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

 Question 1

#### $2 Hint: A possible fee, but not the largest possible fee. Check the other choices to see which are factors of all three numbers. ####$7

Hint:
A possible fee, but not the largest possible fee. Check the other choices to see which are factors of all three numbers.

#### $14 Hint: This is the greatest common factor of 70, 126, and 266. ####$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

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

#### 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 $$\large 28 \dfrac{4}{7}$$ mlHint: 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 $$\large 28 \dfrac{1}{4}$$ mlHint: 49 ml of medicine requires 28 ml of saline. How much saline does the extra ml require? C $$\large 28 \dfrac{1}{7}$$ mlHint: 49 ml of medicine requires 28 ml of saline. How much saline does the extra ml require? D $$\large 87.5$$ mlHint: 49 ml of medicine requires 28 ml of saline. How much saline does the extra ml require?
Question 6 Explanation:
Topic: Apply proportional thinking to estimate quantities in real world situations (Objective 0019).
 Question 7

#### The graph above represents the equation $$\large 3x+Ay=B$$, where A and B are integers.  What are the values of A and B?

 A $$\large A = -2, B= 6$$Hint: Plug in (2,0) to get B=6, then plug in (0,-3) to get A=-2. B $$\large A = 2, B = 6$$Hint: Try plugging (0,-3) into this equation. C $$\large A = -1.5, B=-3$$Hint: The problem said that A and B were integers and -1.5 is not an integer. Don't try to use slope-intercept form. D $$\large A = 2, B = -3$$Hint: Try plugging (2,0) into this equation.
Question 7 Explanation:
Topic: Find a linear equation that represents a graph (Objective 0022).
 Question 8

#### 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 8 Explanation:
Topic: Using number lines (Objective 0017)
 Question 9

#### 2,000

Hint:
The answer is bigger than 7,000.

#### 20,000

Hint:
Estimate 896/216 first.

#### 3,000

Hint:
The answer is bigger than 7,000.

#### 30,000

Hint:
$$\dfrac{896}{216} \approx 4$$ and $$7154 \times 4$$ is over 28,000, so this answer is closest.
Question 9 Explanation:
Topics: Estimation, simplifying fractions (Objective 0016, overlaps with other objectives).
 Question 10

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

 A $$\large 20{}^\circ$$Hint: Angles ACD and ADC are congruent since they are base angles of an isosceles triangle. Since the angles of a triangle sum to 180, they sum to 80, and they are 40 deg each. Thus angle ADE is 140 deg, since it makes a straight line with angle ADC. Angles DAE and DEA are base angles of an isosceles triangle and thus congruent-- they sum to 40 deg, so are 20 deg each. B $$\large 25{}^\circ$$Hint: If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal. C $$\large 30{}^\circ$$Hint: If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal. D $$\large 40{}^\circ$$Hint: Make sure you're calculating the correct angle.
Question 10 Explanation:
Topic: Classify and analyze polygons using attributes of sides and angles, including real-world applications. (Objective 0024).
 Question 11

#### 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 11 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 12

#### M is odd.

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

#### M is a multiple of 3.

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

#### M is 26.

Hint:
1 x 26 is a multiple of 26.

#### M is 0.

Hint:
0 x 26 is a multiple of 26.
Question 12 Explanation:
Topic: Characteristics of composite numbers (Objective 0018).
 Question 13

#### 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 13 Explanation:
Topic: Recognize and apply the concept of conditional probability (Objective 0026).
 Question 14

#### 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 14 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 15

#### Which of the graphs below represent functions?

I. II. III. IV.

#### I and IV only.

Hint:
There are vertical lines that go through 2 points in IV .

#### 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).

#### II and III only.

Hint:
Learn about the vertical line test.

#### I, II, and IV only.

Hint:
There are vertical lines that go through 2 points in II.
Question 15 Explanation:
Understand the definition of function and various representations of functions (e.g., input/output machines, tables, graphs, mapping diagrams, formulas). (Objective 0021).
 Question 16

#### 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 16 Explanation:
Topic: Interpret the meaning of the slope and the intercepts of a linear equation that models a real-world situation (Objective 0022).
 Question 17

#### 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 17 Explanation:
Topic: Solve a variety of measurement problems (e.g., time, temperature, rates, average rates of change) in real-world situations (Objective 0023).
 Question 18

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

#### 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 19 Explanation:
Topic: Laws of Exponents (Objective 0019).
 Question 20

#### 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 20 Explanation:
Topic: Calculate the probabilities of simple and compound events and of independent and dependent events (Objective 0026).
 Question 21

#### 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.

#### 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.

#### 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.

#### 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 21 Explanation:
Topic: Use qualitative graphs to represent functional relationships in the real world (Objective 0021).
 Question 22

#### 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 22 Explanation:
Topic: Demonstrate knowledge of divisibility rules (Objective 0018).
 Question 23

#### I, II, and III

Hint:
The integers are ...-3, -2, -1, 0, 1, 2, 3, ....
Question 23 Explanation:
Topic: Characteristics of Integers (Objective 0016)
 Question 24

#### A family has four children.  What is the probability that two children are girls and two are boys?  Assume the the probability of having a boy (or a girl) is 50%.

 A $$\large \dfrac{1}{2}$$Hint: How many different configurations are there from oldest to youngest, e.g. BGGG? How many of them have 2 boys and 2 girls? B $$\large \dfrac{1}{4}$$Hint: How many different configurations are there from oldest to youngest, e.g. BGGG? How many of them have 2 boys and 2 girls? C $$\large \dfrac{1}{5}$$Hint: Some configurations are more probable than others -- i.e. it's more likely to have two boys and two girls than all boys. Be sure you are weighting properly. D $$\large \dfrac{3}{8}$$Hint: There are two possibilities for each child, so there are $$2 \times 2 \times 2 \times 2 =16$$ different configurations, e.g. from oldest to youngest BBBG, BGGB, GBBB, etc. Of these configurations, there are 6 with two boys and two girls (this is the combination $$_{4}C_{2}$$ or "4 choose 2"): BBGG, BGBG, BGGB, GGBB, GBGB, and GBBG. Thus the probability is 6/16=3/8.
Question 24 Explanation:
Topic: Apply knowledge of combinations and permutations to the computation of probabilities (Objective 0026).
 Question 25

#### 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 25 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 26

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

 A $$\large \dfrac{3}{100}=\dfrac{d}{753}$$Hint: Units on both side are inches/mile, and both numerators and denominators correspond -- this one is correct. B $$\large \dfrac{3}{100}=\dfrac{753}{d}$$Hint: Unit on the left is inches per mile, and on the right is miles per inch. The proportion is set up incorrectly (which is what we wanted). Another strategy is to notice that one of A or B has to be the answer because they cannot both be correct proportions. Then check that cross multiplying on A gives part D, so B is the one that is different from the other 3. C $$\large \dfrac{3}{d}=\dfrac{100}{753}$$Hint: Unitless on each side, as inches cancel on the left and miles on the right. Numerators correspond to the map, and denominators to the real life distances -- this one is correct. D $$\large 100d=3\cdot 753$$Hint: This is equivalent to part A.
Question 26 Explanation:
Topic: Analyze the relationships among proportions, constant rates, and linear functions (Objective 0022).
 Question 27

#### 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 27 Explanation:
Topic: Recognize and apply connections between algebra and geometry (e.g., the use of coordinate systems, the Pythagorean theorem) (Objective 0024).
 Question 28

#### $$\large A-B+C\div D\times E$$?

 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 28 Explanation:
Topic: Justify algebraic manipulations by application of the properties of order of operations (Objective 0020).
 Question 29

#### 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 29 Explanation:
Topic: Analyze computational algorithms (Objective 0019).
 Question 30

#### An equiangular triangle that is not equilateral.

Hint:
The AAA property of triangles states that all triangles with corresponding angles congruent are similar. Thus all triangles with three equal angles are similar, and are equilateral.

#### An equiangular quadrilateral that is not equilateral.

Hint:
A rectangle is equiangular (all angles the same measure), but if it's not a square, it's not equilateral (all sides the same length).

#### An equilateral quadrilateral that is not equiangular.

Hint:
This rhombus has equal sides, but it doesn't have equal angles:

#### An equiangular hexagon that is not equilateral.

Hint:
This hexagon has equal angles, but it doesn't have equal sides:
Question 30 Explanation:
Topic: Classify and analyze polygons using attributes of sides and angles (Objective 0024).
 Question 31

#### 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 31 Explanation:
Topic: Identify and analyze direct and inverse relationships in tables, graphs, algebraic expressions and real-world situations (Objective 0021)
 Question 32

#### 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 32 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 33

#### 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 33 Explanation:
Topic: Understand and apply principles of number theory (Objective 0018).
 Question 34

#### Which of the following inequalities describes all values of x  with $$\large \dfrac{x}{2}\le \dfrac{x}{3}$$?

 A $$\large x < 0$$Hint: If x =0, then x/2 = x/3, so this answer can't be correct. B $$\large x \le 0$$ C $$\large x > 0$$Hint: If x =0, then x/2 = x/3, so this answer can't be correct. D $$\large x \ge 0$$Hint: Try plugging in x = 6.
Question 34 Explanation:
Topics: Inequalities, operations (Objective 0019) (not exactly sure how to classify, but this is like one of the problems on the official sample test).
 Question 35

#### 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 35 Explanation:
Topic: Classify and analyze three-dimensional figures using attributes of faces, edges, and vertices (Objective 0024).
 Question 36

#### 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 36 Explanation:
Topic: Analyze and interpret various graphic and nongraphic data representations (e.g., frequency distributions, percentiles) (Objective 0025).
 Question 37

#### 100

Hint:
6124/977 is approximately 6.

#### 200

Hint:
6124/977 is approximately 6.

#### 1,000

Hint:
6124/977 is approximately 6. 155 is approximately 150, and $$6 \times 150 = 3 \times 300 = 900$$, so this answer is closest.

#### 2,000

Hint:
6124/977 is approximately 6.
Question 37 Explanation:
Topics: Estimation, simplifying fractions (Objective 0016).
 Question 38

#### 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 38 Explanation:
Topic: Understand meaning and models of operations on fractions (Objective 0019).
 Question 39

#### 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 39 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 40

#### 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 40 Explanation:
Topic: Ordering decimals and integers (Objective 0017).
 Question 41

#### 0.38

Hint:
If you are just writing the numerator next to the denominator then your technique is way off, but by coincidence your answer is close; try with 2/3 and 0.23 is nowhere near correct.

#### 0.125

Hint:
This is 1/8, not 3/8.

#### 0.83

Hint:
3/8 is less than a half, and 0.83 is more than a half, so they can't be equal.
Question 41 Explanation:
Topic: Converting between fractions and decimals (Objective 0017)
 Question 42

#### 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 42 Explanation:
Topic: Find the greatest common factor of a set of numbers (Objective 0018).
 Question 43

#### 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 43 Explanation:
Topic: Match three-dimensional figures and their two-dimensional representations (e.g., nets, projections, perspective drawings) (Objective 0024).
 Question 44

#### 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 44 Explanation:
Topics: Calculate measurements and derive and use formulas for calculating the areas of geometric shapes and figures (Objective 0023).
 Question 45

#### A sphere

Hint:
All views would be circles.

#### A cone

Hint:
Two views would be triangles, not rectangles.

#### A pyramid

Hint:
How would one view be a circle?
Question 45 Explanation:
Topic: Match three-dimensional figures and their two-dimensional representations (e.g., nets, projections, perspective drawings) (Objective 0024).
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