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


Your answers are highlighted below.
Question 1

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

A
\( \large \dfrac{17}{24}\)
Hint:
You might try adding segments so each quadrant is divided into 6 pieces with equal area -- there will be 24 regions, not all the same shape, but all the same area, with 17 of them shaded (for the top left quarter, you could also first change the diagonal line to a horizontal or vertical line that divides the square in two equal pieces and shade one) .
B
\( \large \dfrac{3}{4}\)
Hint:
Be sure you're taking into account the different sizes of the pieces.
C
\( \large \dfrac{2}{3}\)
Hint:
The bottom half of the picture is 2/3 shaded, and the top half is more than 2/3 shaded, so this answer is too small.
D
\( \large \dfrac{17}{6} \)
Hint:
This answer is bigger than 1, so doesn't make any sense. Be sure you are using the whole picture, not one quadrant, as the unit.
Question 1 Explanation: 
Topic: Models of Fractions (Objective 0017)
Question 2

In which table below is y a function of x?

A
Hint:
If x=3, y can have two different values, so it's not a function.
B
Hint:
If x=3, y can have two different values, so it's not a function.
C
Hint:
If x=1, y can have different values, so it's not a function.
D
Hint:
Each value of x always corresponds to the same value of y.
Question 2 Explanation: 
Topic: Understand the definition of function and various representations of functions (e.g., input/output machines, tables, graphs, mapping diagrams, formulas) (Objective 0021).
Question 3

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 3 Explanation: 
Identify variables and derive algebraic expressions that represent real-world situations (Objective 0020).
Question 4

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 4 Explanation: 
Topic: Apply LCM in "real-world" situations (according to standardized tests....) (Objective 0018).
Question 5

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

 
A

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

Hint:
Part of the figure would move below the x-axis with these transformations.
B

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

Hint:
See what happens to the point (5,1) under this set of transformations.
C

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

D

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

Hint:
See what happens to the point (3,3) under this set of transformations.
Question 5 Explanation: 
Topic:Analyze and apply geometric transformations (e.g., translations, rotations, reflections, dilations) (Objective 0024).
Question 6

The picture below represents a board with pegs on it, where the closest distance between two pegs is 1 cm.  What is the area of the pentagon shown?

A
\( \large 8\text{ c}{{\text{m}}^{2}} \)
Hint:
Don't just count the dots inside, that doesn't give the area. Try adding segments so that the slanted lines become the diagonals of rectangles.
B
\( \large 11\text{ c}{{\text{m}}^{2}}\)
Hint:
Try adding segments so that the slanted lines become the diagonals of rectangles.
C
\( \large 11.5\text{ c}{{\text{m}}^{2}}\)
Hint:
An easy way to do this problem is to use Pick's Theorem (of course, it's better if you understand why Pick's theorem works): area = # pegs inside + half # pegs on the border - 1. In this case 8+9/2-1=11.5. A more appropriate strategy for elementary classrooms is to add segments; here's one way.

There are 20 1x1 squares enclosed, and the total area of the triangles that need to be subtracted is 8.5
D
\( \large 12.5\text{ c}{{\text{m}}^{2}}\)
Hint:
Try adding segments so that the slanted lines become the diagonals of rectangles.
Question 6 Explanation: 
Topics: Calculate measurements and derive and use formulas for calculating the areas of geometric shapes and figures (Objective 0023).
Question 7

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 7 Explanation: 
Topic: Classify and analyze three-dimensional figures using attributes of faces, edges, and vertices (Objective 0024).
Question 8

An above-ground swimming pool is in the shape of a regular hexagonal prism, is one meter high, and holds 65 cubic meters of water.  A second pool has a base that is also a regular hexagon, but with sides twice as long as the sides in the first pool.  This second pool is also one meter high.  How much water will the second pool hold?

A
\( \large 65\text{ }{{\text{m}}^{3}}\)
Hint:
A bigger pool would hold more water.
B
\( \large 65\cdot 2\text{ }{{\text{m}}^{3}}\)
Hint:
Try a simpler example, say doubling the sides of the base of a 1 x 1 x 1 cube.
C
\( \large 65\cdot 4\text{ }{{\text{m}}^{3}}\)
Hint:
If we think of the pool as filled with 1 x 1 x 1 cubes (and some fractions of cubes), then scaling to the larger pool changes each 1 x 1 x 1 cube to a 2 x 2 x 1 prism, or multiplies volume by 4.
D
\( \large 65\cdot 8\text{ }{{\text{m}}^{3}}\)
Hint:
Try a simpler example, say doubling the sides of the base of a 1 x 1 x 1 cube.
Question 8 Explanation: 
Topic: Determine how the characteristics (e.g., area, volume) of geometric figures and shapes are affected by changes in their dimensions (Objective 0023).
Question 9

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 9 Explanation: 
Topics: Identify prime and composite numbers and demonstrate knowledge of divisibility rules (Objective 0018).
Question 10

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 10 Explanation: 
Topic: Ordering decimals and integers (Objective 0017).
Question 11

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 11 Explanation: 
Topic: Analyze the relationships among proportions, constant rates, and linear functions (Objective 0022).
Question 12

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 12 Explanation: 
Topic: Recognize and apply the concepts of variable, equality, and equation to express relationships algebraically (Objective 0020).
Question 13

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 13 Explanation: 
Topic: Understand and apply principles of number theory (Objective 0018).
Question 14

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

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

A
\( \large \dfrac{7}{16}\)
Hint:
Multiplication comes before addition and subtraction in the order of operations.
B
\( \large \dfrac{1}{2}\)
Hint:
Addition and subtraction are of equal priority in the order of operations -- do them left to right.
C
\( \large \dfrac{3}{4}\)
Hint:
\( \dfrac{3}{4}-\dfrac{1}{8}+\dfrac{2}{8}\times \dfrac{1}{2}\)=\( \dfrac{3}{4}-\dfrac{1}{8}+\dfrac{1}{8}\)=\( \dfrac{3}{4}+-\dfrac{1}{8}+\dfrac{1}{8}\)=\( \dfrac{3}{4}\)
D
\( \large \dfrac{3}{16}\)
Hint:
Multiplication comes before addition and subtraction in the order of operations.
Question 16 Explanation: 
Topic: Operations on Fractions, Order of Operations (Objective 0019).
Question 17

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 17 Explanation: 
Topic: Analyze geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to concepts of symmetry (Objective 0024).
Question 18

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 18 Explanation: 
Topic: Least Common Multiple (Objective 0018)
Question 19

Here is a student’s work solving an equation:

\( x-4=-2x+6\)

\( x-4+4=-2x+6+4\)

\( x=-2x+10\)

\( x-2x=10\)

\( x=10\)

Which of the following statements is true?

A

The student’s solution is correct.

Hint:
Try plugging into the original solution.
B

The student did not correctly use properties of equality.

Hint:
After \( x=-2x+10\), the student subtracted 2x on the left and added 2x on the right.
C

The student did not correctly use the distributive property.

Hint:
Distributive property is \(a(b+c)=ab+ac\).
D

The student did not correctly use the commutative property.

Hint:
Commutative property is \(a+b=b+a\) or \(ab=ba\).
Question 19 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 20

Taxicab fares in Boston (Spring 2012) are $2.60 for the first \(\dfrac{1}{7}\) of a mile or less and $0.40 for each \(\dfrac{1}{7}\) of a mile after that.

Let d represent the distance a passenger travels in miles (with \(d>\dfrac{1}{7}\)). Which of the following expressions represents the total fare?

A
\( \large \$2.60+\$0.40d\)
Hint:
It's 40 cents for 1/7 of a mile, not per mile.
B
\( \large \$2.60+\$0.40\dfrac{d}{7}\)
Hint:
According to this equation, going 7 miles would cost $3; does that make sense?
C
\( \large \$2.20+\$2.80d\)
Hint:
You can think of the fare as $2.20 to enter the cab, and then $0.40 for each 1/7 of a mile, including the first 1/7 of a mile (or $2.80 per mile).

Alternatively, you pay $2.60 for the first 1/7 of a mile, and then $2.80 per mile for d-1/7 miles. The total is 2.60+2.80(d-1/7) = 2.60+ 2.80d -.40 = 2.20+2.80d.
D
\( \large \$2.60+\$2.80d\)
Hint:
Don't count the first 1/7 of a mile twice.
Question 20 Explanation: 
Topic: Identify variables and derive algebraic expressions that represent real-world situations (Objective 0020), and select the linear equation that best models a real-world situation (Objective 0022).
Question 21

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 21 Explanation: 
Topic: Identify and analyze direct and inverse relationships in tables, graphs, algebraic expressions and real-world situations (Objective 0021)
Question 22

Which of the graphs below represent functions?

I. 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 22 Explanation: 
Understand the definition of function and various representations of functions (e.g., input/output machines, tables, graphs, mapping diagrams, formulas). (Objective 0021).
Question 23

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 23 Explanation: 
Topic: Venn Diagrams (Objective 0025)
Question 24

Below is a portion of a number line:

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

 
A
\( \large 0.645\)
Hint:
That point is marked on the line, to the right.
B
\( \large 0.6421\)
Hint:
That point is to the left of point B.
C
\( \large 0.6422\)
Hint:
That point is to the left of point B.
D
\( \large 0.6425\)
Question 24 Explanation: 
Topic: Using Number Lines (Objective 0017)
Question 25

The following story situations model \( 12\div 3\):

I)  Jack has 12 cookies, which he wants to share equally between himself and two friends.  How many cookies does each person get?

II) Trent has 12 cookies, which he wants to put into bags of 3 cookies each.  How many bags can he make?

III) Cicely has $12.  Cookies cost $3 each.  How many cookies can she buy?

Which of these questions illustrate the same model of division, either partitive (partioning) or measurement (quotative)?

A

I and II

B

I and III

C

II and III

Hint:
Problem I is partitive (or partitioning or sharing) -- we put 12 objects into 3 groups. Problems II and III are quotative (or measurement) -- we put 12 objects in groups of 3.
D

All three problems model the same meaning of division

Question 25 Explanation: 
Topic: Understand models of operations on numbers (Objective 0019).
Question 26

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 26 Explanation: 
Topic: Demonstrate knowledge of divisibility rules (Objective 0018).
Question 27

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 27 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 28

What is the greatest common factor of 540 and 216?

A
\( \large{{2}^{2}}\cdot {{3}^{3}}\)
Hint:
One way to solve this is to factor both numbers: \(540=2^2 \cdot 3^3 \cdot 5\) and \(216=2^3 \cdot 3^3\). Then take the smaller power for each prime that is a factor of both numbers.
B
\( \large2\cdot 3\)
Hint:
This is a common factor of both numbers, but it's not the greatest common factor.
C
\( \large{{2}^{3}}\cdot {{3}^{3}}\)
Hint:
\(2^3 = 8\) is not a factor of 540.
D
\( \large{{2}^{2}}\cdot {{3}^{2}}\)
Hint:
This is a common factor of both numbers, but it's not the greatest common factor.
Question 28 Explanation: 
Topic: Find the greatest common factor of a set of numbers (Objective 0018).
Question 29

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

A
\( \large {83}.0{3}\cdot 0.{761}\)
Hint:
This equation gives less than the number of yen per dollar, but 1 Euro is worth more than 1 dollar.
B
\( \large \dfrac{0.{761}}{{83}.0{3}}\)
Hint:
Number is way too small.
C
\( \large \dfrac{{83}.0{3}}{0.{761}}\)
Hint:
One strategy here is to use easier numbers, say 1 dollar = .5 Euros and 100 yen, then 1 Euro would be 200 Yen (change the numbers in the equations and see what works). Another is to use dimensional analysis: we want # yen per Euro, or yen/Euro = yen/dollar \(\times\) dollar/Euro = \(83.03 \times \dfrac {1}{0.761}\)
D
\( \large \dfrac{1}{0.{761}}\cdot \dfrac{1}{{83}.0{3}}\)
Hint:
Number is way too small.
Question 29 Explanation: 
Topic: Analyze the relationships among proportions, constant rates, and linear functions (Objective 0022).
Question 30

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 30 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 31

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 31 Explanation: 
Topic: Identifying rational and irrational numbers (Objective 0016).
Question 32

A family on vacation drove the first 200 miles in 4 hours and the second 200 miles in 5 hours.  Which expression below gives their average speed for the entire trip?

A
\( \large \dfrac{200+200}{4+5}\)
Hint:
Average speed is total distance divided by total time.
B
\( \large \left( \dfrac{200}{4}+\dfrac{200}{5} \right)\div 2\)
Hint:
This seems logical, but the problem is that it weights the first 4 hours and the second 5 hours equally, when each hour should get the same weight in computing the average speed.
C
\( \large \dfrac{200}{4}+\dfrac{200}{5} \)
Hint:
This would be an average of 90 miles per hour!
D
\( \large \dfrac{400}{4}+\dfrac{400}{5} \)
Hint:
This would be an average of 180 miles per hour! Even a family of race car drivers probably doesn't have that average speed on a vacation!
Question 32 Explanation: 
Topic: Solve a variety of measurement problems (e.g., time, temperature, rates, average rates of change) in real-world situations (Objective 0023).
Question 33

Below are several expressions

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 33 Explanation: 
Topic: Converting between fractions, decimals, and percents (Objective 0017)
Question 34

Use the graph below to answer the question that follows:

 

The graph above best matches which of the following scenarios:

A

George left home at 10:00 and drove to work on a crooked path. He was stopped in traffic at 10:30 and 10:45. He drove 30 miles total.

Hint:
Just because he ended up 30 miles from home doesn't mean he drove 30 miles total.
B

George drove to work. On the way to work there is a little hill and a big hill. He slowed down for them. He made it to work at 11:15.

Hint:
The graph is not a picture of the roads.
C

George left home at 10:15. He drove 10 miles, then realized he’d forgotten something at home. He turned back and got what he’d forgotten. Then he drove in a straight line, at many different speeds, until he got to work around 11:15.

Hint:
A straight line on a distance versus time graph means constant speed.
D

George left home at 10:15. He drove 10 miles, then realized he’d forgotten something at home. He turned back and got what he’d forgotten. Then he drove at a constant speed until he got to work around 11:15.

Question 34 Explanation: 
Topic: Use qualitative graphs to represent functional relationships in the real world (Objective 0021).
Question 35

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 35 Explanation: 
Topic: Derive and use formulas for calculating the lengths, perimeters, areas, volumes, and surface areas of geometric shapes and figures (Objective 0023).
Question 36

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 36 Explanation: 
Converting between fractions, decimals, and percents (Objective 0017).
Question 37

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 37 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 38

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 38 Explanation: 
Topic: Analyze and justify standard and non-standard computational techniques (Objective 0019).
Question 39

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

A
\( \large \dfrac{1}{7}\)
Hint:
It doesn't matter what day the first person was born on. The probability that the second person will match is 1/7 (just designate one person the first and the other the second). Another way to look at it is that if you list the sample space of all possible pairs, e.g. (Wed, Sun), there are 49 such pairs, and 7 of them are repeats of the same day, and 7/49=1/7.
B
\( \large \dfrac{1}{14}\)
Hint:
What would be the sample space here? Ie, how would you list 14 things that you pick one from?
C
\( \large \dfrac{1}{42}\)
Hint:
If you wrote the seven days of the week on pieces of paper and put the papers in a jar, this would be the probability that the first person picked Sunday and the second picked Monday from the jar -- not the same situation.
D
\( \large \dfrac{1}{49}\)
Hint:
This is the probability that they are both born on a particular day, e.g. Sunday.
Question 39 Explanation: 
Topic: Calculate the probabilities of simple and compound events and of independent and dependent events (Objective 0026).
Question 40

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

A

M is odd.

Hint:
All multiples of 26 are also multiples of 2, so they must be even.
B

M is a multiple of 3.

Hint:
3 x 26 is a multiple of both 3 and 26.
C

M is 26.

Hint:
1 x 26 is a multiple of 26.
D

M is 0.

Hint:
0 x 26 is a multiple of 26.
Question 40 Explanation: 
Topic: Characteristics of composite numbers (Objective 0018).
Question 41

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

A
\( \large \dfrac{1}{3}\)
Hint:
This is the probability that the first child picks a red gumball, but not that the first two children pick red gumballs.
B
\( \large \dfrac{1}{8}\)
Hint:
Are you adding things that you should be multiplying?
C
\( \large \dfrac{1}{9}\)
Hint:
This would be the probability if the gumballs were returned to the bag.
D
\( \large \dfrac{1}{15}\)
Hint:
The probability that the first child picks red is 2/6 = 1/3. Then there are 5 gumballs in the bag, one red, so the probability that the second child picks red is 1/5. Thus 1/5 of the time, after the first child picks red, the second does too, so the probability is 1/5 x 1/3 = 1/15.
Question 41 Explanation: 
Topic: Calculate the probabilities of simple and compound events and of independent and dependent events (Objective 0026).
Question 42

Use the graph below to answer the question that follows.

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

A
Hint:
Try following the point (1,4) to see where it goes after each transformation.
B
C
Hint:
Make sure you're reflecting in the correct axis.
D
Hint:
Make sure you're rotating the correct direction.
Question 42 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 43

Which of the following is equivalent to

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

A
\( \large A-B-\dfrac{C}{DE} \)
Hint:
In the order of operations, multiplication and division have the same priority, so do them left to right; same with addition and subtraction.
B
\( \large A-B+\dfrac{CE}{D}\)
Hint:
In practice, you're better off using parentheses than writing an expression like the one in the question. The PEMDAS acronym that many people memorize is misleading. Multiplication and division have equal priority and are done left to right. They have higher priority than addition and subtraction. Addition and subtraction also have equal priority and are done left to right.
C
\( \large \dfrac{AE-BE+CE}{D}\)
Hint:
Use order of operations, don't just compute left to right.
D
\( \large A-B+\dfrac{C}{DE}\)
Hint:
In the order of operations, multiplication and division have the same priority, so do them left to right
Question 43 Explanation: 
Topic: Justify algebraic manipulations by application of the properties of order of operations (Objective 0020).
Question 44

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 44 Explanation: 
Topic: Using number lines (Objective 0017)
Question 45

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 45 Explanation: 
Topic: Derive and use formulas for calculating the lengths, perimeters, areas, volumes, and surface areas of geometric shapes and figures (Objective 0023).
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.
12345
678910
1112131415
1617181920
2122232425
2627282930
3132333435
3637383940
4142434445
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.