Category II: Math

Correct Answer: A Option (A) is correct. The question requires an understanding of the basic properties of real numbers. The distributive property of multiplication over addition states that for any real numbers a,b, and c, a(b+c),is equal to ab+ac. In other words, adding the numbers within the parentheses and then multiplying by the number outside the parentheses yields the same result as multiplying each term within the parentheses by the number outside the parentheses and then adding the two products together.

15(4+3)=15×4+15×3 Question: The equation shown demonstrates which of the following? A. The distributive property of multiplication over addition B. The commutative property of multiplication C. The associative property of multiplication D. The additive inverse and additive identity properties

Correct Answer: D Option (D) is correct. The question requires an understanding of average (or arithmetic mean) and the ability to set up and solve several computations. An average of 77 points in four games means that they scored a total of 77 times 4, or 308 points. Since the scores for the first three games are given as 70, 76, and 82 points, it is necessary to add these up (228 points) and subtract from the four-game total of 308 points. This leaves 80 points for the last game's score.

The Clearbrook Wildcats basketball team scored an average of 77 points in four games. In the first three games, the team scored 70, 76, and 82 points. How many points did they score in their last game? A. 70 B. 76 C. 77 D. 80

Correct Answer: C Option (C) is correct. This question requires an understanding of basic probability. If a coin is fair, the probability of tossing heads is the same as the probability of tossing tails. There are only two possible outcomes, so the probability of tossing tails is 1/2. The number 1/2 an also be written as 0.5. Please note that since each toss of the coin is an independent event, which means that the previous results do not affect the current toss of the coin, each time the coin is tossed, the probabilities of H or T would always be the same, 0.5.

Wai tossed a fair coin 9 times with an outcome of H H T T T T H H H, where H means heads and T means tails. What is the probability that the next toss will be T ? A. 0.2 B. 0.4 C. 0.5 D. 1.0

Correct Answer: C Option (C) is correct. The question requires an understanding of prime factorization of a number.

What is the greatest odd factor of the number 2,112 ? A. 3 B. 21 C. 33 D.111

Correct Answer: D Option (D) is correct. The question requires an understanding of how to solve problems using the order of operations.

8+24÷4×(6−4) Which of the following is equivalent to the expression shown? A. 5 B.11 C.16 D.20

Correct Answer: B Option (B) is correct. The question requires an understanding of how to use proportional relationships to solve ratio and percent problems. If 3 pounds of raisins are used in the 10-pound mixture, then 7 pounds of nuts are used in the mixture, giving a ratio of pounds of nuts to pounds of total mixture of 7:10. So 70% of the total number of pounds in the mixture consists of nuts. Since the ratio of pounds of nuts to pounds of total mixture in the 25-pound mixture is the same, then 70% of 25, or 17.5, gives the number of pounds of nuts in the 25-pound mixture. The problem could also be solved by setting up a proportion using x to represent the number of pounds of nuts in the 25-pound mixture. Then 7/10=x2/5 , and solving for x yields 17.5.

A wholesale nut company makes 10-pound and 25-pound bags of trail mix. For the 10-pound bag, the company uses 3 pounds of raisins, and the rest is nuts. If the proportion of raisins to nuts is the same in the 25-pound bag as in the 10-pound bag, how many pounds of nuts does the company need for the 25-pound bag? A. 7.5 B.17.5 C.18.5 D.22.0

Correct Answer: B Option (B) is correct. The question requires an understanding of equations and the ability to translate a word problem into an equation. If x represents the number of glasses and y represents the number of plates that Nicholas bought, then 2x+3.5y=18. Both x and y must be integers. Therefore, 18−2x must be a multiple of 3.5. The possible multiples of 3.5 for this problem are 3.5, 7, 10.5, 14, and 17.5. The only multiple of 3.5 that is equivalent to 18−2x as an integer is 14. Thus, x=2.

At a yard sale, Tenille sold drinking glasses for $2.00 each and plates for $3.50 each. Nicholas spent a total of $18.00 on drinking glasses and plates at Tenille's yard sale. If Nicholas bought at least one glass and one plate, how many drinking glasses did he buy? A.1 B.2 C.3 D.4

Correct Answer: A Option (A) is correct. The question requires an understanding of how to round multidigit numbers to any place value. To round to the nearest thousand, one must look at the digit in the hundreds place first. The digit in the hundreds place is 2, which is less than 5. Therefore, the digit in the thousands place is not changed when rounding to the nearest thousand.

Carlos makes an annual salary of $65,295. Which of the following is Carlos' salary rounded to the nearest thousand? A.$65,000 B.$65,300 C.$66,000 D.$70,000

Correct Answer: B Option (B) is correct. The question requires an understanding of computing percent increase. The increase in the population of the city is 50,600−50,000=600 people. The value of the fraction 600/50,000 gives the percent increase based on the population before the increase occurred. The fraction is equivalent to the decimal 0.012, which is equivalent to 1.2 percent.

The population of a certain city was 50,000 people. One year later, the population of the same city grew to 50,600. What was the percent increase in the city's population in that one-year period? A. 0.6% B. 1.2% C. 6% D. 12%

Correct Answer: C Option (C) is correct. The question requires an understanding of algebraic properties. The associative property of multiplication concerns the order in which the multiplications are performed when three or more numbers are multiplied, which can be changed by inserting or removing grouping symbols such as parentheses.

Which of the following is an example of the associative property of multiplication? A.ab+c=ba+c B.ab+c=c+ab C.(ab)c=a(bc) D.a(b+c)=ab+ac

Correct Answer: C Option (C) is correct. The question requires an understanding of factors of natural numbers. The question requires a determination of the number that has two even factors and one odd factor. The even numbers need not be unique. In (C), 20=2×2×5; 20 can be written as the product of 2, 2, and 5, so 20 can be written as the product of two even numbers and one odd number. In (A), 15=3×5, and in (D), 21=3×7; 15 and 21 do not have any even factors. In (B), 16=2×2×2×2; 16 does not have any odd factors.

Which of the following is the product of two even numbers and an odd number, each of which is greater than 1 ? A.15 B.16 C.20 D.21

Correct Answer: C Option (C) is correct. The question requires an understanding of place value. To compare the four numbers in the options, the digits that determine place value must be compared moving left to right beginning with the digit in the tenths place.

Which of the following numbers is least? A. 0.103 B. 0.1041 C. 0.1005 D. 0.11

Correct Answer: B,C Options (B) and (C) are correct. The question requires an understanding of bar graphs and the ability to read and interpret them. The scale is in billions of dollars and rises in increments of $0.5 billion.The exports from Country C decreased a small amount from 1995 to 1996, so the statement in (A) cannot be inferred from the graph. The statements in (B) and (C) can be inferred, since Country B had the greatest yearly exports, for a three-year total of about $11.5 billion. Also, the exports from Country A more than doubled, going from $2 billion to just over $4 billion.

Which of the following statements can be inferred from the graph shown? Select all that apply. A.For each country shown, exports to the United States increased each year from the previous year. B.The country that had the greatest yearly exports to the United States for each of the years shown had a three-year export total between $11 billion and $12 billion. C.The exports from Country A to the United States more than doubled from 1995 to 1997.

Correct Answer: C Option (C) is correct. The question requires an understanding of how to determine how changes in data affect measures of center or range. Placing the first 6 quiz scores in order gives 85, 90, 90, 90, 90, and 95. Since the two middle numbers are both 90, it is easy to see that the median is 90. The mode and the mean are also 90, and the range is 10. After adding 95 to the list, the median remains 90, the mode remains 90, and the range remains 10. Only the mean is affected by adding 95 to the list. Since 95 is greater than 90, that is, the mean of the first 6 quiz scores, the mean of the 7 quiz scores is greater than the mean of the first 6 quiz scores. Therefore, of the given statements, the only true statement is that the two medians are equal.

90, 90, 95, 90, 85, 90 Caleb's scores for the first 6 quizzes in his algebra class are shown above. If he receives a score of 95 on the 7thseventh quiz, which of the following statements will be true? A.The average (arithmetic mean) of the 7 quiz scores is less than the average of the first 6 quiz scores. B.The mode of the 7 quiz scores is greater than the mode of the first 6 quiz scores. C.The median of the 7 quiz scores is equal to the median of the first 6 quiz scores. D.The range of the 7 quiz scores is greater than the range of the first 6 quiz scores.

Correct Answer: A Option (A) is correct. The question requires an understanding of how to determine how changes in data affect measures of center or range. The range is the difference between the greatest and the least values. Removing the greatest and least values will decrease the range. The mean is the sum of the daily values divided by the number of days. Removing the greatest and the least values will likely affect the mean by shifting it toward the value of the two that was closer to the mean before the removal. The median is the value of the data point that is in the middle when the values in the data set are arranged in numerical order. Since the least value is at the left of the median and the greatest value is at the right of the median, removing those values will not affect the median.

A salesperson records data consisting of total sales each day for one year. If the least and greatest total sales values are deleted from the data set, which of the following is most likely true about the effect of the deletion? A.The range and mean of the data set will change, but not the median. B.The range and median of the data set will change, but not the mean. C.The median and mean of the data set will change, but not the range. D.The mean, median, and range of the data set will all change.

Correct Answer: B Option (B) is correct. The question requires an understanding of how to differentiate between dependent and independent variables in formulas. Since the number of hours Heewon expects to be racing does not depend on how many water bottles she prepares, the number of hours h is the independent variable. Since the number of bottles prepared depends on how many hours Heewon expects to be racing, the number of water bottles n is the dependent variable.

Heewon is filling water bottles for a bicycle race. The number of bottles, n, needed for the race is n=2h+1, where h is the number of hours she expects to be racing. Which of the following statements is true about the variables n and h ? A.n is the independent variable, and h is the dependent variable. B.h is the independent variable, and n is the dependent variable. C.Both n and h are dependent variables. D.Both n and h are independent variables.

Correct Answer: B Option (B) is correct. The question requires an understanding of the coordinate plane. Since points in the second quadrant have a negative x-coordinate and a positive y-coordinate, the point with coordinates (⎯8,2) is located in quadrant II.

In which quadrant is the point (⎯8,2) located? A. Quadrant I B. Quadrant II C. Quadrant III D. Quadrant IV

Correct Answer: D Option (D) is correct. The question requires an understanding of how to evaluate the reasonableness of a solution to a contextual word problem. The solution to the given word problem must belong to the set of all natural numbers because the unit is pieces of clothing, which is a positive and discrete unit.

Sara went to the store to buy some clothes. She bought six shirts, half as many pairs of pants as shirts, and a fourth as many sweaters as shirts. How many pieces of clothing did Sara buy? Which of the following statements about the solution to the word problem shown must be true? A.Because of the real-world context, the solution must belong to the set of all rational positive numbers; therefore, the solution is acceptable. B.Because of the real-world context, the solution must belong to the set of all rational positive numbers; therefore, the solution is not acceptable. C.Because of the real-world context, the solution must belong to the set of all natural numbers; therefore, the solution is acceptable. D.Because of the real-world context, the solution must belong to the set of all natural numbers; therefore, the solution is not acceptable.

Correct Answer: A Option (A) is correct. The question requires an understanding of how to use mental math, estimation, and rounding strategies to solve problems and determine reasonableness of results. The total cost of the trip can be calculated by multiplying the hourly rate by trip duration, in hours. The cost of the bus per hour is best estimated as $34, and the duration of the trip is best estimated as 5 hours. Therefore, the best expression to estimate the total cost using a mental calculation is 34×5.

The cost to rent a bus for a field trip is $34.25 per hour, and the duration of the trip is 4 hours and 45 minutes. Which of the following expressions is best for doing a mental calculation to closely estimate the total cost, in dollars, of renting the bus for the trip? A .34 × 5 B. 34 × 4.75 C. 34.25 × 4.75 D. 35 × 5

Correct Answer: C Option (C) is correct. The question requires an understanding of how to use formulas to determine unknown quantities. Since V=9 V=9 volts and R=4 ohms, I=V/R=9/4=2.25=2.25 amps.

The formula V=IR relates the voltage V, in volts, to the current I, in amps, and the resistance R, in ohms, in a circuit. What is the current produced by a 9-volt battery in a circuit with 4 ohms of resistance? A.1.50 amps B.2.00 amps C.2.25 amps D.2.50 amps

Correct Answer: D Option (D) is correct. The question requires an understanding of how to use the distributive property to generate equivalent linear expressions. The expression shown can be simplified using the distributive property of multiplication over addition in two different approaches.

(x−2)(y+1)+(y−1)(x−2) Which of the following expressions is equivalent to the expression shown? A.2y B.(x−2) C.2(x−2) D.2y(x−2)

Correct Answer: A, C Options (A) and (C) are correct. The question requires an understanding of how to make conjectures, predictions, or generalizations based on patterns. In the pattern described in option (A), the first term is 1, and if the constant described in the pattern is 2, then the first 3 terms of the sequence are 1, 2, and 4. Hence, the pattern described in option (A) could produce the three terms 1, 2, and 4, and, if extended, the fourth term would be 8. In the pattern described in option (C), the first term is 1, and if the quantity that is added is initially 1, then the first three terms of the sequence are 1, 2, and 4. Hence the pattern described in option (C) could produce the three terms 1, 2, and 4, and, if extended, the fourth term would be 7. In option (B), the quantity that would add to the first term, 1, to produce the second term, 2, must be 1. Hence the first three terms produced by the pattern are 1, 2, and 3. The third term is 3, not 4, as required. In option (D), the terms produced by the pattern would be 1, 2, and 5. The third term is 5, not 4, as required. In option (E), the first three terms are 1, 4, and 9. The second and third term values, respectively, are not 2 and 4, as required

1,2,4,... The first three terms of a certain sequence are shown. Which of the following mathematical relationships could describe the terms of the sequence? Select all that apply. A.The first term is 1. Each subsequent term of the sequence is obtained by multiplying a constant quantity by the preceding term. B.The first term is 1. Each subsequent term of the sequence is obtained by adding a constant quantity to the preceding term. C.The first term is 1. Each subsequent term of the sequence is obtained by adding the preceding term and a quantity that increases by 1 with each new term. D.The first term is 1. Each subsequent term of the sequence is obtained by adding 1 to the square of the preceding term. E.The first term is 1. Each subsequent term of the sequence is obtained by squaring a number that is one greater than the number that was squared to obtain the preceding term.

Correct Answer: C Option (C) is correct. The question requires an understanding of how to solve unit-rate problems. One must first find the unit rate by dividing 18 pounds (lbs) by 3 hours, resulting in 6 lbs/hour. Then one must divide 72 lbs by 6 lbs/hour to determine how many hours it will take to process 72 lbs of fruit. Since 72÷6=12, it will take 12 hours to process 72 lbs of fruit.

A machine that works at a constant rate processes 18 pounds of fruit every 3 hours. At this rate, how many hours does it take the machine to process 72 pounds of fruit? A. 4 B.10 C.12 D.15

Correct Answer: C Option (C) is correct. The question requires an understanding of the metric system. Since 1 centimeter equals 10 millimeters, 18 centimeters equal 180 millimeters.

A pencil is 18 centimeters in length. How long is the pencil in millimeters? A. 0.18 B. 1.8 C. 180 D.1,800

Correct Answer: B Option (B) is correct. The question requires an understanding of three-dimensional geometry. Triangular pyramids and rectangular prisms have four and six faces, respectively. Triangular prisms have five faces: two are triangles, and three are quadrilaterals. Rectangular pyramids also have five faces: one is a rectangle, and four are triangles. Triangular prisms have nine edges, while rectangular pyramids only have eight edges.

A two-dimensional net of a certain three-dimensional figure includes five faces, nine edges, and six vertices. Which of the following three-dimensional figures is represented by the net? A. Triangular pyramid B. Triangular prism C. Rectangular pyramid D. Rectangular prism

Correct Answer: D Option (D) is correct. The question requires an understanding of how changes to dimensions change area and volume of three-dimensional shapes. Since the volume of cube A is 1 cubic inch, its sides must have a length of 1 inch. The length of each side in cube B is 1 inch greater than the 1 inch length of each side in cube A, so each side in cube B has length 2 inches. The volume of cube B is thus 8 cubic inches, which is 8 times the volume of cube A.

Cube A has a volume of 1 cubic inch. The length of each side of cube B is 1 inch greater than the length of each side of cube A. The volume of cube B is how many times the volume of cube A? A.2 times B.3 times C.4 times D.8 times

Correct Answer: C Option (C) is correct. The question requires an understanding of patterns and the ability to find and use a pattern rule. A geometric sequence is a sequence for which the ratio between consecutive terms is constant.

If the geometric sequence below continues to increase in the same way, what is the next number in the sequence? 2, 6, 18, 54, 162, ... A.243 B.324 C.486 D.729

Correct Answer: 8 The correct answer is 8. The question requires an understanding of how to find the volume of a right rectangular prism. If 27 number cubes fit in the cubical box, there must be 3 layers of number cubes, with each layer consisting of 3 rows of 3 number cubes.

In a set of number cubes, the length of the edge of each number cube is 2/3 inch. It takes 27 of these number cubes to completely fill a box in the shape of a cube. What is the volume, in cubic inches, of the box?

Correct Answer: D Option (D) is correct. The question requires an understanding of place value systems. Reading the number from left to right, the digit 4 is in the ones position and has a value of 4. The digit 9 is in the tens position and has a value of 90. The digit 8 is in the hundreds position and has a value of 800. The digit 7 is in the thousands position and has a value of 7,000. The digit 6 is in the ten-thousands position and has a value of 60,000.

In the number 567,894, what is the value of the digit 6 ? A. Sixty B. Ten thousand C. Six thousand D. Sixty thousand

Correct Answer: A Option (A) is correct. The question requires an understanding of how to solve problems by plotting points and drawing polygons in the coordinate plane. The formula for the area of a triangle is A=1/2×b×h where b is the length of a side of the triangle that is used as the base and h is the length of the height relative to the side chosen as the base. Since the coordinates of points A and B are given and the segment AB is parallel to the x-axis, the length of side AB can be easily computed, making AB the best choice for the base of the triangle. The length of segment AB is [5-2], or 3 units.Substituting the value of 9 for A and the value of 3 for b in the formula yields 9=12×3×h ; that is, h=6 . Therefore, the length of the height of the triangle relative to side AB must be 6. Since AB is parallel to the x-axis, point C must be on the line with equation y=7 or on the line with equation y=−5 ; that is, it must have a y-coordinate of either 7 or −5 .

In the xy-plane, point A has coordinates (2,1) and point B has coordinates (5,1). Which of the following could be the coordinates for point C so that the area of triangle ABC is equal to 9 square units? A.(5,7) B.(5,4) C.(3,4) D.(4,6)

Correct Answer: D Option (D) is correct. The question requires an understanding of how to compose and decompose two-dimensional shapes. A regular hexagon can be inscribed in a circle. When segments are drawn from the center of the circle to the vertices of the hexagon, 6 triangles are formed. The triangles are congruent because of the SSS theorem. Moreover, the triangles are isosceles because, in each triangle, two of the sides are radii of the circle. In each triangle, the angle that is opposed to the base measures 360°÷6=60° . Then, each base angle measures (180°-60°)÷2=120°÷2=60° . It follows that, in each triangle, all angles have the same measure; thus, the 6 triangles are equilateral.

Into how many equilateral triangles can a regular hexagon be decomposed? A.3 B.4 C.5 D.6

Correct Answer: B Option (B) is correct. The question requires an understanding of how to recognize which measure of center best describes a set of data. The median salary is $1 million and represents nearly all (80%) of the players' salaries on the team. The mean is $3.4 million and does not represent a typical yearly salary since the five salaries that are very high cause the mean to be far from the typical player salary. The mode is not unique, and the range is not a measure of center.

On a major league roster of 25 players, three players have a yearly salary of $15 million each, two players have a yearly salary of $10 million each, and the remaining players have yearly salaries of $1 million each. Which of the following best represents a typical player's yearly salary? A.Mean B.Median C.Mode D.Range

Correct Answer: A Option (A) is correct. The question requires an understanding of how to convert units within the metric system. Since 1 liter is equivalent to 1,000 milliliters, it follows that 1 milliliter is equivalent to 0.001 liter. Therefore, 1,500 milliliters are equivalent to 1,500×0.001 , or 1.5 liters.

Tony made 1,500 milliliters of lemonade for a party. Which of the following represents the amount of lemonade, in liters, that Tony made? A. 1.5 B. 15 C. 150 D.15,000

Correct Answer: A Option (A) is correct. The question requires an understanding of dependent and independent variables within various formulas. The input of a function is referred to as the independent variable because the input can be any number. In this instance, the output, referred to as the dependent variable, is the number of gallons g remaining in the pool, because the volume depends on the input variable d, or the number of days since the pool started to leak.

The community pool has a capacity of 50,000 gallons. It is leaking at a rate of 450 gallons per day. The equation g=50,000⎯450d can be used to find the number of gallons g remaining in the pool after d days. Which of the following statements is true? A. g is the dependent variable because the volume is dependent on the number of days d. B. g is the independent variable because it is what needs to be found. C. d is the dependent variable because it is being multiplied by the independent rate of 450. D. Dependent and independent variables cannot be determined in this situation because the equation is linear.

Correct Answer: D Option (D) is correct. The question requires an understanding of place value. The digit 5 is in the fourth place to the right of the decimal point, and the places in order from the decimal point to the right are the tenths, the hundredths, the thousandths, and the ten thousandths.

The digit 5 is in what place in the number 3.14159 ? A.Thousands place B.Thousandths place C.Hundredths place D.Ten-thousandths place

Correct Answer: B Option (B) is correct. The question requires an understanding of how to find factors and multiples of numbers.

What is the least common multiple of 12, 20, and 30? A. 2 B. 60 C.240 D.360

Correct Answer: D Option (D) is correct. The question requires an understanding of how to round multidigit numbers to any place value. The tenths place is the first digit after the decimal and would be rounded to 3 based on the next digit, 7, which is in the hundredths place. The thousandths place is the third digit after the decimal and would be rounded to 5 based on the next digit, 6, which is in the ten-thousandths place.

Which of the following pairs of numbers estimates the number 3.27461 to the nearest tenth and to the nearest thousandth, respectively? A.3.2 and 3.274 B.3.2 and 3.275 C.3.3 and 3.274 D.3.3 and 3.275

Correct Answer: A Option (A) is correct. The question requires an understanding of how to classify angles based on their measure. The sum of the measures of the angles in a triangle is 180°. Since the right angle measures 90°, the other two angles must add up to the remaining 90°. Since both together make 90°, each individually must be less than 90° and therefore be acute.

Which of the following statements must be true about the two non-right angles of a right triangle? A.Both angles are acute. B.One angle is acute and one is obtuse. C.The angles are congruent. D.Both angles are obtuse.

Correct Answer: A Option (A) is correct. The question requires an understanding of how to classify angles based on their measure. An equilateral triangle is also equiangular; that is, all its angles have the same measure. Therefore, each angle has a measure of 180÷3 180 ÷ 3 , or 60 degrees. An acute angle is an angle that measures less than 90 degrees. Therefore, the angles of an equilateral triangle are all acute.

Which word describes each angle in an equilateral triangle? A.Acute B.Obtuse C.Right D.Straight