## 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

Question 1 |

#### The “houses” below are made of toothpicks and gum drops.

#### Which of the following does not represent the number of gumdrops in a row of h houses?

\( \large 2+3h\) Hint: Think of this as start with 2 gumdrops on the left wall, and then add 3 gumdrops for each house. | |

\( \large 5+3(h-1)\) Hint: Think of this as start with one house, and then add 3 gumdrops for each of the other h-1 houses. | |

\( \large h+(h+1)+(h+1)\) Hint: Look at the gumdrops in 3 rows: h gumdrops for the "rooftops," h+1 for the tops of the vertical walls, and h+1 for the floors. | |

\( \large 5+3h\) Hint: This one is not a correct equation (which makes it the correct answer!). Compare to choice A. One of them has to be wrong, as they differ by 3. |

Question 2 |

#### 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?

## 4Hint: The card blocks more than half of the circles, so this number is too small. | |

## 5Hint: The card blocks more than half of the circles, so this number is too small. | |

## 8Hint: The card blocks more than half of the circles, so this number is too small. | |

## 12Hint: 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 3 |

#### Below are front, side, and top views of a three-dimensional solid.

#### Which of the following could be the solid shown above?

## A sphereHint: All views would be circles. | |

## A cylinder | |

## A coneHint: Two views would be triangles, not rectangles. | |

## A pyramidHint: How would one view be a circle? |

Question 4 |

#### Below is a pictorial representation of \(2\dfrac{1}{2}\div \dfrac{2}{3}\):

#### Which of the following is the best description of how to find the quotient from the picture?

## The quotient is \(3\dfrac{3}{4}\). There are 3 whole blocks each representing \(\dfrac{2}{3}\) and a partial block composed of 3 small rectangles. The 3 small rectangles represent \(\dfrac{3}{4}\) of \(\dfrac{2}{3}\). | |

## The quotient is \(3\dfrac{1}{2}\). There are 3 whole blocks each representing \(\dfrac{2}{3}\) and a partial block composed of 3 small rectangles. The 3 small rectangles represent \(\dfrac{3}{6}\) of a whole, or \(\dfrac{1}{2}\).Hint: We are counting how many 2/3's are in 2 1/2: the unit becomes 2/3, not 1. | |

## The quotient is \(\dfrac{4}{15}\). There are four whole blocks separated into a total of 15 small rectangles.Hint: This explanation doesn't make much sense. Probably you are doing "invert and multiply," but inverting the wrong thing. | |

## This picture cannot be used to find the quotient because it does not show how to separate \(2\dfrac{1}{2}\) into equal sized groups.Hint: Study the measurement/quotative model of division. It's often very useful with fractions. |

Question 5 |

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

\( \large 2\) Hint: Write \(8^3\) as a power of 2. | |

\( \large \dfrac{1}{2}\) Hint: \(8^3 \cdot {2}^{-10}={(2^3)}^3 \cdot {2}^{-10}\) =\(2^9 \cdot {2}^{-10} =2^{-1}\) | |

\( \large 16\) Hint: Write \(8^3\) as a power of 2. | |

\( \large \dfrac{1}{16}\) Hint: Write \(8^3\) as a power of 2. |

Question 6 |

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

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

## The median number of pairs of footware owned is between 50 and 60 pairs.Hint: The same number of data points are less than the median as are greater than the median -- but on this histogram, clearly more than half the students own less than 50 pairs of shoes, so the median is less than 50. | |

## The mode of the number of pairs of footware owned is 20.Hint: The mode is the most common number of pairs of footwear owned. We can't tell it from this histogram because each bar represents 10 different numbers-- perhaps 8 students each own each number from 10 to 19, but 40 students own exactly 6 pairs of shoes.... or perhaps not.... | |

## The mean number of pairs of footware owned is less than the median number of pairs of footware owned.Hint: This is a right skewed distribution, and so the mean is bigger than the median -- the few large values on the right pull up the mean, but have little effect on the median. | |

## The median number of pairs of footware owned is between 10 and 20.Hint: There are approximately 230 students represented in this survey, and the 41st through 120th lowest values are between 10 and 20 -- thus the middle value is in that range. |

Question 7 |

#### Here is a mental math strategy for computing 26 x 16:

#### Step 1: 100 x 16 = 1600

#### Step 2: 25 x 16 = 1600 ÷ 4 = 400

#### Step 3: 26 x 16 = 400 + 16 = 416

#### Which property best justifies Step 3 in this strategy?

## 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 8 |

#### Here are some statements:

#### I) 5 is an integer II)\( -5 \) is an integer III) \(0\) is an integer

#### Which of the statements are true?

## I only | |

## I and II only | |

## I and III only | |

## I, II, and IIIHint: The integers are ...-3, -2, -1, 0, 1, 2, 3, .... |

Question 9 |

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

\( \large \dfrac{1}{2},\quad \dfrac{1}{3},\quad \dfrac{1}{4},\quad \dfrac{1}{5}\) Hint: This is ordered from greatest to least. | |

\( \large \dfrac{1}{3},\quad \dfrac{2}{7},\quad \dfrac{3}{8},\quad \dfrac{4}{11}\) Hint: 1/3 = 2/6 is bigger than 2/7. | |

\( \large \dfrac{1}{4},\quad \dfrac{2}{5},\quad \dfrac{2}{3},\quad \dfrac{4}{5}\) Hint: One way to look at this: 1/4 and 2/5 are both less than 1/2, and 2/3 and 4/5 are both greater than 1/2. 1/4 is 25% and 2/5 is 40%, so 2/5 is greater. The distance from 2/3 to 1 is 1/3 and from 4/5 to 1 is 1/5, and 1/5 is less than 1/3, so 4/5 is bigger. | |

\( \large \dfrac{7}{8},\quad \dfrac{6}{7},\quad \dfrac{5}{6},\quad \dfrac{4}{5}\) Hint: This is in order from greatest to least. |

Question 10 |

#### 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?

\( \large \dfrac{3}{5}p-2\) Hint: The company gets 3/5=60% of the price, minus the $2 per item. | |

\( \large \dfrac{3}{5}\left( p-2 \right)\) Hint: This is sensible, but not what the problem states. | |

\( \large \dfrac{2}{5}p+2\) Hint: The company pays the extra $2; it doesn't collect it. | |

\( \large \dfrac{2}{5}p-2\) Hint: This has the company getting 2/5 = 40% of the price of each item, but that's what the representative gets. |

Question 11 |

#### Which of the following is an irrational number?

\( \large \sqrt[3]{8}\) 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. | |

\( \large \sqrt{8}\) Hint: It is not trivial to prove that this is irrational, but you can get this answer by eliminating the other choices. | |

\( \large \dfrac{1}{8}\) Hint: 1/8 is the RATIO of two integers, so it is rational. | |

\( \large -8\) Hint: Negative integers are also rational, -8 = -8/1, a ratio of integers. |

Question 12 |

#### 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?

\( \large \dfrac{34}{43}\) | |

\( \large \dfrac{34}{71}\) Hint: This is the probability that a randomly chosen junior or senior is a commuter student. | |

\( \large \dfrac{34}{147}\) Hint: This is the probability that a randomly chosen student is a junior or senior who is a commuter. | |

\( \large \dfrac{71}{147}\) Hint: This is the probability that a randomly chosen student is a junior or a senior. |

Question 13 |

#### The equation \( \large F=\frac{9}{5}C+32\) is used to convert a temperature measured in Celsius to the equivalent Farentheit temperature.

#### A patient’s temperature increased by 1.5° Celcius. By how many degrees Fahrenheit did her temperature increase?

## 1.5°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 14 |

#### The prime factorization of n can be written as n=pqr, where p, q, and r are distinct prime numbers. How many factors does n have, including 1 and itself?

\( \large3\) Hint: 1, p, q, r, and pqr are already 5, so this isn't enough. You might try plugging in p=2, q=3, and r=5 to help with this problem. | |

\( \large5\) Hint: Don't forget pq, etc. You might try plugging in p=2, q=3, and r=5 to help with this problem. | |

\( \large6\) Hint: You might try plugging in p=2, q=3, and r=5 to help with this problem. | |

\( \large8\) Hint: 1, p, q, r, pq, pr, qr, pqr. |

Question 15 |

#### Use the table below to answer the question that follows:

#### Each number in the table above represents a value W that is determined by the values of x and y. For example, when x=3 and y=1, W=5. What is the value of W when x=9 and y=14? Assume that the patterns in the table continue as shown.

\( \large W=-5\) Hint: When y is even, W is even. | |

\( \large W=4\) Hint: Note that when x increases by 1, W increases by 2, and when y increases by 1, W decreases by 1. At x=y=0, W=0, so at x=9, y=14, W has increased by \(9 \times 2\) and decreased by 14, or W=18-14=4. | |

\( \large W=6\) Hint: Try fixing x or y at 0, and start by finding W for x=0 y=14 or x=9, y=0. | |

\( \large W=32\) Hint: Try fixing x or y at 0, and start by finding W for x=0 y=14 or x=9, y=0. |

Question 16 |

#### Elena is going to use a calculator to check whether or not 267 is prime. She will pick certain divisors, and then find 267 divided by each, and see if she gets a whole number. If she never gets a whole number, then she’s found a prime. Which numbers does Elena NEED to check before she can stop checking and be sure she has a prime?

## All natural numbers from 2 to 266.Hint: She only needs to check primes -- checking the prime factors of any composite is enough to look for divisors. As a test taking strategy, the other three choices involve primes, so worth thinking about. | |

## All primes from 2 to 266 .Hint: Remember, factors come in pairs (except for square root factors), so she would first find the smaller of the pair and wouldn't need to check the larger. | |

## All primes from 2 to 133 .Hint: She doesn't need to check this high. Factors come in pairs, and something over 100 is going to be paired with something less than 3, so she will find that earlier. | |

## All primes from \( \large 2\) to \( \large \sqrt{267}\).Hint: \(\sqrt{267} \times \sqrt{267}=267\). Any other pair of factors will have one factor less than \( \sqrt{267}\) and one greater, so she only needs to check up to \( \sqrt{267}\). |

Question 17 |

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

\( \large \dfrac{1}{25}\) Hint: 1/25=4/100, so this is equal to 4% (be sure you read the question correctly). | |

\( \large \dfrac{4}{100}\) Hint: 4/100=4% (be sure you read the question correctly). | |

\( \large 0.4\) Hint: 0.4=40% so this is not equal to 4% | |

\( \large 0.04\) Hint: 0.04=4/100, so this is equal to 4% (be sure you read the question correctly). |

Question 18 |

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

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

\( \large 1\) | |

\( \large 2\) Hint: The ellipse has 2 lines of reflective symmetry (horizontal and vertical, through the center) and the triangle has 3. The other two figures have rotational symmetry, but not reflective symmetry. | |

\( \large 3\) | |

\( \large 4\) Hint: All four have rotational symmetry, but not reflective symmetry. |

Question 19 |

#### Use the table below to answer the question that follows:

#### Gordon wants to buy three pounds of nuts. Each of the stores above ordinarily sells the nuts for $4.99 a pound, but is offering a discount this week. At which store can he buy the nuts for the least amount of money?

## Store AHint: This would save about $2.50. You can quickly see that D saves more. | |

## Store BHint: This saves 15% and C saves 25%. | |

## Store C | |

## Store DHint: This is about 20% off, which is less of a discount than C. |

Question 20 |

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

\( \large \dfrac{8}{{{\left( 56 \right)}^{4}}}\) Hint: The bases are whole numbers, and the exponents are negative. How can the numerator be 8? | |

\( \large \dfrac{64}{{{\left( 56 \right)}^{4}}}\) Hint: The bases are whole numbers, and the exponents are negative. How can the numerator be 64? | |

\( \large \dfrac{1}{8\cdot {{\left( 56 \right)}^{4}}}\) Hint: \(8^{-6}=8^{-4} \times 8^{-2}\) | |

\( \large \dfrac{1}{64\cdot {{\left( 56 \right)}^{4}}}\) |

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)?

Hint: d(x) is 12 divided by x, not x divided by 12. | |

Hint: When x=2, what should d(x) be? | |

Hint: When x=2, what should d(x) be? | |

Question 22 |

#### On a map the distance from Boston to Detroit is 6 cm, and these two cities are 702 miles away from each other. Assuming the scale of the map is the same throughout, which answer below is closest to the distance between Boston and San Francisco on the map, given that they are 2,708 miles away from each other?

## 21 cmHint: How many miles would correspond to 24 cm on the map? Try adjusting from there. | |

## 22 cmHint: How many miles would correspond to 24 cm on the map? Try adjusting from there. | |

## 23 cmHint: 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 cmHint: 4 groups of 6 cm is over 2800 miles on the map, which is too much. |

Question 23 |

#### 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)

#### Which of the following statements can be inferred from the above chart?

## 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 24 |

#### 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?

## 7.5 metersHint: 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. | |

## 22.5 metersHint: Draw a picture. | |

## 30 metersHint: Draw a picture. | |

## 45 metersHint: Draw a picture. |

Question 25 |

#### Use the expression below to answer the question that follows.

#### \(\large \dfrac{\left( 155 \right)\times \left( 6,124 \right)}{977}\)

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

## 100Hint: 6124/977 is approximately 6. | |

## 200Hint: 6124/977 is approximately 6. | |

## 1,000Hint: 6124/977 is approximately 6. 155 is approximately 150, and \( 6 \times 150 = 3 \times 300 = 900\), so this answer is closest. | |

## 2,000Hint: 6124/977 is approximately 6. |

Question 26 |

#### The polygon depicted below is drawn on dot paper, with the dots spaced 1 unit apart. What is the perimeter of the polygon?

\( \large 18+\sqrt{2} \text{ units}\) Hint: Be careful with the Pythagorean Theorem. | |

\( \large 18+2\sqrt{2}\text{ units}\) Hint: There are 13 horizontal or vertical 1 unit segments. The longer diagonal is the hypotenuse of a 3-4-5 right triangle, so its length is 5 units. The shorter diagonal is the hypotenuse of a 45-45-90 right triangle with side 2, so its hypotenuse has length \(2 \sqrt{2}\). | |

\( \large 18 \text{ units}
\) Hint: Use the Pythagorean Theorem to find the lengths of the diagonal segments. | |

\( \large 20 \text{ units}\) Hint: Use the Pythagorean Theorem to find the lengths of the diagonal segments. |

Question 27 |

#### 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 28 |

#### 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?

\( \large 137\) Hint: Doesn't include the senior athletes who are not vegetarians. | |

\( \large 167\) | |

\( \large 197\) Hint: That's all seniors, including vegetarians. | |

\( \large 279\) Hint: Includes all athletes who are not vegetarians, some of whom are not seniors. |

Question 29 |

#### In which table below is y a function of x?

Hint: If x=3, y can have two different values, so it's not a function. | |

Hint: If x=3, y can have two different values, so it's not a function. | |

Hint: If x=1, y can have different values, so it's not a function. | |

Hint: Each value of x always corresponds to the same value of y. |

Question 30 |

#### What is the least common multiple of 540 and 216?

\( \large{{2}^{5}}\cdot {{3}^{6}}\cdot 5\) Hint: This is the product of the numbers, not the LCM. | |

\( \large{{2}^{3}}\cdot {{3}^{3}}\cdot 5\) 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 for each prime that's a factor of either number, use the largest exponent that appears in one of the factorizations. You can also take the product of the two numbers divided by their GCD. | |

\( \large{{2}^{2}}\cdot {{3}^{3}}\cdot 5\) Hint: 216 is a multiple of 8. | |

\( \large{{2}^{2}}\cdot {{3}^{2}}\cdot {{5}^{2}}\) Hint: Not a multiple of 216 and not a multiple of 540. |

Question 31 |

#### If two fair coins are flipped, what is the probability that one will come up heads and the other tails?

\( \large \dfrac{1}{4}\) Hint: Think of the coins as a penny and a dime, and list all possibilities. | |

\( \large \dfrac{1}{3} \) Hint: This is a very common misconception. There are three possible outcomes -- both heads, both tails, and one of each -- but they are not equally likely. Think of the coins as a penny and a dime, and list all possibilities. | |

\( \large \dfrac{1}{2}\) Hint: The possibilities are HH, HT, TH, TT, and all are equally likely. Two of the four have one of each coin, so the probability is 2/4=1/2. | |

\( \large \dfrac{3}{4}\) Hint: Think of the coins as a penny and a dime, and list all possibilities. |

Question 32 |

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

\( \large x < 0\) Hint: If x =0, then x/2 = x/3, so this answer can't be correct. | |

\( \large x \le 0\) | |

\( \large x > 0\) Hint: If x =0, then x/2 = x/3, so this answer can't be correct. | |

\( \large x \ge 0\) Hint: Try plugging in x = 6. |

Question 33 |

#### What is the length of side \(\overline{BD}\) in the triangle below, where \(\angle DBA\) is a right angle?

\( \large 1\) Hint: Use the Pythagorean Theorem. | |

\( \large \sqrt{5}\) Hint: \(2^2+e^2=3^2\) or \(4+e^2=9;e^2=5; e=\sqrt{5}\). | |

\( \large \sqrt{13}\) Hint: e is not the hypotenuse. | |

\( \large 5\) Hint: Use the Pythagorean Theorem. |

Question 34 |

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

\( \large 6x\) Hint: Use the Pythagorean Theorem. | |

\( \large 3x+5{{x}^{2}}\) Hint: Don't forget to take square roots when you use the Pythagorean Theorem. | |

\( \large 3x+\sqrt{5}{{x}^{2}}\) Hint: \(\sqrt {5 x^2}\) is not \(\sqrt {5}x^2\). | |

\( \large 3x+\sqrt{5}{{x}^{{}}}\) 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 35 |

#### 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?

\( \large d=20t\) Hint: Try plugging t=7 into the equation, and see how it matches the graph. | |

\( \large d=30t\) Hint: Try plugging t=7 into the equation, and see how it matches the graph. | |

\( \large d=40t\) | |

\( \large d=50t\) Hint: Try plugging t=7 into the equation, and see how it matches the graph. |

Question 36 |

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

## aHint: The slope of line a is negative. | |

## bHint: Wrong slope and wrong intercept. | |

## cHint: The intercept of line c is positive. | |

## dHint: 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 37 |

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

## TetrahedronHint: All the faces of a tetrahedron are triangles. | |

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

## Triangular PyramidHint: A pyramid has one base, not two. | |

## TrigonHint: A trigon is a triangle (this is not a common term). |

Question 38 |

#### The “houses” below are made of toothpicks and gum drops.

#### How many toothpicks are there in a row of 53 houses?

## 212Hint: Can the number of toothpicks be even? | |

## 213Hint: One way to see this is that every new "house" adds 4 toothpicks to the leftmost vertical toothpick -- so the total number is 1 plus 4 times the number of "houses." There are many other ways to look at the problem too. | |

## 217Hint: Try your strategy with a smaller number of "houses" so you can count and find your mistake. | |

## 265Hint: Remember that the "houses" overlap some walls. |

Question 39 |

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

## 1.6 cmHint: This is more the height of a Lego toy college student -- less than an inch! | |

## 16 cmHint: Less than knee high on most college students. | |

## 160 cmHint: 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. | |

## 1600 cmHint: This college student might be taller than some campus buildings! |

Question 40 |

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

## 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. | |

## 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. | |

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

## 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 41 |

#### 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?

\( \large C\le 300\) Hint: Find the LCM. | |

\( \large 300 < C \le 500 \) Hint: Find the LCM. | |

\( \large 500 < C \le 700 \) Hint: Find the LCM. | |

\( \large C>700\) Hint: The LCM is 900, which is the smallest number of chairs. |

Question 42 |

#### 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?

## 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 43 |

#### 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)?

\( \large 343.2\times 60\times 10\) Hint: In kilometers, not meters. | |

\( \large 343.2\times 60\times 10\times \dfrac{1}{1000}\) Hint: Units are meters/sec \(\times\) seconds/minute \(\times\) minutes \(\times\) kilometers/meter, and the answer is in kilometers. | |

\( \large 343.2\times \dfrac{1}{60}\times 10\) Hint: Include units and make sure answer is in kilometers. | |

\( \large 343.2\times \dfrac{1}{60}\times 10\times \dfrac{1}{1000}\) Hint: Include units and make sure answer is in kilometers. |

Question 44 |

#### 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?

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

\( \large A = 0, B = 0, C=4\) Hint: Not divisible by 10, since it doesn't end in 0. | |

\( \large A = 4, B = 2, C=0\) 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. | |

\( \large A =1, B=0, C=0\) 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 45 |

#### Use the graph below to answer the question that follows.

#### Which of the following is a correct equation for the graph of the line depicted above?

\( \large y=-\dfrac{1}{2}x+2\) Hint: The slope is -1/2 and the y-intercept is 2. You can also try just plugging in points. For example, this is the only choice that gives y=1 when x=2. | |

\( \large 4x=2y\) Hint: This line goes through (0,0); the graph above does not. | |

\( \large y=x+2\) Hint: The line pictured has negative slope. | |

\( \large y=-x+2\) Hint: Try plugging x=4 into this equation and see if that point is on the graph above. |

List |

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