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MTEL General Curriculum Mathematics Practice


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Question 1

The table below gives the result of a survey at a college, asking students whether they were residents or commuters:

Based on the above data, what is the probability that a randomly chosen commuter student is a junior or a senior?

 
A
\( \large \dfrac{34}{43}\)
B
\( \large \dfrac{34}{71}\)
Hint:
This is the probability that a randomly chosen junior or senior is a commuter student.
C
\( \large \dfrac{34}{147}\)
Hint:
This is the probability that a randomly chosen student is a junior or senior who is a commuter.
D
\( \large \dfrac{71}{147}\)
Hint:
This is the probability that a randomly chosen student is a junior or a senior.
Question 1 Explanation: 
Topic: Recognize and apply the concept of conditional probability (Objective 0026).
Question 2

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 2 Explanation: 
Topic: Use qualitative graphs to represent functional relationships in the real world (Objective 0021).
Question 3

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

A

A

Hint:
Rise is more than 30 inches.
B

B

Hint:
Run is almost 24 feet, so rise can be almost 2 feet.
C

C

Hint:
Run is 12 feet, so rise can be at most 1 foot.
D

D

Hint:
Slope is 1:10 -- too steep.
Question 3 Explanation: 
Topic: Interpret meaning of slope in a real world situation (Objective 0022).
Question 4

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 4 Explanation: 
Topic: Solve linear equations (Objective 0020).
Question 5

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

A

A

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

B

Hint:
Estimate with simpler fractions.
C

C

Hint:
Estimate with simpler fractions.
D

D

Hint:
Estimate with simpler fractions.
Question 5 Explanation: 
Topic: Understand meaning and models of operations on fractions (Objective 0019).
Question 6

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 6 Explanation: 
Topic: Understand divisibility rules and why they work (Objective 018).
Question 7

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 7 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 8

Each individual cube that makes up the rectangular solid depicted below has 6 inch sides.  What is the surface area of the solid in square feet?

 
A
\( \large 11\text{ f}{{\text{t}}^{2}}\)
Hint:
Check your units and make sure you're using feet and inches consistently.
B
\( \large 16.5\text{ f}{{\text{t}}^{2}}\)
Hint:
Each square has surface area \(\dfrac{1}{2} \times \dfrac {1}{2}=\dfrac {1}{4}\) sq feet. There are 9 squares on the top and bottom, and 12 on each of 4 sides, for a total of 66 squares. 66 squares \(\times \dfrac {1}{4}\) sq feet/square =16.5 sq feet.
C
\( \large 66\text{ f}{{\text{t}}^{2}}\)
Hint:
The area of each square is not 1.
D
\( \large 2376\text{ f}{{\text{t}}^{2}}\)
Hint:
Read the question more carefully -- the answer is supposed to be in sq feet, not sq inches.
Question 8 Explanation: 
Topics: Use unit conversions to solve measurement problems, and derive and use formulas for calculating surface areas of geometric shapes and figures (Objective 0023).
Question 9

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

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

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

\( \large {x}^{2}+9=10\)

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

\( \large {x}^{2}=1\)

\( \large x=1\text{ or }x=-1\)

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

A

The commutative property is used incorrectly.

Hint:
The commutative property is \(a+b=b+a\) or \(ab=ba\).
B

The associative property is used incorrectly.

Hint:
The associative property is \(a+(b+c)=(a+b)+c\) or \(a \times (b \times c)=(a \times b) \times c\).
C

Order of operations is done incorrectly.

D

The distributive property is used incorrectly.

Hint:
\((x+3)(x+3)=x(x+3)+3(x+3)\)=\(x^2+3x+3x+9.\)
Question 9 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 10

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 10 Explanation: 
Topic: Classify and analyze three-dimensional figures using attributes of faces, edges, and vertices (Objective 0024).
Question 11

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

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 13 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 14

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 14 Explanation: 
Topic: Analyze the relationships among proportions, constant rates, and linear functions (Objective 0022).
Question 15

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

A

P divides 30

Hint:
2, 3, and 5 are the prime factors of 240, and all divide 30.
B

P divides 48

Hint:
P=5 doesn't work.
C

P divides 75

Hint:
P=2 doesn't work.
D

P divides 80

Hint:
P=3 doesn't work.
Question 15 Explanation: 
Topic: Find the prime factorization of a number and recognize its uses (Objective 0018).
Question 16

Which property is not shared by all rhombi?

A

4 congruent sides

Hint:
The most common definition of a rhombus is a quadrilateral with 4 congruent sides.
B

A center of rotational symmetry

Hint:
The diagonal of a rhombus separates it into two congruent isosceles triangles. The center of this line is a center of 180 degree rotational symmetry that switches the triangles.
C

4 congruent angles

Hint:
Unless the rhombus is a square, it does not have 4 congruent angles.
D

2 sets of parallel sides

Hint:
All rhombi are parallelograms.
Question 16 Explanation: 
Topic: Classify and analyze polygons using attributes of sides and angles, and symmetry (Objective 0024).
Question 17

Which of the following is closest to the height of a college student in centimeters?

A

1.6 cm

Hint:
This is more the height of a Lego toy college student -- less than an inch!
B

16 cm

Hint:
Less than knee high on most college students.
C

160 cm

Hint:
Remember, a meter stick (a little bigger than a yard stick) is 100 cm. Also good to know is that 4 inches is approximately 10 cm.
D

1600 cm

Hint:
This college student might be taller than some campus buildings!
Question 17 Explanation: 
Topic: Estimate and calculate measurements using customary, metric, and nonstandard units of measurement (Objective 0023).
Question 18

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 18 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 19

A family went on a long car trip.  Below is a graph of how far they had driven at each hour.

Which of the following is closest to their average speed driving on the trip?

 
A
\( \large d=20t\)
Hint:
Try plugging t=7 into the equation, and see how it matches the graph.
B
\( \large d=30t\)
Hint:
Try plugging t=7 into the equation, and see how it matches the graph.
C
\( \large d=40t\)
D
\( \large d=50t\)
Hint:
Try plugging t=7 into the equation, and see how it matches the graph.
Question 19 Explanation: 
Topic: Select the linear equation that best models a real-world situation (Objective 0022).
Question 20

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 20 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).
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