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

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 1 Explanation:

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

Question 2

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

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

Question 2 Explanation:

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

Question 3

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 3 Explanation:

Topic: Least Common Multiple (Objective 0018)

Question 4

The letters A, B, and C represent digits (possibly equal) in the twelve digit number x=111,111,111,ABC. For which values of A, B, and C is x divisible by 40?

A	$\large A = 3, B = 2, C=0$ Hint: Note that it doesn't matter what the first 9 digits are, since 1000 is divisible by 40, so DEF,GHI,JKL,000 is divisible by 40 - we need to check the last 3.
B	$\large A = 0, B = 0, C=4$ Hint: Not divisible by 10, since it doesn't end in 0.
C	$\large A = 4, B = 2, C=0$ Hint: Divisible by 10 and by 4, but not by 40, as it's not divisible by 8. Look at 40 as the product of powers of primes -- 8 x 5, and check each. To check 8, either check whether 420 is divisible by 8, or take ones place + twice tens place + 4 * hundreds place = 18, which is not divisible by 8.
D	$\large A =1, B=0, C=0$ Hint: Divisible by 10 and by 4, but not by 40, as it's not divisible by 8. Look at 40 as the product of powers of primes -- 8 x 5, and check each. To check 8, either check whether 100 is divisible by 8, or take ones place + twice tens place + 4 * hundreds place = 4, which is not divisible by 8.

Question 4 Explanation:

Topic: Understand divisibility rules and why they work (Objective 018).

Question 5

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

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

Question 5 Explanation:

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

Question 6

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 6 Explanation:

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

Question 7

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

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

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

Question 7 Explanation:

Topic: Venn Diagrams (Objective 0025)

Question 8

Which property is not shared by all rhombi?

A	4 congruent sides Hint: The most common definition of a rhombus is a quadrilateral with 4 congruent sides.
B	A center of rotational symmetry Hint: The diagonal of a rhombus separates it into two congruent isosceles triangles. The center of this line is a center of 180 degree rotational symmetry that switches the triangles.
C	4 congruent angles Hint: Unless the rhombus is a square, it does not have 4 congruent angles.
D	2 sets of parallel sides Hint: All rhombi are parallelograms.

Question 8 Explanation:

Topic: Classify and analyze polygons using attributes of sides and angles, and symmetry (Objective 0024).

Question 9

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 9 Explanation:

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

Question 10

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 10 Explanation:

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

Question 11

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 11 Explanation:

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

Question 12

Which of the numbers below is the decimal equivalent of $\dfrac{3}{8}?$

A	0.38 Hint: If you are just writing the numerator next to the denominator then your technique is way off, but by coincidence your answer is close; try with 2/3 and 0.23 is nowhere near correct.
B	0.125 Hint: This is 1/8, not 3/8.
C	0.375
D	0.83 Hint: 3/8 is less than a half, and 0.83 is more than a half, so they can't be equal.

Question 12 Explanation:

Topic: Converting between fractions and decimals (Objective 0017)

Question 13

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 13 Explanation:

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

Question 14

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 14 Explanation:

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

Question 15

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

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

Question 15 Explanation:

Topic: Integers (Objective 0016)

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

A teacher has a list of all the countries in the world and their populations in March 2012. She is going to have her students use technology to compute the mean and median of the numbers on the list. Which of the following statements is true?

A	The teacher can be sure that the mean and median will be the same without doing any computation. Hint: Does this make sense? How likely is it that the mean and median of any large data set will be the same?
B	The teacher can be sure that the mean is bigger than the median without doing any computation. Hint: This is a skewed distribution, and very large countries like China and India contribute huge numbers to the mean, but are counted the same as small countries like Luxembourg in the median (the same thing happens w/data on salaries, where a few very high income people tilt the mean -- that's why such data is usually reported as medians).
C	The teacher can be sure that the median is bigger than the mean without doing any computation. Hint: Think about a set of numbers like 1, 2, 3, 4, 10,000 -- how do the mean/median compare? How might that relate to countries of the world?
D	There is no way for the teacher to know the relative size of the mean and median without computing them. Hint: Knowing the shape of the distribution of populations does give us enough info to know the relative size of the mean and median, even without computing them.

Question 17 Explanation:

Topic: Use measures of central tendency (e.g., mean, median, mode) and spread to describe and interpret real-world data (Objective 0025).

Question 18

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 18 Explanation:

Topic: Laws of Exponents (Objective 0019).

Question 19

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 19 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 20

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 20 Explanation:

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

Question 21

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 21 Explanation:

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

Question 22

Aya and Kendra want to estimate the height of a tree. On a sunny day, Aya measures Kendra's shadow as 3 meters long, and Kendra measures the tree's shadow as 15 meters long. Kendra is 1.5 meters tall. How tall is the tree?

A	7.5 meters Hint: Here is a picture, note that the large and small right triangles are similar: One way to do the problem is to note that there is a dilation (scale) factor of 5 on the shadows, so there must be that factor on the heights too. Another way is to note that the shadows are twice as long as the heights.
B	22.5 meters Hint: Draw a picture.
C	30 meters Hint: Draw a picture.
D	45 meters Hint: Draw a picture.

Question 22 Explanation:

Topic: Apply geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to similarity, ; and use these concepts to solve problems (Objective 0024) . Fits in other places too.

Question 23

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

A	$\large -0.044,\quad -0.04,\quad 0.04,\quad 0.044$ Hint: These are easier to compare if you add trailing zeroes (this is finding a common denominator) -- all in thousandths, -0.044, -0.040,0 .040, 0.044. The middle two numbers, -0.040 and 0.040 can be modeled as owing 4 cents and having 4 cents. The outer two numbers are owing or having a bit more.
B	$\large -0.04,\quad -0.044,\quad 0.044,\quad 0.04$ Hint: 0.04=0.040, which is less than 0.044.
C	$\large -0.04,\quad -0.044,\quad 0.04,\quad 0.044$ Hint: -0.04=-0.040, which is greater than $-0.044$ .
D	$\large -0.044,\quad -0.04,\quad 0.044,\quad 0.04$ Hint: 0.04=0.040, which is less than 0.044.

Question 23 Explanation:

Topic: Ordering decimals and integers (Objective 0017).

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

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

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

Question 25 Explanation:

Topic: Ordering Fractions (Objective 0017)

Question 26

A map has a scale of 3 inches = 100 miles. Cities A and B are 753 miles apart. Let d be the distance between the two cities on the map. Which of the following is not correct?

A	$\large \dfrac{3}{100}=\dfrac{d}{753}$ Hint: Units on both side are inches/mile, and both numerators and denominators correspond -- this one is correct.
B	$\large \dfrac{3}{100}=\dfrac{753}{d}$ Hint: Unit on the left is inches per mile, and on the right is miles per inch. The proportion is set up incorrectly (which is what we wanted). Another strategy is to notice that one of A or B has to be the answer because they cannot both be correct proportions. Then check that cross multiplying on A gives part D, so B is the one that is different from the other 3.
C	$\large \dfrac{3}{d}=\dfrac{100}{753}$ Hint: Unitless on each side, as inches cancel on the left and miles on the right. Numerators correspond to the map, and denominators to the real life distances -- this one is correct.
D	$\large 100d=3\cdot 753$ Hint: This is equivalent to part A.

Question 26 Explanation:

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

Question 27

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 27 Explanation:

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

Question 28

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

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

Question 28 Explanation:

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

Question 29

A sales companies pays its representatives $2 for each item sold, plus 40% of the price of the item. The rest of the money that the representatives collect goes to the company. All transactions are in cash, and all items cost $4 or more. If the price of an item in dollars is p, which expression represents the amount of money the company collects when the item is sold?

A	$\large \dfrac{3}{5}p-2$ Hint: The company gets 3/5=60% of the price, minus the $2 per item.
B	$\large \dfrac{3}{5}\left( p-2 \right)$ Hint: This is sensible, but not what the problem states.
C	$\large \dfrac{2}{5}p+2$ Hint: The company pays the extra $2; it doesn't collect it.
D	$\large \dfrac{2}{5}p-2$ Hint: This has the company getting 2/5 = 40% of the price of each item, but that's what the representative gets.

Question 29 Explanation:

Topic: Use algebra to solve word problems involving fractions, ratios, proportions, and percents (Objective 0020).

Question 30

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 30 Explanation:

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

Question 31

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 31 Explanation:

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

Question 32

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 32 Explanation:

Topic: Solve linear equations (Objective 0020).

Question 33

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 33 Explanation:

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

Question 34

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 34 Explanation:

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

Question 35

Which of the following is equal to one million three hundred thousand?

A	$\large1.3\times {{10}^{6}}$
B	$\large1.3\times {{10}^{9}}$ Hint: That's one billion three hundred million.
C	$\large1.03\times {{10}^{6}}$ Hint: That's one million thirty thousand.
D	$\large1.03\times {{10}^{9}}$ Hint: That's one billion thirty million

Question 35 Explanation:

Topic: Scientific Notation (Objective 0016)

Question 36

A car is traveling at 60 miles per hour. Which of the expressions below could be used to compute how many feet the car travels in 1 second? Note that 1 mile = 5,280 feet.

A	$\large 60\dfrac{\text{miles}}{\text{hour}}\cdot 5280\dfrac{\text{feet}}{\text{mile}}\cdot 60\dfrac{\text{minutes}}{\text{hour}}\cdot 60\dfrac{\text{seconds}}{\text{minute}}$ Hint: This answer is not in feet/second.
B	$\large 60\dfrac{\text{miles}}{\text{hour}}\cdot 5280\dfrac{\text{feet}}{\text{mile}}\cdot \dfrac{1}{60}\dfrac{\text{hour}}{\text{minutes}}\cdot \dfrac{1}{60}\dfrac{\text{minute}}{\text{seconds}}$ Hint: This is the only choice where the answer is in feet per second and the unit conversions are correct.
C	$\large 60\dfrac{\text{miles}}{\text{hour}}\cdot \dfrac{1}{5280}\dfrac{\text{foot}}{\text{miles}}\cdot 60\dfrac{\text{hours}}{\text{minute}}\cdot \dfrac{1}{60}\dfrac{\text{minute}}{\text{seconds}}$ Hint: Are there really 60 hours in a minute?
D	$\large 60\dfrac{\text{miles}}{\text{hour}}\cdot \dfrac{1}{5280}\dfrac{\text{mile}}{\text{feet}}\cdot 60\dfrac{\text{minutes}}{\text{hour}}\cdot \dfrac{1}{60}\dfrac{\text{minute}}{\text{seconds}}$ Hint: This answer is not in feet/second.

Question 36 Explanation:

Topic: Use unit conversions and dimensional analysis to solve measurement problems (Objective 0023).

Question 37

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 37 Explanation:

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

Question 38

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 38 Explanation:

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

Question 39

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 39 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 40

Use the solution procedure below to answer the question that follows:

$\large {\left( x+3 \right)}^{2}=10$

$\large \left( x+3 \right)\left( x+3 \right)=10$

$\large {x}^{2}+9=10$

$\large {x}^{2}+9-9=10-9$

$\large {x}^{2}=1$

$\large x=1\text{ or }x=-1$

Which of the following is incorrect in the procedure shown above?

A	The commutative property is used incorrectly. Hint: The commutative property is $a+b=b+a$ or $ab=ba$ .
B	The associative property is used incorrectly. Hint: The associative property is $a+(b+c)=(a+b)+c$ or $a \times (b \times c)=(a \times b) \times c$ .
C	Order of operations is done incorrectly.
D	The distributive property is used incorrectly. Hint: $(x+3)(x+3)=x(x+3)+3(x+3)$ = $x^2+3x+3x+9.$

Question 40 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 41

P is a prime number that divides 240. Which of the following must be true?

A	P divides 30 Hint: 2, 3, and 5 are the prime factors of 240, and all divide 30.
B	P divides 48 Hint: P=5 doesn't work.
C	P divides 75 Hint: P=2 doesn't work.
D	P divides 80 Hint: P=3 doesn't work.

Question 41 Explanation:

Topic: Find the prime factorization of a number and recognize its uses (Objective 0018).

Question 42

The function d(x) gives the result when 12 is divided by x. Which of the following is a graph of d(x)?

A	Hint: d(x) is 12 divided by x, not x divided by 12.
B	Hint: When x=2, what should d(x) be?
C	Hint: When x=2, what should d(x) be?
D

Question 42 Explanation:

Topic: Identify and analyze direct and inverse relationships in tables, graphs, algebraic expressions and real-world situations (Objective 0021)

Question 43

Use the expression below to answer the question that follows.

$\large \dfrac{\left( 155 \right)\times \left( 6,124 \right)}{977}$

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

A	100 Hint: 6124/977 is approximately 6.
B	200 Hint: 6124/977 is approximately 6.
C	1,000 Hint: 6124/977 is approximately 6. 155 is approximately 150, and $6 \times 150 = 3 \times 300 = 900$ , so this answer is closest.
D	2,000 Hint: 6124/977 is approximately 6.

Question 43 Explanation:

Topics: Estimation, simplifying fractions (Objective 0016).

Question 44

The column below consists of two cubes and a cylinder. The cylinder has diameter y, which is also the length of the sides of each cube. The total height of the column is 5y. Which of the formulas below gives the volume of the column?

A	$\large 2{{y}^{3}}+\dfrac{3\pi {{y}^{3}}}{4}$ Hint: The cubes each have volume $y^3$ . The cylinder has radius $\dfrac{y}{2}$ and height $3y$ . The volume of a cylinder is $\pi r^2 h=\pi ({\dfrac{y}{2}})^2(3y)=\dfrac{3\pi {{y}^{3}}}{4}$ . Note that the volume of a cylinder is analogous to that of a prism -- area of the base times height.
B	$\large 2{{y}^{3}}+3\pi {{y}^{3}}$ Hint: y is the diameter of the circle, not the radius.
C	$\large {{y}^{3}}+5\pi {{y}^{3}}$ Hint: Don't forget to count both cubes.
D	$\large 2{{y}^{3}}+\dfrac{3\pi {{y}^{3}}}{8}$ Hint: Make sure you know how to find the volume of a cylinder.

Question 44 Explanation:

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

Question 45

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 45 Explanation:

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

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MTEL General Curriculum Mathematics Practice

Which of the graphs below represent functions?

I and IV only.

I and III only.

II and III only.

I, II, and IV only.

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

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

The letters A, B, and C represent digits (possibly equal) in the twelve digit number x=111,111,111,ABC. For which values of A, B, and C is x divisible by 40?

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

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

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

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

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

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

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

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

Which property is not shared by all rhombi?

4 congruent sides

A center of rotational symmetry

4 congruent angles

2 sets of parallel sides

Which of the numbers below is a fraction equivalent to 0.ˉ6 0.\bar{6}?

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?

95% of 12 year old boys can do 56 sit-ups in 60 seconds.

At most 25% of 7 year old boys can do 19 or more sit-ups in 60 seconds.

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.

At least 75% of 16 year old boys can only do 51 or fewer sit-ups in 60 seconds.

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?

Which of the numbers below is the decimal equivalent of 38? \dfrac{3}{8}?

0.38

0.125

0.375

0.83

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?

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

Tetrahedron

Triangular Prism

Triangular Pyramid

Trigon

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

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

A teacher has a list of all the countries in the world and their populations in March 2012. She is going to have her students use technology to compute the mean and median of the numbers on the list. Which of the following statements is true?

The teacher can be sure that the mean and median will be the same without doing any computation.

The teacher can be sure that the mean is bigger than the median without doing any computation.

The teacher can be sure that the median is bigger than the mean without doing any computation.

There is no way for the teacher to know the relative size of the mean and median without computing them.

The expression 83⋅2−10 \large{{8}^{3}}\cdot {{2}^{-10}} is equal to which of the following?

Use the table below to answer the question that follows:

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

Store A

Store B

Store C

Store D

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?

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?

Aya and Kendra want to estimate the height of a tree. On a sunny day, Aya measures Kendra's shadow as 3 meters long, and Kendra measures the tree's shadow as 15 meters long. Kendra is 1.5 meters tall. How tall is the tree?

7.5 meters

22.5 meters

30 meters

45 meters

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

A

B

C

D

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

A map has a scale of 3 inches = 100 miles. Cities A and B are 753 miles apart. Let d be the distance between the two cities on the map. Which of the following is not correct?

Use the expression below to answer the question that follows.

(4×103)×(3×104)6×106 \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?

2

20

200

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

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

Which of the numbers below is the decimal equivalent of $\dfrac{3}{8}?$

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

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

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

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

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?

$\dfrac{4}{7}$ 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.

$7-4=3$ and $8-5=3$ , so the fractions are equal.

$4\times 8=32$ and $7\times 5=35$ . Since $32<35$ , $\dfrac{5}{8}<\dfrac{4}{7}$

$4<5$ and $7<8$ , so $\dfrac{4}{7}<\dfrac{5}{8}$

$\large {\left( x+3 \right)}^{2}=10$

$\large \left( x+3 \right)\left( x+3 \right)=10$

$\large {x}^{2}+9=10$

$\large {x}^{2}+9-9=10-9$

$\large {x}^{2}=1$

$\large x=1\text{ or }x=-1$

$\large \dfrac{\left( 155 \right)\times \left( 6,124 \right)}{977}$

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