Elementary Education: Multiple Subjects Mathematics
Which of the following is the expanded form of 3,056.453,056 point 4 5 ? A.(3×1,000)+(5×10)+(6×1)+(4×1100)+(5×110) B.(3×1,000)+(5×10)+(6×1)+(4×110)+(5×1100) C.(3×1,000)+(5×100)+(6×10)+(4×110)+(5×1100) D.(3×1,000)+(5×100)+(6×10)+(4×1)+(5×110)
B. (3×1,000)+(5×10)+(6×1)+(4×110)+(5×1100) The question requires an understanding of place value and how to write numbers in expanded form. Expanded form is a decomposition of a number into a sum, where the addends are the values represented by the digits in the number. Therefore, 3,056.45=3,000+50+6+0.4+0.05 =(3×1,000)+(5×10)+(6×1)+(4×110)+(5×1100) .
Which of the following is equivalent to 14÷18 ? A.1/4×1/8 B.1/4×8 C. 4×1/8 D.4×8
B. 1/4×8 The question requires an understanding of various strategies and algorithms used to perform operations on rational numbers. Dividing a fraction by a fraction is equivalent to multiplying the first fraction by the reciprocal of the second fraction. The reciprocal of 1/8 is 8/1, or 8.
462,359 In the number shown, the value represented by the digit 6 is 6×b, where b is a constant. Which of the following equals b? A.10^3 B.10^4 C.10^5 D.10^6
B. 10^4 The question requires an understanding of how to use whole-number exponents to denote powers of 10. The number 462,359 can be written as 4×10^5+6×10^4+2×10^3+3×10^2+5×10^1+9×100. The value represented by the digit 6 is 6×104, so b=104.
The figure presents a sequence of figures in a 3 column table, with 4 rows of data. The heading for column 1 is "Term." The heading for column 2 is "Figure." The heading for column 3 is "Lines Needed." The 4 rows of data are as follows. Row 1. Term 1: 1 square, 4 lines. Row 2. Term 2: 1 square, 1 cross, 1 square, 10 lines. Row 3. Term 3: 1 square, 1 cross, 1 square, 1 cross, 1 square, 16 lines. Row 4: All three columns indicate that the table continues In the sequence of figures shown, the first term is made of a single square and each successive term is made by adding one cross and one square to the preceding term. To form each square, 4 line segments are needed, and to form each cross, 2 line segments are needed. Which of the following expressions can be used to find the number of line segments needed to form the nth term in the sequence of figures? A.n+6 B.6n−2 C.n^2+3n D.(n+1)^2
B. 6n−2 The question requires an understanding of how to identify, extend, describe, or generate number and shape patterns. The nth figure includes n squares made up of 4 line segments and n−1 crosses made up of 2 line segments. Therefore, the number of line segments required for the nth figure is equal to 4n+2(n−1), which simplifies to 6n−2.
{9, 11, 1, 4, 7, 12, 10, 4, 9, 2, 5, 9, 8 } The figure presents a box plot. 15 equally spaced tick marks are indicated on a number line. The letters A, B, C, D, and E are indicated at some of the tick marks on the number line. A is indicated at the third tick mark, B is indicated at the sixth tick mark, C is indicated at the seventh tick mark, D is indicated at the ninth tick mark, and E is indicated at the twelfth tick mark. The whiskers range from point A to point E, and the box ranges from point B to point D, with an interior vertical line segment at point C. The list shows the number of books each of the 13 students in a class read over the summer. The teacher wants to summarize the data using the boxplot shown. What is the value of C on this boxplot? A. 7 B. 8 C. 9 D.10
B. 8 The question requires an understanding of how to identify, construct, and complete a boxplot that correctly represents given data. Point C on the boxplot is the median of the data. To find the median, the data points must be reordered from least to greatest, which yields 1, 2, 4, 4, 5, 7, 8, 9, 9, 9, 10, 11, 12. The median is the 7th data point, which is 8.
−3(x−4)≥9 Which of the following inequalities is equivalent to the inequality shown? A.x≤−1 B.x≤1 C.x≥−1 D.x≥1
B. x≤1 The question requires an understanding of how to solve multistep one-variable linear inequalities. Using the distributive property of multiplication over addition yields −3x+12≥9. Adding −12 to both sides of the inequality and adding like terms yields −3x≥−3. Dividing both sides by −3 yields x≤1 , as the inequality sign must be reversed when multiplying or dividing by a negative number.
Which list of numbers is ordered from least to greatest? A. 9/25, 0.350, 2/5 , 0.¯35 B.0.350 point 3 5, 0.¯35, 9/25, 2/5 C.2/5, 9/25, 0.350, 0.¯35 D.0.¯35, 0.350, 9/25, 2/5
B.0.350, 0.35¯¯¯, 9/25, 2/5 The question requires an understanding of how to compare and order rational numbers. One way to compare the four numbers is by finding the decimal equivalent of the fractions. The fraction 925 is equivalent to 0.360 point 3 6; the fraction 25 is equivalent to 0.40 point 4. The decimal number 0.35¯¯¯ is equivalent to 0.353535... , where the digits 35 repeat indefinitely. Therefore, the order from least to greatest is 0.350 point 3 5, 0.35¯¯¯, 925, 25.
Activity Time Dance class 15 minutes Running 8 minutes Walking 20 minutes The table shows the time it takes Larissa to burn 100 calories while performing each of three activities. Larissa's workout routine includes a 30-minute dance class, 10 minutes of running, and 15 minutes of walking. How many calories did Larissa burn to complete her workout? A.375 B.400 C.425 D.450
B.400 The question requires an understanding of how to solve a unit-rate problem. Using the information in the chart, a 30-minute dance class would burn 100×2, or 200 calories; 10 minutes of the running would burn 100/8×10, or 125 calories; and 15 minutes of walking would burn 100/20×15, or 75 calories. Therefore, Larissa burned a total of 200+125+75, or 400 calories.
−12x−6>4−3x Which of the following is the graph of the solutions to the inequality shown? A.A rightward pointing ray is drawn on the number line beginning with a closed circle at the number 4. B.A rightward pointing ray is drawn on the number line beginning with an open circle at the number 4. C.A leftward pointing ray is drawn on the number line beginning with a closed circle at the number 4. D.A rightward pointing ray is drawn on the number line beginning with an open circle at the number negative 2.
B.A rightward pointing ray is drawn on the number line beginning with an open circle at the number 4. The question requires an understanding of how to interpret solutions of multistep one-variable linear equations and inequalities. The inequality can be solved by first adding 6 and 3x to each side of the inequality and then adding like terms, resulting in 5/2 x>10. Multiplying each side by 2/5 yields x>4. On the number line, this inequality corresponds to a graph of the numbers greater than 4 but not including 4.
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
B.Median 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.
7, 9, 10, 11, 14, 15, 15, 15, 17, 18, 18, 20, 26, 26, 27, 28, 30 The list shows the times spent eating lunch for the 17 students in Ms. Begay's class. Which of the following statements about the times must be true? Select all that apply. A.The mode is 17 minutes. B.The median is 17 minutes. C.The mean is 18 minutes. D.The range is 37 minutes.
B.The median is 17 minutes. C.The mean is 18 minutes. The question requires an understanding of solving problems involving measures of center (mean, median, and mode) and range. The list is already ordered and contains 17 numbers, so the ninth number, which is 17, is the median since it has 8 numbers above it and 8 numbers below it in the ordered list. The sum of all 17 numbers is 306, and the mean is found by dividing 306 by 17. This yields a mean of 18. Hence both options (B) and (C) are correct. The mode is the value that occurs most frequently in the list, so that is 15, and hence option (A) is incorrect. The range is 30−7=23, so option (D) is incorrect.
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.
B.h is the independent variable, and n is the dependent variable. 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.
Rosa left a 20% tip on a restaurant bill of $47.30. What amount did Rosa leave for the tip? A. $0.95 B. $4.73 C. $9.46 D.$14.19
C. $9.46 The question requires an understanding of percent as a rate per 100. The tip is 20% of $47.30, and it can be found by multiplying 47.3047 point 3 0 by 20/100 , which yields $9.46. Alternatively, to find 20% of $47.30, 0.2×47.30=9.46 could be calculated.
3×2+7×2 The expression shown results when the distributive property of multiplication over addition is applied to which of the following expressions? A. 3+3+7+7 B.6+14 C.(3+7)×2 D.(7×2)+(3×2)
C. (3+7)×2 he question requires an understanding of properties of operations. The distributive property of multiplication over addition states that (A+B)×C=(A×C)+(B×C), so per the distributive property, (3+7)×2=3×2+7×2.
What is the prime factorization of 180 ? A.3^2×4×5 B. 2^2×5×9 C.2^2×3^2×5 D.2×3^2×10
C. 2^2×3^2×5 The question requires an understanding of how to identify prime and composite numbers. Since 180=2×90=2×2×45=2×2×3×15=2×2×3×3×5 , the prime factorization of 180 is 22×32×5.
The figure presents a line graph. The horizontal axis is labeled "Number of Days Spent Reading," and the numbers 0 through 6 are indicated. The vertical axis is labeled "Number of Pages Read," and the numbers 0 through 400, in increments of 100, are indicated. The line begins at the origin and extends upward and to the right, through 5 data points. The data represented by the 5 points are as follows. Note that all values are approximate. Point 1: 1 day, 75 pages Point 2: 2 days, 150 pages Point 3: 3 days, 225 pages Point 4: 4 days, 300 pages Point 5: 5 days, 375 pages Alireza has been reading a 600-page novel at a constant rate. The graph shows the number of pages y he has read after x days of reading. If he continues reading at the same rate, how many days will it take him to read the novel from beginning to end? A. 4 B. 6 C. 8 D.10
C. 8 The question requires an understanding of how to use linear relationships represented by a graph to solve problems. The graph shows that Alireza has read 150 pages in 2 days and that he has read 300 pages in 4 days. Therefore, Alireza has been reading the novel at a constant rate of 75 pages per day. If he continues to read at the same rate, he will finish reading the novel in 600÷75, or 8 days.
Jamal bought a pair of headphones priced at $17.99. If the sales tax rate is 8%8 percent, which of the following proportions can be used to find the amount of sales tax x, in dollars, for the headphones? A.8/17.99=x/100 B.8/17.99=100/x C. 8/100=x/17.99 D.8/100=17.99/x
C. 8/100=x/17.99 The question requires an understanding of how to use proportional relationships to solve ratio and percent problems. The taxes are 8% of the price; that is, the ratio x:17.99 is equal to 8:100. Therefore, 8/100=x/17.99.
A basket contains A apples and P pears, and there are two fewer apples than three times the number of pears. Which of the following equations can be used to represent algebraically the relationship between the number of apples and the number of pears in the basket? A.A=2−3P B.A−2=3P C.A+2=3P D.3A=P−2
C. A+2=3P The question requires an understanding of how to translate between verbal statements and algebraic equations. The basket contains two fewer apples than three times the number of pears, so if there were two more apples, there would be as many apples as three times the number of pears. Therefore, A+2=3P.
The figure presents a line. The points A, B, and C are indicated from left to right. Which of the following can be used to name the ray that has endpoint A and goes in the direction of C in the figure shown? A.AB−→ only B.AC−→ only C.AC−→ and AB−→ D.CA−→ and AC−→
C. AC−→ and AB−→ The question requires an understanding of the definition of a ray. Since a ray is a part of a line that has one endpoint and extends in one direction without ending, then both ray ACA C and ray ABA B represent the same ray that has endpoint A and extends in the direction of C.
30M+15R=900 In a certain year, Rodney earned a total of $900 mowing lawns and raking leaves for his neighbors. He received $30 for each lawn he mowed and $15 for each job raking leaves he had. In the formula above, M represents the number of lawns Rodney mowed for his neighbors and R represents the number of jobs raking leaves he had during the year. If Rodney mowed 15 lawns that year, how many jobs raking leaves did he have? A.15 B.20 C.30 D.90
C.30 The question requires an understanding of how to use formulas to determine unknown quantities. Since Rodney mowed 15 lawns, M=15. Plugging the number in the formula yields 30×15+15R=900. Solving for R yields R=900−45015. Therefore, R=30.
Color Probability Red 1/3 Blue x Green 1/6 Yellow 1/12 A spinner is divided into 12 congruent sections. Each section is either red, blue, green, or yellow. The table shows the theoretical probability of landing on a color when spinning the spinner once. How many sections are blue? A.2 B.4 C.5 D.9
C.5 The question requires an understanding of probability. The sample space consists of the possible outcomes of a spin, which are red, blue, green, or yellow. The sum of the probabilities of the possible outcomes in the sample space is 1; that is, 1/3+x+1/6+1/12=1, which is equivalent to 7/12+x=12/12. It follows that the probability x of landing on blue is 5/12. Therefore, 5 out of 12 sections of the spinner are blue.
Tasha has T books, and Aisha has A books. Aisha has twice as many books as Tasha, and altogether they have 30 books. Which of the following proportions can be used to find out how many books Aisha has? A.A:T=2:3 B.T:A=2:3 C.A:(T+A)=2:3 D.T:(T+A)=2:3
C.A:(T+A)=2:3 The question requires an understanding of ratios and unit rates to describe relationships between quantities. Since Aisha has twice as many books as Tasha, the ratio of T:A is equal to 1:2; that is, Tasha has one-third of the books, while Aisha has two-thirds. Therefore, the ratio of Aisha's books, A, and the total T+A is 2:3.
Ms. Gupta is making 3 batches of chocolate chip cookies for a picnic. Each batch requires 1 1/2 cups of sugar and 3 1/4 cups of flour. Which of the following is the best estimate of the total amount of sugar and flour Ms. Gupta uses for the cookies? A.Less than 9 cups B.Between 9 and 12 cups C.Between 12 and 15 cups D.More than 15 cups
C.Between 12 and 15 cups The question requires an understanding of how to use mental math and estimation to solve problems and determine the reasonableness of results. Each batch of cookies requires a total of 1 1/2+3 1/4=4 3/4 cups of sugar and flour. This number is between 4 and 5, so three batches would require between 3×4 and 3×5 cups of flour and sugar, so the total is between 12 and 15 cups.
(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)
D. 2y(x−2) 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. The first approach is based on noticing that (x−2)(y+1)+(y−1)(x−2) can be split into two products, namely (x−2)(y+1) and (y−1)(x−2), each of which has x−2 as a factor. The expression can be seen as the result of the distributive property having been applied to factors x−2 and ((y+1)+(y−1)). Therefore, the expression can be rewritten as (x−2)((y+1)+(y−1)), which is equivalent to (x−2)(y+1+y−1), or 2y(x−2). The second approach is based on applying the distributive property of multiplication over addition to the products in the expression. The product (x−2)(y+1) is equivalent to xy+x−2y−2; the product (y−1)(x−2) is equivalent to xy−2y−x+2. Combining the two results, the expression shown is equivalent to xy+x−2y−2+(xy−2y−x+2), or xy+x−2y−2+xy−2y−x+2. Adding like terms yields 2xy−4y, which is equivalent to 2y(x−2).
Item Cost of Each Package Utensils $1.50 Cups $2.00 Plates $2.00 Tablecloths $4.50 The table shows the costs of supplies for a birthday party. Anton buys 6 packages of utensils, 4 packages of cups, 2 packages of plates, and 2 packages of tablecloths. Anton's total purchase is subject to a 7.5%7 point 5 percent sales tax. How much will the party supplies cost Anton? A.$10.75 B.$23.65 C.$30.00 D.$32.25
D.$32.25 The question requires an understanding of how to solve multistep mathematical and real-world problems using addition, subtraction, multiplication, and division of rational numbers. The cost before tax can be found by simplifying the expression 6×1.50+2×4+2×2+2×4.50, obtaining $30. The taxes are 30×7.5100=2.25 dollars. Therefore, the total cost is 30+2.25 dollars, that is, $32.2532 point 2 5 dollars.. Alternatively, the total cost can be found by 30×1.075=32.25.
The figure presents a polygon in the shape of an upside down L. 4 of the 6 sides are labeled. The top side is horizontal, and its length is not labeled. The left side is vertical, with a length of 1 yard. The bottom side extends 18 inches, to the right, from the left corner. The shape then extends upward an unlabeled length. Next, it extends 1 point 5 feet to the right. Finally, it extends upward 2 feet, where it meets the top right corner of the shape. What is the area, in square inches, of the polygon shown? A. 21 B. 54 C.1,008 D.1,080
D.1,080 The question requires an understanding of relative sizes of United States customary units. To find the area of the polygon in square inches, all measurements must first be converted to equivalent measurements in inches using the conversions that one foot is equivalent to 12 inches and 1 yard is equivalent to 3 feet. The lengths, in inches, of the labeled sides in the polygon are 36 inches, 18 inches, 18 inches, and 24 inches. The polygon can be decomposed into two rectangles: one having a length of 36 inches and a width of 18 inches, and the other having a length of 24 inches and a width of 18 inches. The first has an area of 36×18, or 648 square inches, and the second has an area of 24×18, or 432 square inches. The area of the polygon is the sum of the areas of these two rectangles, which equals 648+432, or 1,080 square inches.
8+24÷4×(6−4) Which of the following is equivalent to the expression shown? A. 5 B.11 C.16 D.20
D.20 The question requires an understanding of how to solve problems using the order of operations. Using the order of operations, the first step is performing the subtraction within the parentheses, which yields 8+24÷4×2. The second step is performing the division, which yields 8+6×2. The third step is performing the multiplication, which yields 8+12. Finally, the fourth step is performing the addition, which yields 20.
Which of the following pairs of numbers estimates the number 3.274613 point 2 7 4 6 1 to the nearest tenth and to the nearest thousandth, respectively? A.3.2 and 3.2743 point 2 3 point 2 7 4 B.3.2 and 3.2753 point 2 3 point 2 7 5 C.3.3 and 3.2743 point 3 3 point 2 7 4 D.3.3 and 3.275
D.3.3 and 3.275 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.
A family of 5 consumes 1/2 gallon of milk a day. How many 1-gallon jugs of milk must the family buy to have enough milk for a week? A.2.52 point 5 B.3 C.3.53 point 5 D.4
D.4 The question requires an understanding of how to recognize the reasonableness of results within the context of a given problem. Since there are 7 days in a week, the family will consume 7×1/2, or 3.53 point 5 gallons of milk. Since each jug has a capacity of 1 gallon of milk, the family must buy 4 jugs to have enough milk for the week.
Into how many equilateral triangles can a regular hexagon be decomposed? A.3 B.4 C.5 D.6
D.6 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 SSSS S S 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.
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
D.8 times 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.
In a certain number, a and b represent digits, and a represents 10^4 times what b represents. Which of the following could be the place value of a and b? A.a is in the thousandths place, and b is in the tens place. B.a is in the tenths place, and b is in the hundreds place. C.a is in the ones place, and b is in the hundredths place. D.a is in the tens place, and b is in the thousandths place.
D.a is in the tens place, and b is in the thousandths place. The question requires an understanding of place value and how a digit in one place represents ten times what it represents in the place to its right and one-tenth what it represents in the place to its left. A digit in the tens place is 10^4 times what a digit in the thousandths place represents. In option (A), a would be 10^2 what b represents. In option (B), a would be 1/10^3 times what b represents. In option (C), a would be 10^2 what b represents.
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?
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. With 3 number cubes, each having edge length 2/3 inch, along one edge of the box, each edge of the box is 3×2/3=2 inches in length. The volume of the box is thus 2^3=8 cubic inches. Another approach is to find the volume of each number cube first. The volume of each number cube is (23)^3, or 8/27 cubic inches. Since there are 27 number cubes in the box, the volume of the box is 8/27×27, or 8 cubic inches.
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 ABCA B C is equal to 9 square units? A.(5,7) B.(5,4) C.(3,4) D.(4,6)
A. (5,7) 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=12×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 ABA B is parallel to the x-axis, the length of side ABA B can be easily computed, making ABA B the best choice for the base of the triangle. The length of segment ABA B 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 ABA B must be 6. Since ABA B 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.
Tony made 1,500 milliliters of lemonade for a party. Which of the following represents the amount of lemonade, in liters, that Tony made? B. 15 C. 150 D. 15,000
A. 1.5 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.
25x^2+6y^3+4x−25y+3x^2−4y^2 Which of the following shows all like terms of the expression above? A.25x^2 and 3x^2 B.25x^2 and 25y C.25x^2, 3x^2, and 4y^2 D.6y^3, 25y, and 4y^2
A. 25x^2 and 3x^2 The question requires an understanding of how to use mathematical terms to identify parts of expressions. In an expression, like terms are terms that meet two conditions. First, the terms contain identical products of variables. Second, the exponent of a variable in a term is identical to the exponent of the same variable in the other terms. The only terms in the expression shown that meet these two conditions are 25x2 and 3x2, since the variable x is identical in each term and the exponent of 2 on the x is identical in each term.
Which of the following is an algebraic equation? A.3x−5=2 B.4x−2y+8 C. 3(9−4)=5(4−1) D.7+5−123
A. 3x−5=2 The question requires an understanding of the difference between algebraic expressions and equations. Neither option (B) nor (D) shows an equation since there is no equal sign in either option; these are instead both expressions. Options (A) and (C) both show equations, but the equation in option (C) is not algebraic since it does not contain any variables. Only the equation in option (A) is algebraic, since it is an equation that contains at least one variable.
(2a−a+3)−(−5a+6a+2) Which of the following is equivalent to the expression shown? A.1 B.5 C.2a+1 D.12a+5
A.1 The question requires an understanding of how to add and subtract linear algebraic expressions. In fact, (2a−a+3)−(−5a+6a+2)=a+3−(a+2)=a+3−a−2=1.
Rodrigo left his workplace at 4:50 P.M., and it took him 3/4 of an hour to get home. At what time did Rodrigo get home? A.5:35 P.M. B.5:45 P.M. C.5:55 P.M. D.6:05 P.M.
A.5:35 P.M. The question requires an understanding of elapsed time. Since there are 60 minutes in an hour, it took Rodrigo 60×3/4, or 45 minutes, to get home. Rodrigo left at 4:50 P.M. Since 50+45=95 minutes, or 1 hour and 35 minutes, Rodrigo got home at 5:35 P.M.
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.
A.Both angles are acute. 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.
The function f(x)=x2+3 is represented by which table of values? A.The 4 rows of data are as follows. Row 1: x, negative 2; f of x, 7. Row 2: x, negative 1; f of x, 4. Row 3: x, 0; f of x, 3. Row 4: x, 3; f of x, 12. B.The 4 rows of data are as follows. Row 1: x, negative 2; f of x, negative 1. Row 2: x, negative 1; f of x, 2. Row 3: x, 0; f of x, 5. Row 4: x, 3; f of x, 12. C.The 4 rows of data are as follows. Row 1: x, negative 2; f of x, negative 1. Row 2: x, negative 1; f of x, 4. Row 3: x, 0; f of x, 3. Row 4: x, 3; f of x, 9. D.The 4 rows of data are as follows. Row 1: x, negative 2; f of x, 7. Row 2: x, negative 1; f of x, 2. Row 3: x, 0; f of x, 3. Row 4: x, 3; f of x, 9.
A.The 4 rows of data are as follows. Row 1: x, negative 2; f of x, 7. Row 2: x, negative 1; f of x, 4. Row 3: x, 0; f of x, 3. Row 4: x, 3; f of x, 12. The question requires an understanding of how to identify a relationship between the corresponding terms of two numerical patterns. A method to find out which table of values may represent the function f(x)=x2+3 is to plug in the x values from each table in the function and verify whether the output matches the value of f(x) given in the table. For x=0, f(0)=3; for x=−2, f(−2)=(−2)2+3=4+3=7; for x=−1, f(−1)=(−1)2+3=1+3=4; finally, for x=3, f(3)=9+3=12. The only table with all four correct pairs of values is the table in option A.
2y^2+17x^2+14x^2y+8xy^2+1 Which THREE of the following statements are correct about the algebraic expression shown? A.The coefficients are 2, 17, 14, 8, and 1. B.The variables are x and y. C.There are like terms. D.There are 4 terms. E.The degree of the expression is 3.
A.The coefficients are 2, 17, 14, 8, and 1. B.The variables are x and y. E.The degree of the expression is 3. The question requires an understanding of how to use mathematical terms to identify parts of expressions and describe expressions. A term in an expression consists of the product of a coefficient, which is a number, and one or more variables raised to positive exponents. The coefficients of the terms in the expression are 2, 17, 14, 8, and 1, so option (A) is correct. The constant term, 1, is a simplified version of 1x^0y^0 , and if written this way, each of the five terms can be seen to contain x and y as variables; so option (B) is correct. The degree of an expression is the greatest degree for any monomial in the expression, and the degree of a monomial is the sum of the exponents of the variables in the monomial. The degrees of the monomials are 2, 2, 3, 3, and 0, so the degree of the expression is 3 and hence option (E) is correct. In an expression, like terms are terms that meet two conditions. First, the terms contain identical products of the variables. Second, the exponent of a variable in a term is identical to the exponent of the same variable in the other terms. No pair of items meets these two conditions, so there are no like terms and option (C) is incorrect. The expression has five terms, not 4, so option (D) is incorrect.
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.
A.The first term is 1. Each subsequent term of the sequence is obtained by multiplying a constant quantity by 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. 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.
The figure presents a net that forms a right triangular prism. The net is composed of 3 rectangles and 2 right triangles. The 3 rectangles are side by side, forming the middle portion of the net. The length of the first rectangle is labeled 6, and its width is labeled 5. The width of the second rectangle is labeled 4. And the width of the third rectangle is labeled 3. The first right triangle is connected to the top of the second rectangle. The second right triangle is connected to the bottom of the second rectangle. The base of each triangle that is connected to the rectangle, has a length of 4. The net shown above forms a right triangular prism. Which of the following represents the surface area of the right triangular prism? A. 84 B. 92 C. 96 D.120
A. 84 The question requires an understanding of how to use nets to determine the surface area of three-dimensional figures. The surface area of the three-dimensional figure is the sum of the areas of its faces. The faces of the right triangular prism are a 6-by-5 rectangle, a 6-by-4 rectangle, a 6-by-3 rectangle, and two congruent right triangles, each of which has sides of lengths 4 and 3. The areas of the rectangular faces are, respectively, 6×5, or 30; 6×4, or 24; and 6×3, or 18. The area of each of the triangular faces is 12×3×4, or 6. Therefore, the surface area of the figure is 30+24+18+2×6, or 84.
The figure presents a model, which consists of 3 boxes. Each box contains a number of counters. The first box has 5 counters with a minus sign, and one counter with a plus sign. There is an arrow pointing from the first box to the second box. The second box has 5 counters with a minus sign, and one counter with a plus sign. 2 of the counters, one with a plus sign and one with a minus sign, are circled with an arrow drawn from the circle, pointing upward and to the right, exiting the box. There is an arrow pointing from the second box to the third box. The third box has 4 counters with a minus sign. Which of the following addition sentences represents the model shown? A.−5+1=−4 B.−3+(−1)=−4 C.5+(−1)=4 D.6+(−2)=4
A. −5+1=−4 The question requires an understanding of how to represent rational numbers and their operations in different ways. The first box shows 5 counters with a minus sign and 1 counter with a plus sign; that is, −5+1 . The middle box shows that −5+1 is equivalent to −4+(−1+1) and that −1+1 can be removed as it is equivalent to 0. The last box shows that −4+0 is equivalent to −4.
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.
A.The range and mean of the data set will change, but not the median. 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.
5 1/4% is equivalent to which of the following? A.5.255 point 2 5 B.0.5250 point 5 2 5 C.0.05250 point 0 5 2 5 D.0.00525
C.0.0525 The question requires an understanding of how to convert between fractions, decimals, and percents. The fraction 1/4 can be converted to 0.250 point 2 5 so that the original percent can be written as 5.25%5 point 2 5 percent. Since 5.25%5 point 2 5 percent is equivalent to 5.25/100, to convert 5.25%5 point 2 5 percent to a decimal number, one must divide 5.255 point 2 5 by 100. This means that 5.25%5 point 2 5 is equivalent to 0.05250 point 0 5 2 5.
8a−3b What is the value of the expression shown when a=2 and b=−5? A. −46 B.−31 C. 1 D. 31
D. 31 The question requires an understanding of how to evaluate simple algebraic expressions. Substituting a=2 and b=−5 into the expression yields 8(2)−3(−5)=16−(−15)=16+15=31.
Which of the following lists shows all the factors of 24 ? A.2, 3 B.3, 4, 6, 8 C.2, 3, 4, 6, 8, 12 D.1, 2, 3, 4, 6, 8, 12, 24
D.1, 2, 3, 4, 6, 8, 12, 24 The question requires an understanding of factors and multiples of a number. Given a natural number n, a natural number less than or equal to n is a factor of n if and only if n is divisible by that number. Since 1×24=24, 2×12=24, 3×8=24, and 4×6=24, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
The digit 5 is in what place in the number 3.141593 point 1 4 1 5 9 ? A.Thousands place B.Thousandths place C.Hundredths place D.Ten-thousandths place
D.Ten-thousandths place 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.