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

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

If loading fails, click here to try again

Your answers are highlighted below.

Question 1

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 1 Explanation:

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

Question 2

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 2 Explanation:

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

Question 3

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 3 Explanation:

Topic: Using number lines (Objective 0017)

Question 4

A biology class requires a lab fee, which is a whole number of dollars, and the same amount for all students. On Monday the instructor collected $70 in fees, on Tuesday she collected $126, and on Wednesday she collected $266. What is the largest possible amount the fee could be?

A	$2 Hint: A possible fee, but not the largest possible fee. Check the other choices to see which are factors of all three numbers.
B	$7 Hint: A possible fee, but not the largest possible fee. Check the other choices to see which are factors of all three numbers.
C	$14 Hint: This is the greatest common factor of 70, 126, and 266.
D	$70 Hint: Not a factor of 126 or 266, so couldn't be correct.

Question 4 Explanation:

Topic: Use GCF in real-world context (Objective 0018)

Question 5

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 5 Explanation:

Topic: Solve linear equations (Objective 0020).

Question 6

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 6 Explanation:

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

Question 7

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 7 Explanation:

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

Question 8

The pattern below consists of a row of black squares surrounded by white squares.

How many white squares would surround a row of 157 black squares?

A	314 Hint: Try your procedure on a smaller number that you can count to see where you made a mistake.
B	317 Hint: Are there ever an odd number of white squares?
C	320 Hint: One way to see this is that there are 6 tiles on the left and right ends, and the rest of the white tiles are twice the number of black tiles (there are many other ways to look at it too).
D	322 Hint: Try your procedure on a smaller number that you can count to see where you made a mistake.

Question 8 Explanation:

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

Question 9

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

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

Question 9 Explanation:

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

Question 10

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

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

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

Question 10 Explanation:

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

Question 11

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 11 Explanation:

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

Question 12

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 12 Explanation:

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

Question 13

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 13 Explanation:

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

Question 14

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 14 Explanation:

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

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

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

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

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

Question 16 Explanation:

Topic: Compare different data sets (Objective 0025).

Question 17

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

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

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

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

Question 17 Explanation:

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

Question 18

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

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

Question 18 Explanation:

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

Question 19

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 19 Explanation:

Topic: Ordering decimals and integers (Objective 0017).

Question 20

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 20 Explanation:

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

Question 21

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 21 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 22

How many factors does 80 have?

A	$ \large8$ Hint: Don't forget 1 and 80.
B	$ \large9$ Hint: Only perfect squares have an odd number of factors -- otherwise factors come in pairs.
C	$ \large10$ Hint: 1,2,4,5,8,10,16,20,40,80
D	$ \large12$ Hint: Did you count a number twice? Include a number that isn't a factor?

Question 22 Explanation:

Topic: Understand and apply principles of number theory (Objective 0018).

Question 23

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 23 Explanation:

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

Question 24

Here is a student's work on several multiplication problems:

For which of the following problems is this student most likely to get the correct solution, even though he is using an incorrect algorithm?

A	58 x 22 Hint: This problem involves regrouping, which the student does not do correctly.
B	16 x 24 Hint: This problem involves regrouping, which the student does not do correctly.
C	31 x 23 Hint: There is no regrouping with this problem.
D	141 x 32 Hint: This problem involves regrouping, which the student does not do correctly.

Question 24 Explanation:

Topic: Analyze computational algorithms (Objective 0019).

Question 25

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 25 Explanation:

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

Question 26

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 26 Explanation:

Topic: Place Value (Objective 0016)

Question 27

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 27 Explanation:

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

Question 28

The speed of sound in dry air at 68 degrees F is 343.2 meters per second. Which of the expressions below could be used to compute the number of kilometers that a sound wave travels in 10 minutes (in dry air at 68 degrees F)?

A	$ \large 343.2\times 60\times 10$ Hint: In kilometers, not meters.
B	$ \large 343.2\times 60\times 10\times \dfrac{1}{1000}$ Hint: Units are meters/sec $\times$ seconds/minute $\times$ minutes $\times$ kilometers/meter, and the answer is in kilometers.
C	$ \large 343.2\times \dfrac{1}{60}\times 10$ Hint: Include units and make sure answer is in kilometers.
D	$ \large 343.2\times \dfrac{1}{60}\times 10\times \dfrac{1}{1000}$ Hint: Include units and make sure answer is in kilometers.

Question 28 Explanation:

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

Question 29

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

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

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

Question 29 Explanation:

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

Question 30

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 30 Explanation:

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

Question 31

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 31 Explanation:

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

Question 32

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 32 Explanation:

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

Question 33

Use the expression below to answer the question that follows:

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

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

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

Question 33 Explanation:

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

Question 34

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 34 Explanation:

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

Question 35

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 35 Explanation:

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

Question 36

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 36 Explanation:

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

Question 37

Use the expression below to answer the question that follows.

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

Which of the following is closest to the expression above?

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

Question 37 Explanation:

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

Question 38

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 38 Explanation:

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

Question 39

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 39 Explanation:

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

Question 40

At a school fundraising event, people can buy a ticket to spin a spinner like the one below. The region that the spinner lands in tells which, if any, prize the person wins.

If 240 people buy tickets to spin the spinner, what is the best estimate of the number of keychains that will be given away?

A	40 Hint: "Keychain" appears on the spinner twice.
B	80 Hint: The probability of getting a keychain is 1/3, and so about 1/3 of the time the spinner will win.
C	100 Hint: What is the probability of winning a keychain?
D	120 Hint: That would be the answer for getting any prize, not a keychain specifically.

Question 40 Explanation:

Topic: I would call this topic expected value, which is not listed on the objectives. This question is very similar to one on the sample test. It's not a good question in that it's oversimplified (a more difficult and interesting question would be something like, "The school bought 100 keychains for prizes, what is the probability that they will run out before 240 people play?"). In any case, I believe the objective this is meant for is, "Recognize the difference between experimentally and theoretically determined probabilities in real-world situations. (Objective 0026)." This is not something easily assessed with multiple choice .

Question 41

Which of the following is an irrational number?

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

Question 41 Explanation:

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

Question 42

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 42 Explanation:

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

Question 43

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 43 Explanation:

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

Question 44

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 44 Explanation:

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

Question 45

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 45 Explanation:

Topic: Ordering Fractions (Objective 0017)

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

There are 45 questions to complete.

←

List

→

Return

Shaded items are complete.

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

Return

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

A	\( \large A = 3, B = 2, C=0\) 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.

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.

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.

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

A	\( \large3\) 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	\( \large 6x\) Hint: Use the Pythagorean Theorem.
B	\( \large 3x+5{{x}^{2}}\) Hint: Don't forget to take square roots when you use the Pythagorean Theorem.
C	\( \large 3x+\sqrt{5}{{x}^{2}}\) Hint: \(\sqrt {5 x^2}\) is not \(\sqrt {5}x^2\).
D	\( \large 3x+\sqrt{5}{{x}^{{}}}\) Hint: To find the hypotenuse, h, use the Pythagorean Theorem: \(x^2+(2x)^2=h^2.\) \(5x^2=h^2,h=\sqrt{5}x\). The perimeter is this plus x plus 2x.

A	\( \large \dfrac{1}{25}\) 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).

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

A	\( \large -0.044,\quad -0.04,\quad 0.04,\quad 0.044\) 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.

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	\( \large {{N}^{2}}\) 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	\( \large8\) Hint: Don't forget 1 and 80.
B	\( \large9\) Hint: Only perfect squares have an odd number of factors -- otherwise factors come in pairs.
C	\( \large10\) Hint: 1,2,4,5,8,10,16,20,40,80
D	\( \large12\) Hint: Did you count a number twice? Include a number that isn't a factor?

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.

A	\( \large 343.2\times 60\times 10\) Hint: In kilometers, not meters.
B	\( \large 343.2\times 60\times 10\times \dfrac{1}{1000}\) Hint: Units are meters/sec \(\times\) seconds/minute \(\times\) minutes \(\times\) kilometers/meter, and the answer is in kilometers.
C	\( \large 343.2\times \dfrac{1}{60}\times 10\) Hint: Include units and make sure answer is in kilometers.
D	\( \large 343.2\times \dfrac{1}{60}\times 10\times \dfrac{1}{1000}\) Hint: Include units and make sure answer is in kilometers.

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

A	\( \large t=s+15\) 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	\( \large 8x+16x=9x+15x\) 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	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	\( \large \dfrac{502}{1222}\) 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	\( \large \dfrac{4}{6}\) Hint: \( 0.\bar{6}=\dfrac{2}{3}=\dfrac{4}{6}\)
B	\( \large \dfrac{3}{5}\) Hint: This is equal to 0.6, without the repeating decimal. Answer is equivalent to choice c, which is another way to tell that it's wrong.
C	\( \large \dfrac{6}{10}\) Hint: This is equal to 0.6, without the repeating decimal. Answer is equivalent to choice b, which is another way to tell that it's wrong.
D	\( \large \dfrac{1}{6}\) Hint: This is less than a half, and \( 0.\bar{6}\) is greater than a half.

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

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

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

A	\( \large 3.10m=400+0.2m\) 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.

MTEL General Curriculum Mathematics Practice

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?

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?

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 biology class requires a lab fee, which is a whole number of dollars, and the same amount for all students. On Monday the instructor collected $70 in fees, on Tuesday she collected $126, and on Wednesday she collected $266. What is the largest possible amount the fee could be?

$2

$7

$14

$70

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

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

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

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

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

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

The 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 pattern below consists of a row of black squares surrounded by white squares.

How many white squares would surround a row of 157 black squares?

314

317

320

322

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

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

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

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

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

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

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

Here is a method that a student used for subtraction:

Which of the following is correct?

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

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

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

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

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

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

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

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

a

b

c

d

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

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

0 years

12 years

18 years

42 years

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

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

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

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

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

It is valid for all rational numbers.

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

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

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?

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

How many factors does 80 have?

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.

Here is a student's work on several multiplication problems:

For which of the following problems is this student most likely to get the correct solution, even though he is using an incorrect algorithm?

58 x 22

16 x 24

31 x 23

141 x 32

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	\( \large \dfrac{1}{2},\quad \dfrac{1}{3},\quad \dfrac{1}{4},\quad \dfrac{1}{5}\) 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.

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
End