}$

IV. $ \large 40\% $	V. $ \large 0.25 $	VI. $\large\dfrac{14}{35}$

Which of the lists below includes all of the above expressions that are equivalent to $ \dfrac{2}{5}$?

A	I, III, V, VI Hint: I and V are not at all how fractions and decimals work.
B	III, VI Hint: These are right, but there are more.
C	II, III, VI Hint: These are right, but there are more.
D	II, III, IV, VI

Question 7 Explanation:

Topic: Converting between fractions, decimals, and percents (Objective 0017)

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

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

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

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

Question 9 Explanation:

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

Question 10

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 10 Explanation:

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

Question 11

Below is a portion of a number line:

Point B is halfway between two tick marks. What number is represented by Point B?

A	$ \large 0.645$ Hint: That point is marked on the line, to the right.
B	$ \large 0.6421$ Hint: That point is to the left of point B.
C	$ \large 0.6422$ Hint: That point is to the left of point B.
D	$ \large 0.6425$

Question 11 Explanation:

Topic: Using Number Lines (Objective 0017)

Question 12

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 12 Explanation:

Topic: Place Value (Objective 0016)

Question 13

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 13 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 14

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 14 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 15

Which of the lines depicted below is a graph of $ \large y=2x-5$?

A	a Hint: The slope of line a is negative.
B	b Hint: Wrong slope and wrong intercept.
C	c Hint: The intercept of line c is positive.
D	d Hint: Slope is 2 -- for every increase of 1 in x, y increases by 2. Intercept is -5 -- the point (0,-5) is on the line.

Question 15 Explanation:

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

Question 16

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 16 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 17

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 17 Explanation:

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

Question 18

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 18 Explanation:

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

Question 19

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

Which of the following does not represent the number of gumdrops in a row of h houses?

A	$ \large 2+3h$ Hint: Think of this as start with 2 gumdrops on the left wall, and then add 3 gumdrops for each house.
B	$ \large 5+3(h-1)$ Hint: Think of this as start with one house, and then add 3 gumdrops for each of the other h-1 houses.
C	$ \large h+(h+1)+(h+1)$ Hint: Look at the gumdrops in 3 rows: h gumdrops for the "rooftops," h+1 for the tops of the vertical walls, and h+1 for the floors.
D	$ \large 5+3h$ Hint: This one is not a correct equation (which makes it the correct answer!). Compare to choice A. One of them has to be wrong, as they differ by 3.

Question 19 Explanation:

Topic: Translate among different representations (e.g., tables, graphs, algebraic expressions, verbal descriptions) of functional relationships (Objective 0021).

Question 20

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 20 Explanation:

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

Question 21

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 21 Explanation:

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

Question 22

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 22 Explanation:

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

Question 23

In each expression below N represents a negative integer. Which expression could have a negative value?

A	$ \large {{N}^{2}}$ Hint: Squaring always gives a non-negative value.
B	$ \large 6-N$ Hint: A story problem for this expression is, if it was 6 degrees out at noon and N degrees out at sunrise, by how many degrees did the temperature rise by noon? Since N is negative, the answer to this question has to be positive, and more than 6.
C	$ \large -N$ Hint: If N is negative, then -N is positive
D	$ \large 6+N$ Hint: For example, if $N=-10$, then $6+N = -4$

Question 23 Explanation:

If you are stuck on a question like this, try a few examples to eliminate some choices and to help you understand what the question means. Topic: Characteristics of integers (Objective 0016).

Question 24

The Americans with Disabilties Act (ADA) regulations state that the maximum slope for a wheelchair ramp in new construction is 1:12, although slopes between 1:16 and 1:20 are preferred. The maximum rise for any run is 30 inches. The graph below shows the rise and runs of four different wheelchair ramps. Which ramp is in compliance with the ADA regulations for new construction?

A	A Hint: Rise is more than 30 inches.
B	B Hint: Run is almost 24 feet, so rise can be almost 2 feet.
C	C Hint: Run is 12 feet, so rise can be at most 1 foot.
D	D Hint: Slope is 1:10 -- too steep.

Question 24 Explanation:

Topic: Interpret meaning of slope in a real world situation (Objective 0022).

Question 25

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

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

Question 25 Explanation:

Topics: Laws of exponents (Objective 0019).

Question 26

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

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

Question 26 Explanation:

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

Question 27

The window glass below has the shape of a semi-circle on top of a square, where the side of the square has length x. It was cut from one piece of glass.

What is the perimeter of the window glass?

A	$ \large 3x+\dfrac{\pi x}{2}$ Hint: By definition, $\pi$ is the ratio of the circumference of a circle to its diameter; thus the circumference is $\pi d$. Since we have a semi-circle, its perimeter is $ \dfrac{1}{2} \pi x$. Only 3 sides of the square contribute to the perimeter.
B	$ \large 3x+2\pi x$ Hint: Make sure you know how to find the circumference of a circle.
C	$ \large 3x+\pi x$ Hint: Remember it's a semi-circle, not a circle.
D	$ \large 4x+2\pi x$ Hint: Only 3 sides of the square contribute to the perimeter.

Question 27 Explanation:

Topic: Derive and use formulas for calculating the lengths, perimeters, areas, volumes, and surface areas of geometric shapes and figures (Objective 0023).

Question 28

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

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

Question 28 Explanation:

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

Question 29

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 29 Explanation:

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

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

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 31 Explanation:

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

Question 32

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 32 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 33

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

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

Question 33 Explanation:

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

Question 34

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 34 Explanation:

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

Question 35

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 35 Explanation:

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

Question 36

Which of the following sets of polygons can be assembled to form a pentagonal pyramid?

A	2 pentagons and 5 rectangles. Hint: These can be assembled to form a pentagonal prism, not a pentagonal pyramid.
B	1 square and 5 equilateral triangles. Hint: You need a pentagon for a pentagonal pyramid.
C	1 pentagon and 5 isosceles triangles.
D	1 pentagon and 10 isosceles triangles.

Question 36 Explanation:

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

Question 37

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

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

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

Question 37 Explanation:

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

Question 38

Use the graph below to answer the question that follows:

The graph above represents the equation $ \large 3x+Ay=B$, where A and B are integers. What are the values of A and B?

A	$ \large A = -2, B= 6$ Hint: Plug in (2,0) to get B=6, then plug in (0,-3) to get A=-2.
B	$ \large A = 2, B = 6$ Hint: Try plugging (0,-3) into this equation.
C	$ \large A = -1.5, B=-3$ Hint: The problem said that A and B were integers and -1.5 is not an integer. Don't try to use slope-intercept form.
D	$ \large A = 2, B = -3$ Hint: Try plugging (2,0) into this equation.

Question 38 Explanation:

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

Question 39

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

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

Question 39 Explanation:

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

Question 40

Use the expression below to answer the question that follows.

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

Which of the following is equivalent to the expression above?

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

Question 40 Explanation:

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

Question 41

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 41 Explanation:

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

Question 42

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 42 Explanation:

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

Question 43

Use the graph below to answer the question that follows.

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

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

Question 43 Explanation:

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

Question 44

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

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

Question 44 Explanation:

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

Question 45

The letters A, and B represent digits (possibly equal) in the ten digit number x=1,438,152,A3B. For which values of A and B is x divisible by 12, but not by 9?

A	$ \large A = 0, B = 4$ Hint: Digits add to 31, so not divisible by 3, so not divisible by 12.
B	$ \large A = 7, B = 2$ Hint: Digits add to 36, so divisible by 9.
C	$ \large A = 0, B = 6$ Hint: Digits add to 33, divisible by 3, not 9. Last digits are 36, so divisible by 4, and hence by 12.
D	$ \large A = 4, B = 8$ Hint: Digits add to 39, divisible by 3, not 9. Last digits are 38, so not divisible by 4, so not divisible by 12.

Question 45 Explanation:

Topic: Demonstrate knowledge of divisibility rules (Objective 0018).

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

A	\( \large 6g\) 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	\( \large 8x+16x=9x+15x\) Hint: What would x represent in this case? Make sure you can describe in words what x represents.
B	\( \large x+(x+2)=(x+1)+(x+1)\) Hint: What would x represent in this case? Make sure you can describe in words what x represents.
C	\( \large x+(x+8)=(x+1)+(x+7)\) Hint: x is the number in the top left square, x+8 is one below and to the right, x+1 is to the right of x, and x+7 is below x.
D	\( \large x+8+16=x+9+15\) Hint: What would x represent in this case? Make sure you can describe in words what x represents.

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.

A	\( \large0.26\) Hint: Please reread the question.
B	\( \large0.2625\) Hint: This is one-quarter of the distance between 0.26 and 0.27, which is not what the question asked.
C	\( \large0.265\)
D	\( \large0.27\) Hint: Please read the question more carefully. This answer would be correct if Point A were halfway between the tick marks, but it's not.

I. \(\large \dfrac{1}{2}+\dfrac{1}{3}\)	II. \( \large .400000\)	III. \(\large\dfrac{1}{5}+\dfrac{1}{5}\)

IV. \( \large 40\% \)	V. \( \large 0.25 \)	VI. \(\large\dfrac{14}{35}\)

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

A	\( \large A(n)=n+4\) 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	\( \large 0.645\) Hint: That point is marked on the line, to the right.
B	\( \large 0.6421\) Hint: That point is to the left of point B.
C	\( \large 0.6422\) Hint: That point is to the left of point B.
D	\( \large 0.6425\)

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	\( \large 1\) 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	\( \large 2+3h\) Hint: Think of this as start with 2 gumdrops on the left wall, and then add 3 gumdrops for each house.
B	\( \large 5+3(h-1)\) Hint: Think of this as start with one house, and then add 3 gumdrops for each of the other h-1 houses.
C	\( \large h+(h+1)+(h+1)\) Hint: Look at the gumdrops in 3 rows: h gumdrops for the "rooftops," h+1 for the tops of the vertical walls, and h+1 for the floors.
D	\( \large 5+3h\) Hint: This one is not a correct equation (which makes it the correct answer!). Compare to choice A. One of them has to be wrong, as they differ by 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: \( \left\| {{(1+2)}^{3}} \right\|\)=\( \left \| {3}^3 \right \| \)=\( \left \| 27 \right \| \) = \(27\)

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	\( \large {{N}^{2}}\) Hint: Squaring always gives a non-negative value.
B	\( \large 6-N\) Hint: A story problem for this expression is, if it was 6 degrees out at noon and N degrees out at sunrise, by how many degrees did the temperature rise by noon? Since N is negative, the answer to this question has to be positive, and more than 6.
C	\( \large -N\) Hint: If N is negative, then -N is positive
D	\( \large 6+N\) Hint: For example, if \(N=-10\), then \(6+N = -4\)

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

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

A	\( \large 3x+\dfrac{\pi x}{2}\) Hint: By definition, \(\pi\) is the ratio of the circumference of a circle to its diameter; thus the circumference is \(\pi d\). Since we have a semi-circle, its perimeter is \( \dfrac{1}{2} \pi x\). Only 3 sides of the square contribute to the perimeter.
B	\( \large 3x+2\pi x\) Hint: Make sure you know how to find the circumference of a circle.
C	\( \large 3x+\pi x\) Hint: Remember it's a semi-circle, not a circle.
D	\( \large 4x+2\pi x\) Hint: Only 3 sides of the square contribute to the perimeter.

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	\( \large y={{x}^{2}}+2x+7\) 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	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	\( \large \dfrac{1}{3}\) Hint: This is the probability that the first child picks a red gumball, but not that the first two children pick red gumballs.
B	\( \large \dfrac{1}{8}\) Hint: Are you adding things that you should be multiplying?
C	\( \large \dfrac{1}{9}\) Hint: This would be the probability if the gumballs were returned to the bag.
D	\( \large \dfrac{1}{15}\) Hint: The probability that the first child picks red is 2/6 = 1/3. Then there are 5 gumballs in the bag, one red, so the probability that the second child picks red is 1/5. Thus 1/5 of the time, after the first child picks red, the second does too, so the probability is 1/5 x 1/3 = 1/15.

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

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

MTEL General Curriculum Mathematics Practice

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

Tetrahedron

Triangular Prism

Triangular Pyramid

Trigon

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?

Some children explored the diagonals in 2 x 2 squares on pages of a calendar (where all four squares have numbers in them). They conjectured that the sum of the diagonals is always equal; in the example below, 8+16=9+15.

Which of the equations below could best be used to explain why the children's conjecture is correct?

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.

21 cm

22 cm

23 cm

24 cm

Below is a portion of a number line.

Point A is one-quarter of the distance from 0.26 to 0.28. What number is represented by point A?

Which of the lists below includes all of the above expressions that are equivalent to \( \dfrac{2}{5}\)?

I, III, V, VI

III, VI

II, III, VI

II, III, IV, VI

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?

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

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

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?

Below is a portion of a number line:

Point B is halfway between two tick marks. What number is represented by Point B?

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

Use the problem below to answer the question that follows:

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

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

In which table below is y a function of x?

Which of the lines depicted below is a graph of \( \large y=2x-5\)?

a

b

c

d

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

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

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?

1.5°

1.8°

2.7°

Not enough information.

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

Which of the following does not represent the number of gumdrops in a row of h houses?

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

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?

In each expression below N represents a negative integer. Which expression could have a negative value?

A

B

C

D

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

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

A

B

C

D

The window glass below has the shape of a semi-circle on top of a square, where the side of the square has length x. It was cut from one piece of glass.

What is the perimeter of the window glass?

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

It is too low by a factor of 10

It is too low by a factor of 100

It is too high by a factor of 10

It is too high by a factor of 100

Which of the following is equivalent to

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

A	\( \large A = -2, B= 6\) Hint: Plug in (2,0) to get B=6, then plug in (0,-3) to get A=-2.
B	\( \large A = 2, B = 6\) Hint: Try plugging (0,-3) into this equation.
C	\( \large A = -1.5, B=-3\) Hint: The problem said that A and B were integers and -1.5 is not an integer. Don't try to use slope-intercept form.
D	\( \large A = 2, B = -3\) Hint: Try plugging (2,0) into this equation.

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

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

A	\( \large t=s+15\) 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	\( \large y=-\dfrac{1}{2}x+2\) Hint: The slope is -1/2 and the y-intercept is 2. You can also try just plugging in points. For example, this is the only choice that gives y=1 when x=2.
B	\( \large 4x=2y\) Hint: This line goes through (0,0); the graph above does not.
C	\( \large y=x+2\) Hint: The line pictured has negative slope.
D	\( \large y=-x+2\) Hint: Try plugging x=4 into this equation and see if that point is on the graph above.

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

A	\( \large A = 0, B = 4\) Hint: Digits add to 31, so not divisible by 3, so not divisible by 12.
B	\( \large A = 7, B = 2\) Hint: Digits add to 36, so divisible by 9.
C	\( \large A = 0, B = 6\) Hint: Digits add to 33, divisible by 3, not 9. Last digits are 36, so divisible by 4, and hence by 12.
D	\( \large A = 4, B = 8\) Hint: Digits add to 39, divisible by 3, not 9. Last digits are 38, so not divisible by 4, so not divisible by 12.

1	2	3	4	5
6	7	8	9	10
11	12	13	14	15
16	17	18	19	20
21	22	23	24	25
26	27	28	29	30
31	32	33	34	35
36	37	38	39	40
41	42	43	44	45
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