Math Knowledge ASVAB
B. 5 miles Think of a triangle. When it goes east then travels north its creating a right triangle. We have the two sides. 3 and 4 and we need the last one. We need to use Pythagorean theorem. C² = a²+b² 3²+4² = 9 +16 = 25 c² = 25, square root both sides. C = 5
A women travels 3 miles directly east and then travels 4 miles directly North. How many miles is she from her starting point? A. 7 miles B. 5 miles C. 25 miles D. 3 1/2 miles
B. 0.78
What is 0.7849 rounded to the nearest hundredth? A. 0.8 B. 0.78 C. 0.785 D. 0.79
Answer C. Do the fractions first. Eventually you will reduce the 16s and be left with 3.
1/2 x 16 x 3/8 = A. 1/4 B. 2 (5/16) C. 3 D. 4 (1/4)
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?
C. 15.68 Remember PEMDAS. You go left to right when its just addition and subtraction. First do 12.00 - 0.92 = 11.08 Then we add 11.08 to 4.60 which will give is 15.68 which is C.
12 - 0.92 + 4.6 A. 17.52 B. 16.68 C. 15.68 D. 8.4
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. The slope formula is m=(y2-y1)/(x2-x1) (-5-7)/(-1-(-3)) = -6 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(x-(-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)?
C. 40 0.40 x 20 = .20x 8= .20x 8/.20 = 40
40 percent of 20 is 20 percent of what number? A. 10 B. 20 C. 40 D. 80
B. 12 We are dealing with Volume here. Volume of a quadrilateral = Length x width x height. First box is 2 inches x 3 inches x 4 inches = 24cubic inches Second box is 3 inches x 8inches x 12inches = 288 cubic inches. How many of those small boxes fit into the family size? 288/24 = 12
A cereal manufacturer packages breakfast cereal in individual-sized boxes measuring 2 inches by 3 inches by 4 inches. The same product is also packaged in large family sized boxes measuring 3 inches by 8 inches by 12 inches. The contents of how many of the individual-sized boxes would be required to fill one family-sized box? A. 6 B. 12 C. 10 D. 8
A. 4% 10 ounces 20% x 10 = 2 ounces of fruit juice 90% = 8 ounces of water Add 40 ounces of water 2 ounces of fruit/48 ounces of water = 0.04 or 4%
A cook is mixing fruit juice from concentrate for a catered event. Ten ounces of liquid contain 20 percent fruit juice and 90 percent water. He then further dilutes the mixture by adding 40 additional ounces of water. What is the percent of fruit juice in the new solution? A. 4% B. 10% C. 14% D. 18%
D. 54 square inches Area if a cube is 6a² In this case each edge is 3 inches as the area of one side is 9 = 3x3 and 9x3 = 27 So, using the formula where a = value of one edge 6a² = 6(3)² = 54
A cube has a volume of 27 cubic inches. What is the surface area? A. 18 square inches B. 36 square inches C. 45 square inches D. 54 square inches
(C) 54 square inches First if the volume is 27 cubic inches we need to find the number of each side. What times itself 3 times gives us 27? 3x3x3 = 27 So each side is 3. SA of a cube = 6a² 6(3)² = 54
A cube has a volume of 27 cubic inches. What's its surface area? (A) 9 square inches (B) 6 square inches (C) 54 square inches (D) 4.5 square inches
If a rectangle has a length of 24 inches and a width of 4 inches, what is the perimeter of the rectangle in feet? A. 2 (1/3) feet B. 4 (2/3) feet C. 48 feet D. 56 feet
B. 4 (2/3) feet First, add the two lengths and the two widths to find the perimeter in inches: P = 24 + 24 + 4 + 4 = 56 inches. Now, convert inches to feet. There are 12 inches in 1 foot: 56/12 = 14/3 = 4 (2/3)
D. 9x²-30x+25 1) (3x-5)² = (3x-5)x(3x-5) 2) FOIL = First, outer, inner, last 3) 9x² -15x-15x+25 4) 9x² -30x+25 answer D
Expand (3x-5)² A. 9x²+30x-25 B. 9x²-15x+25 C. x²-30x+25 D. 9x²-30x+25
If m∠ 2=120°, then m∠ 1 = 60 because they are a linear pair (add to 180°) m∠ 1 = m∠ 3 because they are corresponding angles. So, m∠ 3 = 60°
In the figure above m is parallel to n. If m∠ 2=120°, then what is m∠ 3? A. 120° B. 60° C. 30° D. 90°
C.1/6 A circle is 360°; 60° is 1/6 of 360°.
In the figure above, m∠ AOB = 60°. If O is the center of the circle, then minor arc AB is what part of the circumference of the circle? A. 1/2 B. 1/3 C.1/6 D.1/8
C. 40º If the sides are parallel, the angles are congruent. It will be the same. 40º
In the figure above, the sides of ABC are respectively parallel to the sides of DEF. If the complement of C is 40°, then the complement of F is A. 20º B. 50º C. 40º D. 60º
A. 1/7 2x/7 = 2x² - multiply both sides by 7 to get rid of denominator. 14x² = 2x - divide both sides by 2 14x²/2x = 7x=1, x = 1/7
Solve for x: 2x/7 = 2x² A. 1/7 B. 2/7 C. 2 D. 7
C. x = -5, x = 3 This is a quadratic equation. So we have to move everything to the left and make it equal to zero. 1) x² + 2x = 15 , subtract 15 to get it to the left. x² + 2x - 15 = 0 Now the x² is split into 2. (x ) (x ) = 0 What 2 numbers when multipled give you 15. 1 and 15 could work but remember that when we FOIL we must also be getting 2x which is the middle term. There is trail and error in this so be quick. In the end you will find that +5 and -3 are the factors to choose for -15 (x+5)(x-3) = 0 x+5 =0, x = -5 x-3 = 0, x = 3
Solve for x: x² + 2x = 15 A. x = 3, x = 5 B. x = -3, x= 5 C. x = -5, x = 3 D. x = -15, x =1
C. 300 square feet Perimeter of the square is 40 yards means each side is equal to 10 since its 4 sides. P = 2l+2W 2 (10)+2(10) = 40 Now area is Length x width. 10 x 10 = 100 square yards. Since this is not one of the answers lets convert it to feet. 1 yard = 3 feet 100 yd²/1 = 3ft/1 yd = 300 ft²
The area of a square with an outside perimeter measurement of 40 yards is A. 100 square feet B. 180 square yards C. 300 square feet D. 300 square yards.
(B) (90 -m)° If two angles are complementary, the sum of their measures is 90°. Because the measure of angle P is m°, you find the complement of angle P by subtracting its measure from 90°.
The measure of angle P is m°. What is the measure of the complement of angle P ? (A) (180 -m)° (B) (90 -m)° (C) (m -90)° (D) (m -180)°
B. 4r Perimeter = r + r + r + r = 4r
The perimeter of a square with side r = A. r²+r² B. 4r C. r⁴ D. None of the above
D. 7 square of 525 is 525 x 525 = 275,625 Second digit is 7
The second digit of the square of 525 is A. 2 B. 5 C. 6 D. 7
(A) 360 degrees. All quadrilaterals (four-sided figures) have angles that total 360 degrees.
The sum of the measures of the angles of a trapezoid is (A) 360 degrees. (B) 540 degrees. (C) 180 degrees. (D) 720 degrees.
A. 540° The formula for the sum of the measures of the interior angles of any polygon is 180(n - 2), where n = number of sides of the figure. Since a pentagon has 5 sides, 180(5 - 2) = 540°
The sum of the measures of the interior angles of a pentagon is A. 540° B. 720° C. 900° D. 1,080°
Answer C. The formula for the area of a circle is pie x r^2. Find the area of the larger circle first. 1) pi x 10^2 = 100 pi Find the rea of the smaller circle 2) pi x 7^2 = 49 pi To find the part of the larger circle that the smaller oen doesn't touch, subtract the two areas. 100-49 =51 pi square inches which is C.
Two circles have the same center. If their radii are 7 inches and 10 inches, find the area that is part of the larger circle but not the smaller one. A. 3 square inches B. 17 square inches C. 51 pi inches D. 70 pi square inches
A. 360 108 is 30% Let x represent the total number of apartments. What (total amount) times 30% will give us 108. Thus, 30%x = 108 .30x = 108 x = 108/.30 1080/30 = 360 is 100% that she can build.
Under the terms of a federal subsidy, a real estate developer is required to rent at least 30 percent of the apartments she builds to low-income families. If she plans on having 108 low-income apartments, what is the maximum number of apartments of all types that she can build? A. 360 B. 252 C. 324 D. 396
Answer C 1 ounce = 1/16 of a pound Four ounces = 4/16 reduce by 4 = 1/4
Four ounces is what fraction of a pound? A. 1/3 B. 3/8 C. 1/4 D. 1/6
D. 27 (a/b)/c = a/(b×c) a/(b/c) = (a×c)/b 1) 2/(1/3) = (2×3)/1 = 6 2) (8/9)/4 = 8/ (9×4) = 8/36 3) 6 ÷ 8/36 4) 6/1 x 36/8 = 27
If p = 1/3 and q =8/9, then which is equal to 2/p ÷ q/4? A. 21 B. 23 C. 25 D. 27
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) 10 PEMDAS (9-3 x 2)² - 0.5(-2) (9-6)²-0.5(-2) 3²-0.5 (-2) 9-(-1) = 10
(9-3 x 2)² - 0.5(-2) = (A) 10 (B) 145 (C) 8 (D) 143
Answer B Remember to make something into a percent. Move the decimal 2 times to the right.
0.925 is equal to what percent? A. 925% B. 92.5% C. 9.25% D. 0.0925%
B. 18.75 First get 25 percent of 750 .25 x 750 = 187.50 187.50 x 10% = Move decimal to the left 1 time 18.75
10 percent of 25 percent of 750 is A. 3.00 B. 18.75 C. 187.5 D. 300.0
C. 3 quarts 12 quarts. 25 percent of it is antifreeze so lets take 25% of 12. 25% is 1/4 12 x 1/4 = 3 quarts of antifreeze. Now let x equal the quarts of water we need to add. The total mixture will be 12 + x. 20 percent of the new total mixture will be the 3 quarts of antifreeze still present in the mixture. Thus, 20% (12+ x) = 3 2.4+0.2x = 3 Subtract 2.4 both sides 0.2x = 0.6 x = 3
12 quarts of a radiator coolant contains 25 percent antifreeze and 75 percent water. How many quarts of water must be added to change the mixture to one containing 20 percent antifreeze? A. 1 quart B. 2 quarts C. 3 quarts D. 4 quarts
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?
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
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
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?
Answer A. Simply make 1 (4/5) into improper 1 (4/5) = 9/5 4/1-9/5 - make the denominator the same 20/5 - 9/5 = 11/5 -improper only choice is A.
4 - 1(4/5)= A. 2 (1/5) B. 2 (4/5) C. 3 (3/10) D. 3 (1/5)
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?
B. 0.25 Remember of means multiplication First don't get confused with the left to right bullshit. The question is not asking 5%/5%/100. Its asking Take 5% of the 5% of 100. Lets first take 5% of 100 100 x 5% = 5 Now lets take 5% of 5 5 x 0.05 = 0.25
5% of 5% of 100 is A. 25 B. 0.25 C. 2.5 D. 10
Answer A. Simply reduce. 5/8 x 4/16 - reduce 1/2 x 1/3 = 1/6 which is A.
5/8 x 4/15 A. 1/6 B. 2/5 C. 9/15 D. 7/45
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
B. 6 feet Think of a triangle. Pythagorean theorem C² = a² + b² 8²+ x² = 10² 64+x² = 100 x² = 36 x=6
A 10 foot high ladder is resting against an 8-foot-high wall around a recreation area. If the top of the ladder is exactly even with the top of the wall, how far is the base of the ladder from the wall? A. 18 feet B. 6 feet C. 12 feet D. 9 feet
B. 150 Right now. 120 miles for every 12 gallons. 120/12 = 10 mpg. 80% reduction to the needed gas. 12 x 80% = 9.6 gallons to the 120. 120/9.6 = 12.5 mpg. 12 gallons x 12.5 mpg = 150 miles on 12 gallons.
A car owner finds he needs 12 gallons of gas for each 120 miles he drives. If he has his carburetor adjusted, he will need only 80% as much gas. How many miles will 12 gallons of gas then last him? A. 90 B. 150 C. 96 D. 160
(B) 3 cm Circumference = π x diameter They give us the circumference 9.24 = 3.14d - solve for d 9.24/3.14 = 3
A circle has a circumference of 9.24 centimeters. What is the diameter of the circle? (A) 1.5 cm (B) 3 cm (C) 3.14 cm (D) 6 cm
D. 10 gallons ' = inches Volume of circle = π x r² x height π = 22/7 22/7 x 49 x 15 22/7 x 49 = 154 154 x 15 = 2,310 cubic inches 231 cubic inches in a gallon so 2310/231 = 10 gallons
A cylindrical container has a radius of 7" and a height 15". How many gallons of hydraulic fluid can it hold? (There are 231 cubic inches in a gallon.) A. 15 gallons B. 14 gallons C. 140 gallons D. 10 gallons
D. 38 feet If the fence fits around the rectangle and the square field that means their perimeters are equal. Perimeter = 2w +2l, 2 (40+36) = 152 feet Since we need to know the length of one side and its a square we divide 152 by 4 152/4 = 38 feet.
A fence that had been installed around a rectangular field 40 feet long and 36 feet wide is torn down. The entire fence is the reused to completely enclose a square field. What is the length in feet of a side of the square field? A. 76 feet B. 19 feet. C. 42 feet D. 38 feet
The correct answer is C. With the passage of time, the ratio of the ages of A and B decreases. Pick a pair of ages and try for yourself. A is 4; B is 2; the ratio of their ages is 4:2 or 2:1. In two years, A is 6 and B is 4. The ratio of their ages is 6:4 or 3:2.
A is older than B. With the passage of time, the A. ratio of the ages of A and B remains unchanged. B. ratio of the ages of A and B increases. C. ratio of the ages of A and B decreases. D. difference in their ages varies
D. 1 foot, 6 inches 5 feet, 3 inches - 3 feet, 9 inches. _____________________ We can't take 9 inches from the 3 inches so lets take 1 foot from the 5 convert it to inches which will be 12 and give it to the 3 so that it can compute. 4 feet, 15 inches - 3 feet, 9 inches _______________________ 1 feet, 6 inches
A length of chain is 5 feet, 3 inches long. if a piece 3 feet, 9 inches in length is cut from the chain, what the length of the remaining chain? A. 2 feet, 6 inches B. 1 foot, 1 inch C. 1 foot, 4 inches D. 1 foot, 6 inches
B. T - 1,000 If the man uses $1,000 of his T dollars, he has T - $1,000 remaining.
A man has T dollars to invest; after he invests $1,000, how much money does he have remaining? A. T + 1,000 B. T - 1,000 C. 1,000 - T D. 1,000T
B. 44 miles per hour 3 x 40 = 120 miles 2 x 50 = 100 miles Add miles 220 miles Add hours 3+2 = 5 hours 220/5 = 44 miles
A motorist travels 3 hours at 40 miles per hour and then travels 2 more hours at 50 miles per hour. What is her average rate of speed in miles per hour for the entire trip? A. 45 miles per hour B. 44 miles per hour C. 43 miles per hour D. 90 miles per hour
D. 154 kilometers Area of a circle = πr² π = 22/7 A = 22/7 x (14)² = 616 square kilometers. Since the area is locked at 90 degrees of the circle Circle is 360 degrees. 90/360 =1/4 The fraction of the whole. 616 x 1/4 = 154 square kilometers.
A new wildlife preserve is laid out in a perfect circle with a radius of 14 kilometers. The lion habitat is shaped like a wedge and has an 8-foot-high razor wire fence around it. Two inner sides of the fence meet at a 90 degree angle in the center of the base. How much ground space (area) does the lion habitat have? A. 140 square kilometers B. 3.5 square kilometers C. 210 kilometers D. 154 kilometers
A. both pairs of its opposite sides are parallel.
A quadrilateral is a parallelogram only if A. both pairs of its opposite sides are parallel. B. it has four right angles. C. it has four right angles and four equal sides. D. it has at least one pair of parallel sides.
C. 31.4 The radar is capable of covering a complete circle whose radius is 10 miles. First find the area of this circle Area of a circle = πr² Area = 3.14 x 10² = 314 square miles There are 260 degrees of rotation in the complete circle. If the radar is used to cover a portion of this, it covers 36/360 or 1/10 1/10 x 314 = 31.4
A radar device is capable of detecting objects within the area around it up to a radius of 10 miles. If it is used to cover a 36 degree angular portion of this area how many square miles of area will it cover? A. 360 B. 6.28 C. 31.4 D. 3.6
A. 19π Remember! Radius is always half the diameter. 20 inches in diameter = 10 inches Radius 18 inches in diameter = 9 inches in Radius πr² formula for both We dont need to count for π π(10) ²= 100π π(9)² = 81π 100 - 81 = 19π
A round armored vehicle hatch is 18 inches in diameter. It has connected to it a circular hatch cover that measures 20 inches across at the middle. How much greater is the area of the hatch cover than the area of the hatch? A. 19π B. 20π C. 40π D. 100π
B. $20.40 1) Make 25.50 whole - 2 steps. 2) Make .20 whole - 2 steps. 3) 4 steps total to move back. 4) 2550 x 20 or 255x2 = 51000 5) Move back 4 times. 6) 5.10 7) Subtract 5.10 from 25.50 = 20.40 which is B.
A shirt normally costs $25.50. How much do you need to pay if it is purchased at a 20% discount? A. $12.50 B. $20.40 C. $21.40 D. $24.00
(B) 16 inches Volume equals length times width times height V lwh . In this case, V= 64, so one edge of the box is 4 inches long (because 4 is the cube root of 64: 64 = 4x4x4). Find the perimeter by adding the four sides together: 4+4+4+4 = 16
A square box has a volume of 64 cubic inches. What's the perimeter of one of its faces? (A) 8 inches (B) 16 inches (C) 64 inches (D) 32 inches
D. decreases by 1%. Assign arbitrary values to solve this problem: Area of square 10 ft. × 10 ft. = 100 sq. ft. Area of rectangle 9 ft. × 11 ft. = 99 sq. ft. 100 − 99 = 1; 1/ 100 = 1%
A square is changed into a rectangle by increasing its length 10% and decreasing its width 10%. Its area A. remains the same. B. decreases by 10%. C. increases by 1%. D. decreases by 1%.
A. 24 Area of rectangle = 9 × 4 = 36 Area of square = 36. Therefore, each side = 6 and the perimeter of the square = 6 + 6 + 6 + 6 = 24.
A square is equal in area to a rectangle whose length is 9 and whose width is 4. Find the perimeter of the square. A. 24 B. 26 C. 34 D. 36
D. He cannot achieve a 75 average. Two tests were 60 Third test was a 70. What does he need on the fourth for average 75? Treat problem with variable. (60+60+70+x)/4 = 75 190+x/4 = 75 Multiply both sides by 4 190+x = 300 Subtract both sides by 190 x = 110. He needs a 110 to get a 75. Not possible.
A student has grades of 60 on each of two tests and a grade of 70 on a third test. What grade must he get on a fourth test to raise his average to 75? A. 95 B. 85 C. 100 D. He cannot achieve a 75 average.
A. 44 inches One rotation will move the wheel along by a distance equal to the circumference. C = 2πr Or since we are giving the diameter C = π x D C = 22/7 x 14 = 44 inches
A wheel has a diameter of 14 inches. How many inches will the wheel roll along the ground during one rotation? (Use 22/7 as the value of π) A. 44 inches B. 22 inches C. 14 inches D. 28 inches
B. 8/15 Add the portions. 1/3+1/5 = 5/15 + 3/15 = 8/15 If they work together they can do 8/15 of the job.
A worker can do 1/3 of a job by herself in one day, and her helper can do 1/5 of the job by himself in one day. What portion of the job can they do if they work together for one day? A. 1/4 B. 8/15 C. 1/8 D. 2/15
Simplify: 2√24 - 3√54 A. -5√6 B. 5√6 C. -√2 D. 0
A. -5√6 √24 - Find 2 numbers and one of those numbers can be squared into a whole number 6 x 4 √4 = 2 and √6 cant be squared so we leave it alone 2 x 2√6 √54 = 9x 6 √9 = 3 and √6 stays same 3√6 2 x 2√6 -3x3√6 4√6-9√6 = -5√6
B. 86 (78+86+96+94+x)/5 =88 354+x/5 = 88 - multiply 5 both sides 354+x = 440 - subtract 354 x = 86
Amanda took five midterm tests for five different college classes; her average for all five tests was 88. That night at home, she could remember only her first four scores: 78,86,94, and 96. What was her score on the fifth test? A. 82 B. 86 C. 84 D. 88
D. 251 kilometers There asking for circumference. Diameter is not giving so we will use 2π x r to get the radius 2π = 6.28 r = 40 40 x 6.28 = 251 or D.
An F/A-18E Super Hornet is flying a circular or "racetrack" orbit around a 4,000 meter high mountaintop. Assuming that the pilot flies a perfectly circular course and that it is 40 kilometers from the mountaintop to the outer edge of his racetrack, what is the distance in kilometers he travels during each orbit? (use π = 22/7) A. 13 kilometers B. 25 kilometers C. 126 kilometers D. 251 kilometers
D. 150° Supplement means 180° Let x equal the measure of the supplement. Let 5x equal the measure of the angle. x +5x = 180° 6x = 180° x = 30 5(30) = 150°
An angle has a measure five times that of its supplement. What is the measure of the angle? A. 60° B. 75° C. 120° D. 150°
C. 20 The square has 4 sides. Each side measures 15 so 15 x 4 = 60 inches Since we need to know the side of one side of the triangle with the same perimeter. We simply divide 60 by 3 60/3 = 20
An equilateral triangle has the same perimeter as a square whose sides each measure 15 inches. What is the length of one side of the triangle? A. 9 inches B. 12 C. 20 D. 30
What is the product of (a+2)(a-5)(a+3) A. a³+2a²+15a-30 B. a³+6a²-49 C. a³-19a-30 D. a³+2a²-15a+30
C. a³-19a-30 (a+2)(a-5)(a+3) 1) First FOIL the first two (a+2)(a-5) a²-5a+2a-10 2) a²-3a-10 (a+3) - Now we multiply a binomial to a trinomial. BOX method (Multiply each one to the trinomial) a² -3a -10 (a) a³ -3a² -10a (+3) 3a² -9a -30 Combine like terms a³-19a-30
In circle O, the radius is 9 units long. Q, R, S, and T are points on the circle, and QRST is a square. Find the length of a diagonal of square QRST. A. 6√2 B. 12 C.12√2 D. 18
D. 18 In order for QRST to be a square, the four vertex points must be evenly spaced around the circle. The diagonal connects opposite corners, so the diagonal is also a diameter of the circle. The diagonal of the square is double the radius, or 2 × 9 = 18.
D. (v²-u²)2s What its asking is get (a) by itself from v² = u²+2as 1) Solve for a 2) subtract u² to both sides 3) v² - u²=2as 3) Since 2as is multiplication lets divide both sides by 2s to get a by itself 4) a = (v²-u²)/2s which is D.
Express a in term of u, v , and s: v² = u²+2as A. (v²+u²)/2s B. (u²-v²)/2s C. (v²-u²)/s D. (v²-u²)2s
B. (z-5)(x+y) 1) Factor out common terms 2) xz-5x = x( z-5) 3) yz-5y = y (z-5) 4) x( z-5) + y (z-5) = (z-5) 5) (z-5)(x+y) which is B
Factorize completely: xz-5x+yz-5y A. (z+5)(x+y) B. (z-5)(x+y) C. (x-5)(x+z) D. (x+5)(x-y)
D. 45 Circumference = 2πr Substitute 90π into the equation 90π =2πr Divide both sides by 2π r = 45
Find the radius of a circle whose circumference is 90π. A. 45π B. 90π C. 90 D. 45
C. -0.0001 Do not trick yourself into just focusing on the negative and positives. 0.1 is 1/10 or 10-¹. If we're talking about -0.1⁴. That just means were moving to the left 4 times since its negative.
Find the value of -x⁴ if x =0.1 A. -0.1 B. 0.0001 C. -0.0001 D. -0.4
A. 11x²-9x These are binomials. Meaning when you subtract or add you need to put parentheses around them and distribute if needed. Align the variables in descending order. Thus, (8x²-7x)-(-3x²+2x) - distribute the negative. 8x²-7x +3x²-2x - combine line terms. 8x²+3x² = 11x² -7x-2x = -9x 11x²-9x
From 8x²-7x subtract 2x-3x² A. 11x²-9x B. 5x²-5x C. 6x²-4x D.10x²-10x
D. A = 2r
Given the formulas d = rt and A = r + d/t which formula below correctly expresses the value of A without using t? A. A = dr B. A = r +d/t C. A = 2r + d D. A = 2r
A. 2 cubic yards ' symbol = feet '' symbol = inches 1 foot = 3 yards Lets convert all feet into yards 9 feet = 3 yards. 12 feet =4 yards 6 inches= 1/6 yards To find volume L x W x H 3 x 4 x 1/6 12/6 = 2 cubic yards
How many cubic yards of concrete are needed to make a cement floor that measures 9' x 12' x 6''? A. 2 cubic yards B. 18 cubic yards C. 54 cubic yards D. 210 cubic yards
(A) Four The factors of a number are all the numbers, including the number and 1, that divide into the number without a remainder. The number 51 has four factors: 1, 3, 17, and 51
How many factors does the number 51 have? (A) Four (B) Three (C) Two (D) One
D. (36y+ i)/12 1 yard = 3 feet 1 foot = 12 inches 1 yard = 36 inches So in y yards there are y times as many, or 36y. The total length of y yards and i inches, expressed inches is 36 y + i There are 12 inches in 1 foot. To find the number of feet in 36y + i inches, divide it by 12 (36y+ i)/12
How many feet are there in a length of y yards and i inches? A. 3y + i B. 3y+12i C.(y+12i)/3 D. (36y+ i)/12
B. 2 2 - x = x - 2 Solve for x Move 2 to the right -x = x-4 Move x to the left -2x = -4 Divide -2 both sides x = 2
If 2 - x = x - 2, then x = A. -2 B. 2 C. 0 D. 1/2
C. 1/12 If 3/4 are absent means we have 1/4 left. 2/3 of that 1/4 leave the room. Which means we have 1/3 left. 1/3 x 1/4 = 1/12
If 3/4 of a class is absent and 2/3 of those present leave the room, what fraction of the original class remains in the room? A. 1/24 B. 1/4 C. 1/12 D. 1/8
(C) either a positive or negative number. In this scenario y = 4 Now it doesn't matter if its -4 or just 4. Square both of them and you still get 80 in the equation. Hence why C is the answer.
If 5y² = 80 then y is (A) a positive number. (B) a negative number. (C) either a positive or negative number. (D) an imaginary number.
A. 60MT To find how many tons fall in a given number of minutes, multiply the number of tons that fall in 1 minute by the number of minutes. There are 60 seconds in 1 minute, and T tons fall in 1 second. In M minutes, the amount of snow that falls is 60MT.
If T tons of snow fall in 1 second, how many tons fall in M minutes? A. 60MT B. MT + 60 C. MT D. 60M/T
B. x solve for b a³ + b³ = a³ + x³ 1) Subtract a³ noth sides b³ = x³ Cube root both sides b = x
If a³ + b³ = a³ + x³, then b = A. b³-a³ B. x C. a³-b³ D. a
A. 0 f(x²) = 144 1) x² = 144 - square root both sides 2) x = 12 3) plug in 12 back to the equation. 4) 1/4 x 12 - 3 5)12/4 - 3 6) 3-3= 0 which is A.
If f(x²) = (1/4)x-3, what is the value of f(144)? A. 0 B. 1 C. 4 D. 8
A. 80º ∠ 2 and ∠ 1 are supplementary angles. m∠ 1 = 100°. ∠ 1 and ∠ 4 are supplementary. m∠ 4 =80°.
If m∠ 2 = 80° in figure above, m∠ 4 = A. 80º B. 100º C. 120º D. None of the above
Every right triangle contains an angle of 90 degrees. This particular right triangle also has an angle of 30 degrees. To find the third angle, subtract the sum of these two angles by 180 degrees. 1) 90+30 = 120 2) 180 -120 = 60 degrees. The other 2 angles are 60 and 90 which is C.
If one of the angles of a right triangle is 30 degrees, what are the other 2 angles. A. 30 degrees, 120 degrees B. 60 degrees, 45 degrees C. 60 degrees, 90 degrees D. 45 degrees, 90 degrees
B. 10 inches area of triangle = 1/2 (base)(height) 45 = 1/2(9)h 45 = 9/2 h Multiply both sides by 2 9x = 90 x = 10
If the area of a triangle is 45 square inches and its base is 9 inches, what is the length of the altitude to that base? A. 5 inches B. 10 inches C. 18 inches D. 22.5 inches
C. 2 The formula to find the circumference of a circle is 2πr. The formula to find the area of a circle is πr²— the only number that has the same value when multiplied by 2 or squared is 2.
If the circumference of a circle has the same numbered value as its area, then the radius of the circle must be A. 1 B. 5 C. 2 D. 0
C. 6π Circumference = 2π x r If radius is increased by 3 C = 2(r + 3) x π C = (2r + 6) x π = 2πr + 6π
If the radius of a circle is increased by 3, the circumference is increased by A. -3 B. 3π C. 6π D. 6
D. 8 8 + 5(x - y) = 8 + 5x - 5y Since x = y, 5x = 5y and 5x - 5y = 0 Substituting: 8 + 0 = 8
If x = y, find the value of 8 + 5(x - y). A. 8 + 5x - 5y B. 8 + 5xy C. 13x - 13y D. 8
D. 11 FOIL (x+2)(2x+5) 2x²+11x+15 When they ask for coefficient, it means the number of what it is worth. x is next to 11 so 11 is the coefficient. If they asked for the coefficient of x². It will be 2 as 2 is next to x²
If you multiply x + 3 by 2x + 5, what will the coefficient of x be? A. 3 B. 6 C. 9 D. 11
C. 9 feet The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. If the unknown length is x, the ratio of x to the length of the hypotenuse must be the same as the ratio 3/7. Thus, x/21 = 3/7 - solve for x Multiply both side by 21 x = 3/7 x 21/1 x = 9
In a right triangle whose hypotenuse has a length of 21 feet, the sine of one of the angles is 3/7. What is the length in feet of the side opposite this angle? A. 6 feet B. 14 feet C. 9 feet D. 10 feet
A. 25° Angles that are complementary add up to 90° If the complement of angle A is 65°, then angle A is 25°. Since angle A and angle D are corresponding angles in similar triangles, then angle D is also 25°.
In the figure above, the sides of ∆ABC are respectively parallel to the sides of ∆DEF. If the complement of A is 65°, then the measure of angle D is A. 25° B. 35° C. 55° D. 65°
(A) (7, 0) The equation 3x +7y = 21 is the equation for a line, and a line intersects the x-axis at the point where the y-coordinate is 0 (y=0). Substitute 0 for y in the equation to find the value for x at the intersection point: 3x +7y = 21 3x+7(0)=21 3x=21 x = 7 The points coordinates are (7,0)
In the graph 3x +7y = 21, at what point is the x-axis intersected? (A) (7, 0) (B) (0, 7) (C) (0, 4) (D) (4, 0)
B. 80 degrees Remember isosceles has 2 sides that are equal. Means we have 2 50's =100 degrees. Means we only have 80 left which is B.
One of the equal angles of an isosceles triangle is 50 degrees. What is the angle opposite the unequal side. A. 50 degrees B. 80 degrees C. 130 degrees D. 140 degrees
B. 4 Cancel out the Y's if you're solving for X. 1) 3x+ y = 13 x - 2y = 2 2(3x+y=13) 2) 6x+2y=26 x-2y=2 Combine and cancel 7x = 28, x = 4
Solve the following system of equations for x : 3x+ y = 13 x - 2y = 2 A. 18 B. 4 C. 3 D. 6
B. -4 We need to combine the system of equations. Since were looking for x lets get the Y's to cancel out. 10 x 3 is 30 6 x -5 is -30 1) 3 x ( 4x+10y=24) = 12x+30y=72 2) -5 x ( 10x+6y=-16) = -50x-30y=80 3) 12x+30y=72 -50x-30y=80 the Y's cancel out. 4) Combine like terms. -38x = 152 divide both sides x = -4
Solve the following system of equations for x: 4x+10y=24 10x+6y=-16 A. -6 B. -4 C. 2 D. 1
B. 8 (3/8) First focus on the fractions 1/8 - 3/4 - get common denominator 3/4 x 2 = 6/8 Now 6/8 is larger than 1/8 so we need to take 1 from the 24 to subtract these values. Now when we add a whole number to a fraction 1 + 1/8 = 1/1+1/8 - find common denominator. 8/8 + 1/8 = 9/8 Now we have 23 (9/8) - 15 (6/8) 23-15 = 8 9/8-6/8 = 3/8 8 (3/8)
Subtract 24 (1/8) - 15 (3/4) A. 7 (1/4) B. 8 (3/8) C. 9 (2/8) D. 10 (7/8)
C. 2x+4z 1) 5x+2y - (2y+3x-4z) 2) Distribute the negative 3) 5x +2y - 2y -3x +4z 4) combine like terms 5) 2x+4z which is C
Subtract 2y +3x -4z from 5x +2y A. 2x+4y+4 B. 7x+4 C. 2x+4z D. 2x+4y
(C) 2πr³ Volume of cylinder = πr²h h = 2r in this question Substitute 2r πr²2r Reorder 2πr³
The height, h, of a cylinder is twice the radius r. What is the volume of the cylinder? (A) πr³ (B) 4πr³ (C) 2πr³ (D) 8πr²
B. two One large 64 ounce paint can did 3/4 of the room. Means we need 1/4 left to complete it. 64/16 = 4 small size cans. Let x = small size cans. Let y equal the finished room. 4x=3/4y We need 1y the complete room so lets divide both sides by 3. 4/3x = 1y 4/3 = 1 (1/3). That is more than 1 some we need 2 more small paint cans to complete the room.
The paint that Amanda is using to paint Madalyn's room comes in two sizes, small (16 ounces) and large (one gallon/64 ounces). The only size paint can left at the hardware store is the small size. If she used up an entire large can painting 3/4 of Madlyn's room, how many cans does she need to buy to finish painting? A. one B. two C. three D. four
Answer B. The perimeter of a rectangle is the sum of its four sides. If x equals its width, then x +3 equals the length. (The length is 3 inches more than the width.) From this, you can write an equation to find the perimeter. Use the formula for perimeter. 2L+2W = P 1) x+x+(x+3)+(x+3) = 38 2) x+x+x+x+3+3 = 38 3) 4x+3+3 = 38 4) 4x+6 = 38 - solve for x 5) 4x = 38-6 6) 4x=32 7)x = 8 (inches) which is B.
The perimeter of a rectangle is 38 inches. If the length is 3 inches more than the width, find the width. A. 17 (1/2) inches B. 8 inches C. 11 inches D. 14 (1/2) inches
A. 15 Let x = length of shorter side; let 2x = length of longer. 2x + x + 2x + x = 90; 6x = 90 x = 15
The perimeter of a rectangle is 90. One side of the rectangle is twice the length of the other side. What is the length of the shorter side? A. 15 B. 20 C. 25 D. 30
B. -8 -8 x -8 = 64
The square root of 64 = A. 4 B. -8 C. 9 D. -9
B. 5:40 Just take 2/3 of 60 60 x 2 120/3 = 40 So from 5 it will be 5:40
What is the correct time if the hour hand is exactly 2/3 of the way between 5 and 6? A. 5:25 B. 5:40 C. 5:30 D. 5:45
B. 1/2 0.125 = 1/8 Remember cube root means what times itself 3 times will get you the number inside. Cube root of 1/8 is cube root of 1/ cube root of 8 Cube root of 1 = 1 Cube root of 8 = 2 x 2 x2 = 2 Answer 1/2
What is the cube root of 0.125? A. 5/2 B. 1/2 C. 1/4 D. 3/2
(D) 7 cm Because the triangle is a right triangle, you need the Pythagorean theorem You know the lengths of side a and the hypotenuse (c), so plug those values into the theorem and solve for b: 24² + b² = 25² 576 - b² = 625 b² = 49 b = 7 Use the positive answer because a length is never negative.
What is the length b in the right triangle? 25cm (c) 24cm (a) B = ? (A) 4 cm (B) 5 cm (C) 6 cm (D) 7 cm
B. 6x⁵y²-8x³y When we distribute. Do not confuse (4y²)² and (4y²)x² (4y²)² - Means were multiplying the exponents and squaring the base = 16y⁴ (4y²)y² - Means we treat it as a normal multiplication and add the exponents if they match. 4y² × y² = 4y⁴ Thus, 2x³y(3x²y-4) 2x³y ×3x²y-4 = 6x⁵y²-8x³y
What is the product of 2x³y and (3x²y-4)? A. 6x⁵y²-4 B. 6x⁵y²-8x³y C. 6x⁶y²-8x³y D.6x⁶y-8x³y
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?
D. 113.04 m³ Volume = 4/3π(r)³ 4/3π(3)³ 4/3π(27) 4(3.14)(27)/3 339.12/3 = 113.04 m³
What is the volume of a sphere that has a radius of 3 meters? A. 12.56 m³ B. 37.68 m³ C. 75.36 m³ D. 113.04 m³
B. 40% So what times 60 gives us 24? Let x equal the percentage we need. 60x = 24 divide both side by 60 24/60 = .4 40%
What percentage of 60 is 24? A. 5% B. 40% C. 400% D. 25%
C. 5√3 1) √12 = 4 x 3= √4 = 2 2 √3 2) √27 = 9 x 3 = √9 = 3 3√3 2√3 + 3√3 = 5√3 which is C.
What value is equal to √12 + √27 ? A. 3√3 B. 4√3 C. 5√3 D. 6√3
Use the formula Substitute 20 degrees 1) (9/5 x 20) + 32 2) 36+32 = 68 degrees
When the temperature is 20 degrees C, what is it on the Fahrenheit (F) scale? Use the following formula: F = (9/5 x C) + 32 A. 93 (3/5) degrees B. 78 degrees C. 62 (3/5) degrees D. 68 degrees
Answer D -1/2 is -0.5 not as small as -1 so this one is ruled out. -1 can be answer 0 are not choices at all. -7/6 is more than one. 7 is more than 6 is the number will be greater. Since its negative this one is the smallest.
Which has the smallest value A. -1/2 B. -1 C. 0 D. -7/6
D. Rhombus A quadrilateral has four sides, so choice D is the correct answer.
Which of the following is a quadrilateral? A. Triangle B. Cylinder C. Pentagon D. Rhombus
Answer C. A prime number is a number larger than 1 that has only itself and 1 as factors. (It can be evenly divided only by itself and by 1) 201 is divisible by 3 205 is divisible by 5 214 is divisible by 2 211 is only divisible by itself and 1 so this is the answer.
Which of the following is the smallest prime number greater than 200? A. 201 B. 205 C. 211 D. 214
A. 1 1) If the bases are equal. The exponents must also be equal. 2) 5 x 5 = 25 x5 = 125 x 5 =625 3) So that's 5⁴ 4) 5ⁿ⁺³ = 5⁴ 5) x = 1 so that we can get 5⁴.
Which one is the solution of the equation below? 5ⁿ⁺³ = 625 A. 1 B. 2 C. 3 D. 4
D. -2 1) First thing is FOIL the left side of the problem. 2) (-x-3)(-x-2) = x²+5x+6 = -4-2x 3) Make the problem equal to 0 so lets get everything on the left side. 4) add 4 to the left and add 2x to the left. 5) x²+7x+10=0 6) Factor. Find what numbers multiplied together get you 10 but when added get you 7. It will be 2 and 5 7) (x+2)(x+5)=0 8) Have each set equal to 0 9) x+2 = 0. x = -2 10) x+5=0, x = -5 11) X is either -2 or -5. Only answer with that is D.
Which one is the solution to the equation below? (-x-3)(-x-2)=-4-2x A. -1 B. 2 C. 3 D. -2 E. 5
C. Every square is a rhombus. A quadrilateral is a rhombus only if its four sides are congruent in length. A square has four congruent sides.
Which one of the following is correct? A. Every rhombus is a square. B. Every rectangle is a square. C. Every square is a rhombus. D. Every trapezoid is a rectangle.
C. 22 minutes You cannot simply add them or subtract the whole numbers. We must use their reciprocal form. 1 = the task T = together time 1/40 + 1/50 = 1/T 5/200 +4/200 = 1/T 9/200 = 1/t - cross multiply 9t = 200 t = 200/9 = 22.22
Working alone, Andrew can finish a certain task in 40 minutes. Carol can finish the same task in 50 minutes by herself. How long would it take Andrew and Carol to complete the task if they worked together? Round your answer to the nearest whole number. A. 24 minutes B. 25 minutes C. 22 minutes D. 21 minutes
D. 3√5 The goal of simplifying expressions with square roots is to factor the radicand (45) into a form that has no square factors. We can take out the square root of 9 in this case. √45 = √3² √5 √3² = 3 3√5
√45 A. 5√3 B. 9√5 C. 9√3 D. 3√5
B. 109° Angles that are a linear and/or are supplementary add up to 180°. So, 6x + 19 + 5x − 4 = 180 11x + 15 = 180 11x = 165 x = 15 Plug x back in to find m∠ 1. 6(15) + 19 = 109°.
∠ 1 and ∠ 2 form a linear pair and therefore are supplementary angles. If m∠ 1 = 6x + 19 and m∠ 2 = 5x - 4, then m∠ 1 = A. 71° B. 109° C. 45° D. 91°
D. 88º Remember supplementary means 180° 7x − 6 + 5x + 18 = 12x + 12 = 180° 12x = 180° −12 = 168° x = 14 5 × 14 + 18 = 70 + 18 = 88°
∠ 1 and ∠ 2 form a linear pair and therefore are supplementary angles. If m∠ 1 = 7x - 6 and m∠ 2 = 5x + 18, m∠ 2 = A. 78º B. 82º C. 85º D. 88º
Answer A. Simply make 2 (2/5) into improper fraction 2 (2/5) = 14/5 14/5 ÷ 7 - remember 7 is really 7/1 we can flip it and multiply. 14/5 x 1/7 = 3/5 answer A.
2 (4/5) ÷ 7 = A. 2/5 B. 9 (8/5) C. 5/2 D. 24/35
B. x⁶y⁹ 1) (x²y³)³ 2) We are simply multiplying the exponents. 3) x² = x⁶ 4) y³ = y⁹
(x²y³)³ = ? A. x⁵y⁶ B. x⁶y⁹ C. x⁷y⁸ D. x⁶y^10
C. y^2p 1) remember the rules with dividing quantities with exponents. 2) x^m/x^n = x ^m-n 3) y^p+q-(q-p) - remember that youa are subtracting from the quantity so we must distribute the negative first. 4) - (q-p) = -q+p 5) p+q-q+p = 2p 6) y^2p which is C.
(y^(p+q))/(y^(q-p)) = ? A. y^2q B. y^p+q C. y^2p D. y^p-q
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?
A. 10% First we need to know how many ounces of water and juice there are before the water is added. 1) 80% x 20 ounces = 16 ounces of water. 2) Which leaves 4 ounces of juice. 3) 20 ounces of water is added to the mixture 4) 20+16 = 36 ounces of water. With 4 ounces of fruit juice. 5) We need to find the percent of juice so get that number and put it over the total to get the percent. 6) Total = 20+16+4 = 40 7) 4/40 = .10 or 10% which is A.
20 ounces of a drink mixture contain 20% fruit juice and 80% water. It is further diluted by adding 20 ounces of additional water. What is the percent of fruit juice in the diluted mixture? A. 10% B. 12% C. 40% D. 8%
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? 2y^3 - 128
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.
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
Answer D. Simply put 45% over 100 and reduce 45/100 -reduce with 5 9/20 and it can't be reduced any further so the answer is D.
45% is equal to what fraction? A. 4/5 B. 5/8 C. 25/50 D. 9/20
A. 0.0315 Make the decimal whole by moving to the right 4 times which give us 63. (Remember to move back 4 times). Now it will be 63 x 5 = 315 Move back 4 times 0.0315
5 x 0.0063 A. 0.0315 B. 0.315 C. 3.15 D. 31.5
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
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?
Answer B. The wall, the ladder, and the ground in the tennis court forma right triangle. The ladder is on a slant, and is opposite the right angle formed by the wall and the ground. In this position, the ladder is the "hypotenuse" of the right triangle. In geometry, the Pythagorean Theorem states that the square of the hypotenuse (c^2) equals the sum of the squares of the other two sides (a^2 +b^2). 1) a^2+b^2 = c^2 2) 8^2 + x^2 = 10^2 Solve for x 3) 64 + x^2 = 100 4) x^2 = 100-64 5) x^2 = 36 -square root each side 6) x = 6 which is Answer B.
A 10-foot-high ladder is resting against an 8-foot-high wall surrounding a tennis court. If the top of the ladder is exactly even with the top of the wall, how far is the base of the ladder from the wall? A. 18 feet B. 6 feet C. 12 feet D. 9 feet
Answer D. To find the volume (V) of a cylinder, multiply pi times the square of the radius (r) times the height (h). V = pi x r^2 x h 1) V = 3.18 x 7^2 x 15 2) V = 3.18 x 49 x 15 3) V = 155.82 x 15 4) V = 2337 Now we divide this by 231 to get the gallons 2337/231 = 10 gallons which is D.
A cylindrical can has a radius of 7 inches and a height of 15 inches. How many gallons of milk can it hold? (There are 231 cubic inches in a gallon.) A. 15 gallons B. 14 gallons C. 140 gallons D. 10 gallons
Answer A. First change all measurement to yards. We know 3 feet = 1 yard since were dealing with cubic yards we cube the transfer 9 feet = 3 yards. 6 inches = 1/6 yards Since its 12 ft by 6 feet lets get the area. 1) 12 x 9 = 108 ft^2 Convert to yards 2) 108/9 = 12 yards Now with the inches we divide 6 inches = 1/6 yards 12 x 1/6 = 12/6 which is 2 cubic yards answer A.
How many cubic yards of concrete are needed to make a cement floor that is 9 feet by 12 feet by 6 inches thick? A. 2 cubic yards B. 18 cubic yards C. 54 cubic yards D. 648 cubic yards
Answer D. 40% = x/30 .4 = x/30 -isolate the x x = .4 x 30 4 x 30 = 120 go back 1 X = 12 which is answer D.
If 40 percent is equal to the fraction x/30, what is the value of X? A. 0.4 B. 15 C. 1,200 D. 12
Answer B. Well we need to know what times 28% will give us 42 pages. We can right the problem like this. 1) 28%x = 42, solve for x 2) x = 42/28% 3) x = 42/.28 - make whole 4) x = 4200/28 = 150
Katherine has a written 42 pages of her doctorate thesis. If she has written 28% of her doctorate thesis, how many pages will her finished thesis be? A. 70 pages B. 150 pages C. 162 pages D. 1,175 pages
Answer B When it says to 36th means the denominator will be 36. For 9 to get to 36 we need to multiply by 4. What we do to the bottom we do to the top. So 5x4 = 20 20/36 will be your answer. Which is B.
Raise 5/9 to 36ths. A. 18/36 B. 20/36 C. 24/36 D. 30/36
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?
Answer A. First find how mnay ounces of the original mixture were fruit juice. 1) 10 x 20% = 10 x .2 = 2 ounces Next find the total number of ounces in the new mixture 2) 10+40 = 50 ounces Then find what part of the new mixture is fruit juice, and convert it to a percentage. 3) 2/50 = 0.04 convert to percentage 4% which is A.
Ten ounces of liquid contain 20 percent fruit juice and 80 percent water. The mixture is diluted by adding 40 additional ounces of water. What is the percentage of fruit juice in the new solution? A. 4% B. 10% C. 20% D. 40%
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?
A. 6√3 1) We can simplify √12 to √ 2² x 3 2) Since 2² is by 3 we can rewrite this whole problem. 3(2√3) 3) Distribute the 3 and just multiply 3 to the 2. 4) We are left with 6√3 which is A.
What is another way to write 3√12 A. 6√3 B. 3√4 C. 12√3 D. 12
Answer B We know 4/8 reduced will be 1/2 or .5 1/4 is .25 Half of that 1/8 will be .125 Lets add both .5 and .125 giving us 0.625 which is B.
What is the decimal value of 5/8? A. 0.56 B. 0.625 C. 0.8 D. 0.835
B. (5,6) Midpoint formula: (x1+x2/2, y1+y2/2) 1) (2+8/2), (3+9/2) 2) 10/2 = x = 5, 12/2 = y= 6 3) (5,6) which is B.
What is the mid point of the joining line segment between P(2,3) and Q (8,9)? A. (3,7) B. (5,6) C. (2,8) D. (4,6)
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
Answer D. Simply make 13 (3/4) into an improper fraction. 13 (3/4) = 55/4 55/4 ÷ 5 = 55/4 x 1/5 = 11/4 11/4 is 2 (3/4) which is answer D.
A cement truck must distribute 13 (3/4) tons of cement evenly to five work sites. How many tons should it give to each work site? A. 2(1/4) B. 2 (1/2) C. 2(3/8) D. 2 (3/4)
B. 16.443 Remember when you're adding decimals to decimals or even whole numbers. Just align the decimals and do the arithmetic.
2.36 + 14 + 0.083 A. 14.059 B. 16.443 C. 16.69 D. 17.19
C. 23,900 Remember when you multiply by a multiple of 10 you move the decimal by however many zeros there are. In this case 10,000 has 4 zeros which means we move the decimal 4 times to the right which will give us 23,900 which is answer C.
2.39 x 10,000 A. 239 B. 2,390 C. 23,900 D. 239,000
D. 1.312 Remember you align the decimals. In this case we need to add extra zeros to the 1.5 to fill the problem so it will look like this. 1.500 0.188 - ------------ Solve Answer should be 1.312 which is D
1.5 - 0.188 A. 0.62 B. 1.262 C. 1.27 D. 1.312
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?
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?
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?
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?
Answer D. First find the area of the entire wildlife preserve. Since its a circle. Use the formula for the area of a circle. Area(A) = pi x r^2 1) 3.18 x 14^2 2) 3.18 x 196 3) 318 x 196 = 62,328 go back 2 = 628.28 square miles The lions territory is a wedge formed by a 90 degree angle at the center of the circle. Since a circle has 360 degrees, we can find the part of the preserve inhabited by lions. 4) 90/360 = 1/4 Next find what this equals in square miles 5) 1/4 x 628/1 = 628/4 = 157 (close to 154) answer D.
A wildlife preserve is laid out in the shape of a perfect circle whose radius is 14 miles. The lions' territory in this preserve is shaped like a wedge and has a fence around it. Two inner sides of a fence meet at a 90 degree angle in the center of the preserve. How much territory do the lions have? A. 140 square miles B. 3 (1/2) miles C. 210 square miles D. 154 square miles
Answer B. One way to solve this is to square each of the suggested answers to see which is close to 85. 9.1 x 9.1 = 82.81 9.2 x 9.2 = 84.64 - close 9.3 x 9.3 = 86.49 - too big. Answer has to be 9.2
Find the square root of 85 to the nearest tenth A. 9.1 B. 9.2 C. 9.3 D. 9.4
Answer A. Solve by doing each arithmetic operation and combining answers. Remember that the product of two negatives or two positives numbers is a positive number. The product of a negative and a positive is negative. 1) (-3)^4 = (-3) x (-3) x (-3) x (-3) = 81 2) (-2)^4 = (-2) x (-2) x (-2) x (-2) = 16 3) (-1)^4 = (-1) x (-1) x (-1) x (-1) = 1 81+16+1 = 98 which is A
Find the value of (-3)^4 + (-2)^4 + (-1)^4 A. 98 B. -98 C. -21 D. 21
D. 2.5 L Remember we are looking for 2% of some amount of water that will give us the solution we need. Hence why the 5% does not have the x. 1) Use the equation (5%) (1) = (2%) x or (0.05)(1) = (0.02) x 2) Solve for X 3) x = 2.5L which is D.
How much water must be added to 1 liter of a 5% saline solution to get a 2% saline solution? A. 1 L B. 1.5 L C. 2 L D. 2.5 L
A. 12x²+2xy-30y² Remember FOIL (First, outer, inner, last)
Simplify the following expression (3x+5y)(4x-6y) A. 12x²+2xy-30y² B. 12x²+38xy+30y² C. 6x²-14xy+8y² D. 6x²-30y²
Answer B. The product of all integers from 1 to x is called the x factorial. The product of all numbers from 1 to 5 is 5 factorial. Thus, 5 x 4 x 3 x 2 x 1 = 120 which is answer B.
The expression "5 factorial" equals A. 125 B. 120 C. 25 D. 10
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?
B: 1) move 7 to the right side. -4/5x < -32/5 2) Multiply both sides by -1, this will make them positive and switch the sign from < to > 4/5x > 32/5 3) Multiply both sides by 5 4x > 32 4) Divide both sides by 4 x > 8 Open circle so the line is going right to infinity. (8, infinity) which is B.
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.
B: 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 19.8, Divide both sides by 19.8 so k=1.36. Therefore, the formula for number of representatives given total population in millions is N=1.36 x 11.8 = 16.04 or just 16.
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? A. 12 B. 16 C. 15.8 D. 15 E. 13
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
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
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?
A. 36 | | = Means that anything inside will be positive. Ex. | -7 | = 7 | x | = 7 , x = 7 and x = -7 Thus 1) z-18 = 5 and z-18 = -5 "we must add the sum of all possible values" as stated in the question. 2) Solve for z for both problems. 3) x-18 = 5 = x = 23 4) z-18 = -5 = x = 13 5) 23 + 13 = 36 which is A.
What is the sum of all possible values of z in the following equation? |z-18| = 5 A. 36 B. 38 C. 40 D. 48
Answer B. Well 10% of 60 is just 6 is 12% would be a bit more than that and the only option that leaves is B. Long way would be to do 60/1 x 12/100 -reduce by 10 6/1 x 12/10 - reduce by 2 3/1 x 12/5 = 36/5 = 7.2 but do note this takes more time and we can't afford that in the test.
What is 12% of 60? A. 5 B. 7.2 C. 50 D. 72
C. 16.9 First make the mixed into an improper fraction. 5 1/5% = 26/5% Now in order to get form fraction to decimal we multiply by 1/100 26/5 x 1/100 = 26/500 26/500 = 0.052 - make whole = 52 three steps. Now compute. 325 x 52 = 16900 - go back 3 times. 16.9
What is 5 1/5% of 325? A. 1.69 B. 169 C. 16.9 D. 32.5
Answer C. Remember FOIL First, outer, inner, Last (a+b) x (c+d) = ac+ad+bc+bd It should all come out to get C.
What is the product of (a-5) and (a+3) A. a^2-15 B. a^2 +2a -15 C. a^2 -2a-15 D. a^2 -2
C. 5x^2+7x-8 Remember it says subtracting from the whole. Thus, 1) 8x^2 +2x -9 - (3x^2 - 5x -1) 2) Distribute the negative 3) 8x^2 +2x -9 - 3x^2 + 5x + 1 -- combine like terms 4) C. 5x^2+7x-8
What is the result of subtracting 3x^2 - 5x -1 from 8x^2 +2x -9 A. 5x^2-3x-10 B. -5x-3x-10 C. 5x^2+7x-8 D. -5x^2-7x+8
C. -3 Slope intercept form: y=mx+b m = -3
What is the slope of the line y = -3x+1? A. 8 B. 3 C. -3 D. 8
A: 8 Exponent rule: a^b ×a^c = a ^ b+c The 16 and 16 combine and now were left with just adding the exponents. 1) 1/4+2/4= 3/4 2) 16^3/4 3) Factor 16 to 2^4 4) (2^4)^3/4 5) 4 x 3/4 = 3 6) 2^3 = 8
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?
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?
