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## MTEL General Curriculum Mathematics Practice

 Question 1

#### 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 1 Explanation:
Topic: Calculate the probabilities of simple and compound events and of independent and dependent events (Objective 0026).
 Question 2

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

 A $$\large \dfrac{3}{5}p-2$$Hint: The company gets 3/5=60% of the price, minus the $2 per item. B $$\large \dfrac{3}{5}\left( p-2 \right)$$Hint: This is sensible, but not what the problem states. C $$\large \dfrac{2}{5}p+2$$Hint: The company pays the extra$2; it doesn't collect it. D $$\large \dfrac{2}{5}p-2$$Hint: This has the company getting 2/5 = 40% of the price of each item, but that's what the representative gets.
Question 2 Explanation:
Topic: Use algebra to solve word problems involving fractions, ratios, proportions, and percents (Objective 0020).
 Question 3

#### 1.5°

Hint:
Celsius and Fahrenheit don't increase at the same rate.

#### 1.8°

Hint:
That's how much the Fahrenheit temp increases when the Celsius temp goes up by 1 degree.

#### 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.

#### 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 3 Explanation:
Topic: Interpret the meaning of the slope and the intercepts of a linear equation that models a real-world situation (Objective 0022).
 Question 4

#### P divides 30

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

#### P divides 48

Hint:
P=5 doesn't work.

#### P divides 75

Hint:
P=2 doesn't work.

#### P divides 80

Hint:
P=3 doesn't work.
Question 4 Explanation:
Topic: Find the prime factorization of a number and recognize its uses (Objective 0018).
 Question 5

#### 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 5 Explanation:
Converting between fractions, decimals, and percents (Objective 0017).
 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

#### The chart below gives percentiles for the number of sit-ups that boys of various ages can do in 60 seconds (source , June 24, 2011) #### 95% of 12 year old boys can do 56 sit-ups in 60 seconds.

Hint:
The 95th percentile means that 95% of scores are less than or equal to 56, and 5% are greater than or equal to 56.

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

Hint:
The 25th percentile means that 25% of scores are less than or equal to 19, and 75% are greater than or equal to 19.

#### Half of all 13 year old boys can do less than 41 sit-ups in 60 seconds and half can do more than 41 sit-ups in 60 seconds.

Hint:
Close, but not quite. There's no accounting for boys who can do exactly 41 sit ups. Look at these data: 10, 20, 41, 41, 41, 41, 50, 60, 90. The median is 41, but more than half can do 41 or more.

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

Hint:
The "at least" is necessary due to duplicates. Suppose the data were 10, 20, 51, 51. The 75th percentile is 51, but 100% of the boys can only do 51 or fewer situps.
Question 7 Explanation:
Topic: Analyze and interpret various graphic and nongraphic data representations (e.g., frequency distributions, percentiles) (Objective 0025).
 Question 8

#### 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 8 Explanation:
Topic: Understand divisibility rules and why they work (Objective 018).
 Question 9

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

 A $$\large1.3\times {{10}^{6}}$$ B $$\large1.3\times {{10}^{9}}$$ Hint: That's one billion three hundred million. C $$\large1.03\times {{10}^{6}}$$ Hint: That's one million thirty thousand. D $$\large1.03\times {{10}^{9}}$$Hint: That's one billion thirty million
Question 9 Explanation:
Topic: Scientific Notation (Objective 0016)
 Question 10

#### 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 10 Explanation:
Topic: Derive and use formulas for calculating the lengths, perimeters, areas, volumes, and surface areas of geometric shapes and figures (Objective 0023).
 Question 11

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

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

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

Hint:
In the middle example,the answer is correct.

#### 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.

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

Hint:
Lowest terms is irrelevant.

#### It is valid for all rational numbers.

Hint:
Can't be correct as the first and last examples have the wrong answers.
Question 11 Explanation:
Topic: Analyze Non-Standard Computational Algorithms (Objective 0019).
 Question 12

#### It is too high by a factor of 100

Question 12 Explanation:
Topics: Estimation, Scientific Notation in the real world (Objective 0016).
 Question 13

#### Commutative Property.

Hint:
For addition, the commutative property is $$a+b=b+a$$ and for multiplication it's $$a \times b = b \times a$$.

#### Associative Property.

Hint:
For addition, the associative property is $$(a+b)+c=a+(b+c)$$ and for multiplication it's $$(a \times b) \times c=a \times (b \times c)$$

#### Identity Property.

Hint:
0 is the additive identity, because $$a+0=a$$ and 1 is the multiplicative identity because $$a \times 1=a$$. The phrase "identity property" is not standard.

#### Distributive Property.

Hint:
$$(25+1) \times 16 = 25 \times 16 + 1 \times 16$$. This is an example of the distributive property of multiplication over addition.
Question 13 Explanation:
Topic: Analyze and justify mental math techniques, by applying arithmetic properties such as commutative, distributive, and associative (Objective 0019). Note that it's hard to write a question like this as a multiple choice question -- worthwhile to understand why the other steps work too.
 Question 14

#### 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 14 Explanation:
Topics: Identify prime and composite numbers and demonstrate knowledge of divisibility rules (Objective 0018).
 Question 15

#### I, II, and III

Hint:
The integers are ...-3, -2, -1, 0, 1, 2, 3, ....
Question 15 Explanation:
Topic: Characteristics of Integers (Objective 0016)
 Question 16

#### The histogram below shows the frequency of a class's scores on a 4 question quiz. #### What was the mean score on the quiz?

 A $$\large 2.75$$Hint: There were 20 students who took the quiz. Total points earned: $$2 \times 1+6 \times 2+ 7\times 3+5 \times 4=55$$, and 55/20 = 2.75. B $$\large 2$$Hint: How many students are there total? Did you count them all? C $$\large 3$$Hint: How many students are there total? Did you count them all? Be sure you're finding the mean, not the median or the mode. D $$\large 2.5$$Hint: How many students are there total? Did you count them all? Don't just take the mean of 1, 2, 3, 4 -- you have to weight them properly.
Question 16 Explanation:
Topics: Analyze and interpret various graphic representations, and use measures of central tendency (e.g., mean, median, mode) and spread to describe and interpret real-world data (Objective 0025).
 Question 17

#### 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 17 Explanation:
Topic: Models of Fractions (Objective 0017)
 Question 18

#### 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 18 Explanation:
Topic: Use unit conversions and dimensional analysis to solve measurement problems (Objective 0023).
 Question 19

#### 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.

#### 120

Hint:
The floor is 120 sq feet, and the tiles are smaller than 1 sq foot.

Hint:

#### 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 19 Explanation:
Topic: Estimate and calculate measurements, use unit conversions to solve measurement problems, solve measurement problems in real-world situations (Objective 0023).
 Question 20

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

 A $$\large 2$$Hint: Write $$8^3$$ as a power of 2. B $$\large \dfrac{1}{2}$$Hint: $$8^3 \cdot {2}^{-10}={(2^3)}^3 \cdot {2}^{-10}$$ =$$2^9 \cdot {2}^{-10} =2^{-1}$$ C $$\large 16$$Hint: Write $$8^3$$ as a power of 2. D $$\large \dfrac{1}{16}$$Hint: Write $$8^3$$ as a power of 2.
Question 20 Explanation:
Topic: Laws of Exponents (Objective 0019).
 Question 21

#### 4 congruent sides

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

#### 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.

#### 4 congruent angles

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

#### 2 sets of parallel sides

Hint:
All rhombi are parallelograms.
Question 21 Explanation:
Topic: Classify and analyze polygons using attributes of sides and angles, and symmetry (Objective 0024).
 Question 22

#### Five million

Hint:
Pay attention to the exponents. Adding 3 and 2 doesn't work because they have different place values.

#### Fifty thousand

Hint:
Pay attention to the exponents. Adding 3 and 2 doesn't work because they have different place values.

Hint:

#### Thirty thousand

Hint:
$$3\times {{10}^{4}} = 30,000;$$ the other term is much smaller and doesn't change the estimate.
Question 22 Explanation:
Topics: Place value, scientific notation, estimation (Objective 0016)
 Question 23

#### 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.

Hint:
Draw a picture.

Hint:
Draw a picture.

#### 45 meters

Hint:
Draw a picture.
Question 23 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 24

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

Hint:
Rise is more than 30 inches.

#### B

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

#### C

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

#### D

Hint:
Slope is 1:10 -- too steep.
Question 24 Explanation:
Topic: Interpret meaning of slope in a real world situation (Objective 0022).
 Question 25

#### The 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 25 Explanation:
Topic: Recognize and apply the concept of conditional probability (Objective 0026).
 Question 26

#### 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 26 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 27

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

 A $$\large x=-3$$Hint: $$\left| {{(-3+2)}^{3}} \right|$$=$$\left | {(-1)}^3 \right |$$=$$\left | -1 \right |=1$$ . B $$\large x=0$$Hint: $$\left| {{(0+2)}^{3}} \right|$$=$$\left | {2}^3 \right |$$=$$\left | 8 \right |$$ =$$8$$ C $$\large x=-4$$Hint: $$\left| {{(-4+2)}^{3}} \right|$$=$$\left | {(-2)}^3 \right |$$=$$\left | -8 \right |$$ =$$8$$ D $$\large x=1$$Hint: $$\left| {{(1+2)}^{3}} \right|$$=$$\left | {3}^3 \right |$$=$$\left | 27 \right |$$ = $$27$$
Question 27 Explanation:
Topics: Laws of exponents, order of operations, interpret absolute value (Objective 0019).
 Question 28

#### 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 28 Explanation:
Topic: Solve a variety of measurement problems (e.g., time, temperature, rates, average rates of change) in real-world situations (Objective 0023).
 Question 29

#### The student‘s solution is correct.

Hint:
Try plugging into the original solution.

#### 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.

#### The student did not correctly use the distributive property.

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

#### The student did not correctly use the commutative property.

Hint:
Commutative property is $$a+b=b+a$$ or $$ab=ba$$.
Question 29 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 30

#### Which of the lists below contains only irrational numbers?

 A $$\large\pi , \quad \sqrt{6},\quad \sqrt{\dfrac{1}{2}}$$ B $$\large\pi , \quad \sqrt{9}, \quad \pi +1$$Hint: $$\sqrt{9}=3$$ C $$\large\dfrac{1}{3},\quad \dfrac{5}{4},\quad \dfrac{2}{9}$$Hint: These are all rational. D $$\large-3,\quad 14,\quad 0$$Hint: These are all rational.
Question 30 Explanation:
Topic: Identifying rational and irrational numbers (Objective 0016).
 Question 31

#### The picture below shows identical circles drawn on a piece of paper.  The rectangle represents an index card that is blocking your view of $$\dfrac{3}{5}$$ of the circles on the paper.  How many circles are covered by the rectangle? #### 4

Hint:
The card blocks more than half of the circles, so this number is too small.

#### 5

Hint:
The card blocks more than half of the circles, so this number is too small.

#### 8

Hint:
The card blocks more than half of the circles, so this number is too small.

#### 12

Hint:
2/5 of the circles or 8 circles are showing. Thus 4 circles represent 1/5 of the circles, and $$4 \times 5=20$$ circles represent 5/5 or all the circles. Thus 12 circles are hidden.
Question 31 Explanation:
Topic: Models of Fractions (Objective 0017)
 Question 32

#### 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 32 Explanation:
Topic: Recognize and apply the concept of conditional probability (Objective 0026).
 Question 33

#### 21 cm

Hint:
How many miles would correspond to 24 cm on the map? Try adjusting from there.

#### 22 cm

Hint:
How many miles would correspond to 24 cm on the map? Try adjusting from there.

#### 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.

#### 24 cm

Hint:
4 groups of 6 cm is over 2800 miles on the map, which is too much.
Question 33 Explanation:
Topic: Apply proportional thinking to estimate quantities in real world situations (Objective 0019).
 Question 34

#### 58 x 22

Hint:
This problem involves regrouping, which the student does not do correctly.

#### 16 x 24

Hint:
This problem involves regrouping, which the student does not do correctly.

#### 31 x 23

Hint:
There is no regrouping with this problem.

#### 141 x 32

Hint:
This problem involves regrouping, which the student does not do correctly.
Question 34 Explanation:
Topic: Analyze computational algorithms (Objective 0019).
 Question 35

#### 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 35 Explanation:
Topic: Analyze the relationships among proportions, constant rates, and linear functions (Objective 0022).
 Question 36

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

Hint:
Estimate with simpler fractions.

#### C

Hint:
Estimate with simpler fractions.

#### D

Hint:
Estimate with simpler fractions.
Question 36 Explanation:
Topic: Understand meaning and models of operations on fractions (Objective 0019).
 Question 37

#### 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 37 Explanation:
Topic: Solve linear equations (Objective 0020).
 Question 38

#### 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 38 Explanation:
Topic: Ordering decimals and integers (Objective 0017).
 Question 39

#### 23 flats, 4 rods, 7 little cubes

Hint:
Be sure you read the question carefully: 2300+40+7=2347

#### 2 large cubes, 3 flats, 47 rods

Hint:
2000+300+470 $$\neq$$ 2347

#### 2 large cubes, 34 rods, 7 little cubes

Hint:
Be sure you read the question carefully: 2000+340+7=2347

#### 2 large cubes, 3 flats, 4 rods, 7 little cubes

Hint:
Be sure you read the question carefully: 2000+300+40+7=2347
Question 39 Explanation:
Topic: Place Value (Objective 0016)
 Question 40

#### Store B

Hint:
This saves 15% and C saves 25%.

#### Store D

Hint:
This is about 20% off, which is less of a discount than C.
Question 40 Explanation:
Topic: Understand the meanings and models of integers, fractions, decimals,percents, and mixed numbers and apply them to the solution of word problems (Objective 0017).
 Question 41

#### 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 41 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 42

#### Four children randomly line up, single file.  What is the probability that they are in height order, with the shortest child in front?   All of the children are different heights.

 A $$\large \dfrac{1}{4}$$Hint: Try a simpler question with 3 children -- call them big, medium, and small -- and list all the ways they could line up. Then see how to extend your logic to the problem with 4 children. B $$\large \dfrac{1}{256}$$Hint: Try a simpler question with 3 children -- call them big, medium, and small -- and list all the ways they could line up. Then see how to extend your logic to the problem with 4 children. C $$\large \dfrac{1}{16}$$Hint: Try a simpler question with 3 children -- call them big, medium, and small -- and list all the ways they could line up. Then see how to extend your logic to the problem with 4 children. D $$\large \dfrac{1}{24}$$Hint: The number of ways for the children to line up is $$4!=4 \times 3 \times 2 \times 1 =24$$ -- there are 4 choices for who is first in line, then 3 for who is second, etc. Only one of these lines has the children in the order specified.
Question 42 Explanation:
Topic: Apply knowledge of combinations and permutations to the computation of probabilities (Objective 0026).
 Question 43

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

 A $$\large \dfrac{8}{{{\left( 56 \right)}^{4}}}$$Hint: The bases are whole numbers, and the exponents are negative. How can the numerator be 8? B $$\large \dfrac{64}{{{\left( 56 \right)}^{4}}}$$Hint: The bases are whole numbers, and the exponents are negative. How can the numerator be 64? C $$\large \dfrac{1}{8\cdot {{\left( 56 \right)}^{4}}}$$Hint: $$8^{-6}=8^{-4} \times 8^{-2}$$ D $$\large \dfrac{1}{64\cdot {{\left( 56 \right)}^{4}}}$$
Question 43 Explanation:
Topics: Laws of exponents (Objective 0019).
 Question 44

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

Hint:
All the faces of a tetrahedron are triangles.

#### Triangular Prism

Hint:
A prism has two congruent, parallel bases, connected by parallelograms (since this is a right prism, the parallelograms are rectangles).

#### Triangular Pyramid

Hint:
A pyramid has one base, not two.

#### Trigon

Hint:
A trigon is a triangle (this is not a common term).
Question 44 Explanation:
Topic: Classify and analyze three-dimensional figures using attributes of faces, edges, and vertices (Objective 0024).
 Question 45

#### The commutative property is used incorrectly.

Hint:
The commutative property is $$a+b=b+a$$ or $$ab=ba$$.

#### The associative property is used incorrectly.

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

#### The distributive property is used incorrectly.

Hint:
$$(x+3)(x+3)=x(x+3)+3(x+3)$$=$$x^2+3x+3x+9.$$
Question 45 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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