Math Knowledge - 25 questions- 22min (52sps)

D: Let x be the missing quantity. The problem can be expressed as the following equation: 3(5-x) = x-5. Distributing the 3 results in: 15 -3x = x+5. Subtract 5 from both sides, add 3x to both sides, and then divide both sides by 4. This results in: 10/4 = 5/2 =2.5

20. Triple the difference of five and a number is equal to the sum of that number and 5. What is the number?

A: The probability of .9 is closer to 1 than any of the other answers. The closer a probability is to 1, the greater the likelihood that the event will occur. The probability of 0.05 shows that it is very unlikely that an adult driver will wear their seatbelt because it is close to zero. A zero probability means that it will not occur. The probability of 0.25 is closer to zero than to one, so it shows that it is unlikely an adult will wear their seatbelt. Choice E is wrong because probability must fall between 0 and 1.

26. A study of adult drivers finds that it is likely that an adult driver wears his seatbelt. Which of the following could be the probability that an adult driver wears his seat belt? a. 0.90 b. 0.05 c. 0.25 d. 0 e. 1.5 What is the solution?

A: A proportion should be used to solve this problem. The ratio of tagged to total deer in each instance is set equal, and the unknown quantity is a variable x. The proportion is 300/x = 5/ 400. Cross-multiplying gives 120,000 =5x, and dividing through by 5 results in 24,000.

27. In order to estimate deer population in a forest, biologists obtained a sample of deer in that forest and tagged each one of them. The sample had 300 deer in total. They returned a week later and harmlessly captured 400 deer, and 5 were tagged. Use this information to estimate how many total deer were in the forest. a. 24,000 deer b. 30,000 deer c. 40,000 deer d. 100,000 deer e. 120,000 deer What is the solution?

C: The mode is the time from the data set that occurs most often. The number 23 occurs twice in the data set, while all others occur only once, so the mode is 23.

36. What is the mode of Eva Jane's time? a. 16 minutes b. 20 minutes c. 23 minutes d. 33 minutes e. 25 minutes

C: The mode for a set of data is the value that occurs the most. The grade that appears the most is 95. It's the only value that repeats in the set. The mean is around 84.3.

10. What is the mode for the grades shown in the chart below?

B: The relationship between age and time for attention span is a positive correlation because the general trend for the data is up and to the right. As the age increases, so does attention span.

11. What type of relationship is there between age and attention span as represented in the graph below?

A: The area of the shaded region is calculated in a few steps. First, the area of the rectangle is found using the formula A =length x width = 6 x 2 = 12. Second, the area of the triangle is found using the formula: A = 1/2 x base x height = 1/2 x 3 x 2 = 3. The last step is to take the rectangle area and subtract the triangle area. The area of the shaded region is A = 12 -3 = 9m².

12. What is the area of the shaded region?

D: The volume for a cylinder is found by using the formula: V = (π x r²h = pi(2²) x 3.5 = 43.9in².

13. What is the volume of the cylinder below?

C: There are 0.006 kiloliters in 6 liters because 1 liter=0.001kiloliters. The conversion comes from the chart where the prefix kilo- is found three places to the left of the base unit.

14. How many kiloliters are in 6 liters?

B: The goal is to first isolate the variable. The fractions can easily be cleared by multiplying the entire inequality by 5, resulting in 35 - 4x < 3. Then, subtract 35 from both sides and divide by -4. This results in x>8. Notice the inequality symbol has been flipped because both sides were divided by a negative number. The solution set, all real numbers greater than 8, is written in interval notation as (8, ∞). A parenthesis shows that 8 is not included in the solution set.

19. What is the solution to the following linear inequality?

B: The outlier is 35. When a small outlier is removed from a data set, the mean and the median increase. The first step in this process is to identify the outlier, which is the number that lies away from the given set. Once the outlier is identified, the mean and median can be recalculated. The mean will be affected because it averages all of the numbers. The median will be affected because it finds the middle number, which is subject to change because a number is lost. The mode will most likely not change because it is the number that occurs the most, which will not be the outlier if there is only one outlier.

22. The following set represents the test scores from a university class: {35,79, 80, 87, 87, 90, 92, 95, 95, 98, 99}. If the outlier is removed from this set, which of the following is TRUE? a. The mean and the median will decrease. b. The mean and the median will increase. c. The mean and the mode will increase. d. The mean and the mode will decrease. e. The mean, median, and mode will increase.

C: First, the slope of the line must be found. This is equal to the change in y over the change in x, given the two points. Therefore, the slope is -6. The slope and one of the points are then plugged into the slope-intercept form of a line: y-y₁ =m(x-x₁). This results in y-7=-6 9x+3). The -6 is simplified and the equation is solved for y to obtain y=-6x-11.

24. What is the equation of the line that passes through the two points (-3, 7) and (-1, -5)?

A: The formula for the rate of change is the same as slope: change in y over change in x. The y-value in this case is percentage of smokers and the x-value is year. The change in percentage of smokers from 2000 to 2015 was 8.1 percent. The change in x was 2000-2015 = -15. Therefore, 8.1%/-15 = -0.54%. The percentage of smokers decreased 0.54 percent each year.

25. The percentage of smokers above the age of 18 in 2000 was 23.2 percent. The percentage of smokers above the age of 18 in 2015 was 15.1 percent. Find the average rate of change in the percent of smokers above the age of 18 from 2000 to 2015. a. -.54 percent b. -54 percent c. -5.4 percent d. -15 percent e. -1.5 percent What is the solution?

A: A vertical line has the same x value for any point on the line. Other points on the line would be (1, 3), (1, 5), (1, 9,) etc. Mathematically, this is written as x=1. A vertical line is always of the form x = a for some constant a.

28. Which of the following is the equation of a vertical line that runs through the point (1, 4)?

C: The Pythagorean Theorem can be used to find the missing length x because it is a right triangle. The theorem states that 6²+8²=x², which simplifies into 100=x². Taking the positive square root of both sides results in the missing value x =10.

29. What is the missing length x?

E: First, the common factor 2 can be factored out of both terms, resulting in: 2(y³ - 64) The resulting binomial is a difference of cubes that can be factored using the rule: a³-b³=(a-b)(a²+ab+b²) a =y and b =4, therefore, the results is: 2(y-4)(y²+4y+16)

30. What is the correct factorization of the following binomial?

B: Look on the horizontal axis to find 3:00 p.m. Move up from 3:00p.m. to reach the dot on the graph. Move horizontally to the left to the horizontal axis to between 20 and 25; the best answer choice is 22. The answer of 25 is too high above the projected time on the graph, and the answers of 20 and 16 degrees are too low.

32. Use the graph below entitled "Projected Temperatures for Tomorrow's Winter Storm" to answer the question.

B: The number of representatives varies directly with the population, so the equation necessary is N=k x P, where N is number of representatives, k is the variation constant, and P is total population in millions. Plugging in the information for New York allows k to be solved for. This process gives 27=k x 20, so k=1.35. Therefore, the formula for number of representatives given total population in millions is N=1.35 x P. Plugging in P = 11.6 for Ohio results in N=15.66, which rounds up to 16 total Representatives.

33. The number of members of the House of Representatives varies directly with the total population in a state. If the state of New York has 19,800,000 residents and has 27 total representatives, how many should Ohio have with a population of 11,800,000?

B: This is a statistical question because in order to determine this answer one would need to collect data from each person in the class and it is expected the answers would vary. The other answers do not require data to be collected from multiple sources; therefore, the answers will not vary.

34. Which of these answer choices is a statistical question? a. What was your grade on the last test? b. What were the grades of the students in your class on the last test? c. What kind of car do you drive? d. What was Sam's time in the marathon? e. What textbooks does Marty use this semester?

E: The mean is found by adding all the times together and dividing by the number of times recorded. 25 18 23 28 30 22.5 23 33 20 222.5, divided by 9 24.7. Rounding to the nearest minute, the mean is 25 minutes.

35. What is the mean of Eva Jane's time? a. 26 minutes b. 19 minutes c. 24.5 minutes d. 23 minutes e. 25 minutes

A: To find the median of a data set, you must first list the numbers from smallest to largest, and then find the number in the middle. If there are two numbers in the middle, as in this data set, add the two numbers in the middle together and divide by 2. Putting this list in order from smallest to greatest yields 18, 20, 22.5, 23, 23, 25, 28, 30, and 33, where 23 is the middle number, so 23 minutes is the median.

37. What is Eva Jane's median score? a. 23 minutes b. 17 minutes c. 28 minutes d. 19 minutes e. 25 minutes

D: The area for a rectangle is found by multiplying the length by the width. The area is also measured in square units, so the correct answer is Choice D. The answer of 26 is the perimeter. The answer of 13 is found by adding the two dimensions instead of multiplying.

38. What is the area of the following figure?

B: The volume of a rectangular prism is found by multiplying the length by the width by the height. This formula yields an answer of 144 cubic units. The answer must be in cubic units because volume involves all three dimensions. Each of the other answers have only two dimensions that are multiplied, and one dimension is forgotten, as in D, where 12 and 3 are multiplied, or have incorrect units, as in E.

39. What is the volume of the given figure? a. 36 cm2 b. 144 cm3 c. 72 cm3 d. 36 cm3 e. 144 cm2

D: This is a one-step real-world application problem. The unknown quantity is the number of cases of cola to be purchased. Let be equal to this amount. Because each case costs $3.50, the total number of cases times $3.50 must equal $40. This translates to the mathematical equation 3.5x = 40 Divide both sides by 3.5 to obtain x = 11.4286, which has been rounded to four decimal places. Because cases are sold whole (the store does not sell portions of cases), and there is not enough money to purchase 12 cases, there is only enough money to purchase 11.

4. How many cases of cola can Lexi purchase if each case is $3.50 and she has $40? a. 10 b. 12 c. 11.4 d. 11 e. 12.5

A: Surface area is a type of area, which means it is measured in square units. Cubic units are used to describe volume, which has three dimensions multiplied by one another. Quartic units describe measurements multiplied in four dimensions.

40. What type of units are used to describe surface area? a. Square b. Cubic c. Single d. Quartic e. Volumetric

B: The perimeter is found by adding the length of all the exterior sides. When the given dimensions are added, the perimeter is 22 meters. The equation to find the perimeter can be P=5+1.5+1.2+4.5+3.8+6=22. The last two dimensions can be found by subtracting 1.2 from 5, and adding 1.5 and 4.5, respectively.

41. What is the perimeter of the following figure? a. 13.4 m b. 22 m c. 12.2 m d. 22.5 m e. 24.4 m

A: The surface area for a cylinder is the sum of the areas of the two circle bases and the rectangle formed on the side. This is easily seen in the net of a cylinder. The area of a circle is found by multiplying pi times the radius squared. The rectangle's area is found by multiplying the circumference of the circle by the height. The equation SA=2π x 5 x 10+2(π5²) shows the area of the rectangle as 2πx5x10, which yields 314. The area of the bases is found by π5², which yields 78.5, then multiplied by 2 for the two bases.

42. Which equation correctly shows how to find the surface area of a cylinder?

C: A hexagon can be formed by any combination of the given shapes except for two rectangles. There are no two rectangles that can make up a hexagon.

43. Which shapes could NOT be used to compose a hexagon? a. Six triangles b. One rectangle and two triangles c. Two rectangles d. Two trapezoids e. One rectangle and four

A: First, the variables have to be defined. Let be the first integer; therefore, x = 1 is the second integer. This is a two-step problem. The sum of three times the first and two less than the second is translated into the following expression: 3x + (x + 1 -2). This expression is set equal to 411 to obtain 3x + (x + 1 -2) = 412. The left-hand side is simplified to obtain 4x -1 = 411. The addition and multiplication properties are used to solve for x. First, add 1 to both sides and then divide both sides by 4 to obtain x = 103. The next consecutive integer is 104.

5. Two consecutive integers exist such that the sum of three times the first and two less than the second is equal to 411. What are those integers? a. 103 and 104 b. 104 and 105 c. 102 and 103 d. 100 and 101 e. 101 and 102

A: Let be the unknown, the number of hours Erin can work. We know Katie works , and the sum of all hours is less than 21. Therefore, x + 2x < 21, which simplifies into 3x < 21. Solving this results in the inequality x < 7 after dividing both sides by 3. Therefore, Erin can work less than 7 hours.

6. Erin and Katie work at the same ice cream shop. Together, they always work less than 21 hours a week. In a week, if Katie worked two times as many hours as Erin, how many hours could Erin work? a. Less than 7 hours b. Less than or equal to 7 hours c. More than 7 hours d. Less than 8 hours e. More than 8 hours

A: The chart is a bar chart showing how many men and women prefer each genre of movies. The dark gray bars represent the number of women, while the light gray bars represent the number of men. The light gray bars are higher and represent more men than women for the genres of Comedy and Action.

7. From the chart below, which two types of movies are preferred by more men than women? a. Comedy and Action b. Drama and Comedy c. Action and Horror d. Action and Romance e. Romance and Comedy

E: A line graph represents continuous change over time. The line on the graph is continuous and not broken, as on a scatter plot. Stacked bar graphs are used when comparing multiple variables at one time. They combine some elements of both pie charts and bar graphs, using the organization of bar graphs and the proportionality aspect of pie charts. A bar graph may show change but isn't necessarily continuous over time. A pie graph is better for representing percentages of a whole. Histograms are best used in grouping sets of data in bins to show the frequency of a certain variable.

8. Which type of graph best represents a continuous change over a period of time? a. Stacked bar graph b. Bar graph c. Pie graph d. Histogram e. Line graph

C: The mean for the number of visitors during the first 4 hours is 14. The mean is found by calculating the average for the four hours. Adding up the total number of visitors during those hours gives 12 + 10 + 18 + 16 = 56. Then, 56 divide 4 = 14

9. Using the graph below, what is the mean number of visitors for the first 4 hours?

A: Division can be used to solve this problem. The division necessary is: (5.972 x 10²⁴)/(7.348x10²²) To compute this division, divide the constants first then use algebraic laws of exponents to divide the exponential expression. This results in about 0.8127x10², which written in scientific notation is 8.127x10¹.

What is the solution?

A: First, the distributive property must be used on the left side. This results in 3x + 6 = 14x -5. The addition property is then used to add 5 to both sides, and then to subtract 3x from both sides, resulting in 11 = 11x. Finally, the multiplication property is used to divide each side by 11. Therefore, x = 1 is the solution.

What is the solution?

A: The corresponding expression written using common denominators of the exponents is (16^1/4)(16^2/4), and then the expression is written as: (16 x 16²)^1/4 This can be written in radical notation as: ⁴√16³ = ⁴√4,096 =±8

What is the solution?

B: The slopes of perpendicular lines are negative reciprocals, meaning their product is equal to -1. The slope of the line given needs to be found. Its equivalent form in slope-intercept form is y = -4/7x + 23, so its slope is -4/7, The negative reciprocal of this number is 7/4, only line in the options given with this same slope is y = 7/4x -12.

What is the solution?

C: By switching from a radical expression to rational exponents, ⁴√x⁶ = x^6/4 = x^3/2. Also, properties of exponents can be used to simplify x/x³ into x¹⁻² = x⁻² = 1/x².The other terms can be left alone, resulting in an equivalent expression x^3/2 - 1/x² + x - 2.

What is the solution?

D: First, like terms are collected to obtain 12 - 5x = -5x + 12. Then, the addition principle is used to move the terms with the variable, so 5x is added to both sides and the mathematical statement 12 = 12 is obtained. This is always true; therefore, all real numbers satisfy the original equation.

What is the solution?

D: The exponential rules (ab)^m =(a^m)(b^m) and (a^m)^n = a^(m)(n) can be used to rewrite the expression as

What is the solution?

E: The conversion between feet and centimeters requires a middle term. As there are 2.54 centimeters in 1 inch, the conversion between inches and feet must be found. As there are 12 inches in a foot, the fractions can be set up as follows: 3ft x 12in/1tft x 2.54cm/1in The feet and inches cancel out to leave only centimeters for the answer. The numbers are calculated across the top and bottom to yield: (3x12x2.5)/(1x1) =91.44 The number and units used together form the answer of 91.44 cm.

What is the solution?

E: The distributive property is used on both sides to obtain 4x + 20 + 6 = 4x + 6. Then, like terms are collected on the left, resulting in 4x + 26 = 4x + 6. Next, the addition principle is used to subtract 4x from both sides, and this results in the false statement 26 = 6. Therefore, there is no solution.

What is the solution?

E: Using Descartes' Rule of Signs, count the number of sign changes in coefficients in the polynomial. This results in the number of possible positive zeros. The coefficients are 1, -3, 2, 1, and -3, so the sign changes from 1 to -3, -3 to 2, and 1 to -3, a total of 3 times. Therefore, there are at most 3 positive zeros.

What is the solution?