KPLGRE MATH
Since proper fractions must have an integer in the denominator, how do you convert a fraction with a square root in the denominator to a fraction with an integer in the denominator?
Multiply the numerator and denominator by whatever square root is in the denominator. 6/√(5) = (6/√(5))(√(5)/√(5)) = (6 × √(5)) / (√(5) × √(5)) = 6√(5) / 5.
The sqaure root of the term A^2 equals
On the GRE, the square root sign always designates the positive square root. Therefore the sqare root of A^2 when A is positive, and Sqrt of A^2 is negative A, when A is negative
How do you multiply square roots?
The product of more than one square root is equal to the square root of their total product. For example: √(3)√(5) = √(3 × 15)
How do you divide square roots?
The quotient square roots is equal to the square root of the quotient. For example: √(6) ÷ √(3) = √(6 ÷ 3) = √(2).
Simplify 5.6 x 10^6
When 10 is raised to a positive integer exponent, that exponent tells how many zeroes the number would contain if written out. Ex: 5.6 x 10^6 = 5,600,000 or 5.6 million.
In a word problem, what are some keywords and phrases that indicate the need for an addition symbol (+)?
add, sum, increased by, more than, plus, exceeds, combined
How do you change a fraction to its percent equivalent?
Multiply by 100%
What is the speed formula?
Speed = Distance / Time
what is a coeficcient?
The numerical constant by which one or more variables is multiplied
Simplify ((3^2)^4)
((3^2)^4) = (3 x3)(3x3)(3x3)(3x3) = 3^(2*4) = 3^8
What is the greatest factor of 16 that is also a factor of 80?
16 The greatest factor of any number is itself. Because 16 is also a factor of 80, 16 is the greatest common factor of both numbers.
If x - 15 = 2y, and 6y + 2x = -10, whats the value of y?
6y+2(2y +15) = -10 6y +4y + 30 = -10 10y = -40 y = -4
How far do you drive if you travel for 5 hours at 60 miles per hour?
Distance = (Speed)(Time) Distance = (60 mph)(5 hours) Distance = 300 miles
How do you change a percent into its fractional equivalent?
Divide by 100%
What is division by zero?
Division by zero is undefined. For test purposes that means "it cant't be done)
What is 12 percent of 25?
Find the part: (12/100) × 25 = 3
In a word problem, what are some keywords and phrases indicating a multiplication symbol?
Multiple, product, multiplied by, of, times, twice, double
What is the order of operations represented by PEMDAS?
Parnthases, Exponents, Multiplication, Division, Addition
What is the only operation that can produce a noninteger from two integers?
The only operation that can produce a noninteger from two integers is division. When two integers are multiplied, added, or subtracted, the result will always be an integer.
How long does it take to drive 520 miles at 40 miles per hour?
Time = Distance / Speed Time = 520 / 40 Time = 13 hours
The ratio of boys to girls is 3 to 4. If there are 135 boys, how many girls are there?
When using a ratio to determine an actual number, set up a proportion, with the number of girls represented by x: (3/4) = (135/x). Cross multiply to obtain: 3x = 4(135). Solve for x, x=180.
solve for x = x^2 +6 = 5x
x^2 +6 = 5x X^2 - 5x + 6 = 0 (x-2)(x-3) = 0 x-2 = 0, x-3 = 0 x = 2 , 3
Factor x^2 + 6x + 9
x^2 +6x + 9 = (x +3)(x+3) = (x+3)^2
Change 1/2 into an equivalent fraction with a denominator of 4
(1 *2)(2*2) = 2/4
True or False: (1) You can determine a ratio from two quantities. (2) You need at least one quantity in a ratio to determine the other quantities.
(1) True. Any two quantities form a ratio. (2) True. A ratio can have any number of numerical values. You need at least one value, along with the ratio itself, to determine the other values.
Simplify 4^5 / 4^2=
(4)(4)(4)(4)(4) / (4)(4) = 4^(5-2) = 4^3
Translate one quarter, one fifth and one eights to their decimal equivalents
1/4 = 0.25 1/5 = 0.20 1/6 = 0.667 1/9 = 0.125
Four raised to the power of one half equals
1: Translate into math 4^(1/2) = Sqrt(4) = 2
What is the smallest prime number?
2. A prime number is a positive integer with exactly two factors: 1 and itself. The smallest such number is the number 2.
Factor the expression 2a + 6ac
2a + 6ac = 2a(1 + 3c)
3^0
3^0 = 1 Any non-zero number raised to the exponent 0 = 1 0^0 us undefined
Simplify 4^5 x 4^2 =
4^5 x 4^2 = (4)(4)(4)(4)(4) x (4)(4) = 4^(5+2) = 4^7
If 4x - 7 = 2x + 5, what is the value of x?
4x-7 = 2x + 5 4x = 2x + 12 2x = 12 x = 6
If a right triangle has legs (short sides) of lengths 3 and 4, what will the length of the longest side, or hypotenuse, be?
5. This is a 3:4:5 triangle, a ratio of side lengths that satisfies the Pythagorean theorem. Such ratios are called Pythagorean triples. These special right triangles occur frequently on the test.
Simplify (5 + 2) / (7 - 3)
7/4, or 1.75
8! / (6! × 2!) =
8! / (6! × 2!) = (8 × 7 × 6!) / (6! × 2!) = (8 × 7) / 2 = 28 Note that most factorial problems can be handled more efficiently by factoring and canceling in this way, rather than computing the entire factorial.
What is the ratio of the lengths of the sides of a 30°-60°-90° triangle?
A 30°-60°-90° triangle has sides in the ratio of x : √(3)x : 2x . This type of special right triangle occurs frequently on the test.
What does a combination question ask for?
A combination question asks you how many unordered subgroups can be formed from a larger group.
What is a cube?
A cube is a special rectangular solid in which all edges are of equal length, and therefore all faces are squares. Sugar cubes and dice without rounded corners are real-life examples of cubes.
What are the characteristics of a cylinder?
A cylinder is a uniform solid whose horizontal cross section is a circle—for example, a soup can, or a pipe that is closed at both ends. A cylinder's measurements are generally given in terms of its radius, r, and its height, h.
What is a factorial?
A factorial is a mathematical operation designated by an exclamation point (!). If n is an integer greater than 1, then n factorial, denoted by n!, is defined as the product of all the integers from 1 to n. For example: 2! = 2 × 1 = 2 3! = 3 × 2 × 1 = 6 4! = 4 × 3 × 2 × 1 = 24 By definition, 0! = 1! = 1.
What is the prime factorization of 210?
A good way to factor any number is to use divisibility rules. 210 is even, so it is divisible by two. Therefore 210 = (2)(105). The factor 105 ends in 5, so it is divisible by 5. Thus 210 = (2)(5)(21). The factor 21 is divisible by 3, and so you get 210 = (2)(5)(3)(7). Now, these numbers are all prime, so no further factorization can occur! Note that it is conventional to order the primes, so formally, the answer is: 210 = (2)(3)(5)(7).
What is an angle bisector?
A line or line segment bisects an angle if it splits the angle into two smaller, equal angles. Line segment BD below bisects ∠ABC, so ∠ABD has the same measure as ∠DBC. Angle ABC is bisected by BD. Angles ABD and DBC both measure x degrees.
What is a multiple?
A multiple is the product of a specified number and an integer. For example, 3, 12, and 90 are all multiples of 3: 3 = (3)(1); 12 = (3)(4); and 90 = (3)(30). The number 4, for example, is not a multiple of 3 because there is no integer that can be multiplied by 3 and yield 4.
How can you determine whether a number is divisible by 2? 3? 4? 5? 6? 9? 10?
A number is divisible by... 2 if the number is even; 3 if the sum of the digits is divisible by 3; 4 if the two-digit number formed by the last two digits is divisible by 4; 5 if the last digit is 0 or 5; 6 if the number is divisible by both 2 and 3; 9 if the sum of the digits is divisible by 9; 10 if the last digit is zero.
Simplify 5^-3
A number raised to the exponent -x is the reciprocal of that number raised to the exponent. Ex: 5^-3 = 1 / 5^3 = 1/(5*5*5) = 1/125
What does it mean for a polygon to be inscribed in a circle and for a polygon to be circumscribed about a circle?
A polygon is inscribed in a circle if all the vertices of the polygon lie on the circle. A polygon is circumscribed about a circle if all the sides of the polygon are tangent to the circle. If Square ABCD is inscribed in circle O, you can also say that circle O is circumscribed about square ABCD. If square PQRS is circumscribed about circle O, you can also say that circle O is inscribed in square PQRS.
What is a rectangular solid?
A rectangular solid is a solid with six rectangular faces. All edges meet at right angles. Real-life examples of rectangular solids are cereal boxes, bricks, etc.
What is the surface area of a rectangular solid with the dimensions 4 × 3 × 2?
A uniform rectangular solid has three pairs of equal, rectangular sides. The total surface area is the sum of the areas of these six sides. Therefore, the total surface area would be: 2(l⋅w + l⋅h + w⋅h) = 2(4⋅3 + 4⋅2 + 3⋅2) = 2(12 + 8 + 6) =2(26) = 52
What is a uniform solid, and what is the formula for its volume?
A uniform solid is a solid with the same dimensions throughout the entire object. The overwhelming majority of solids on this test are uniform. Volume = (Area of base)(Height) Be prepared to use the formulas for triangles, circles, and rectangles to calculate the area of the base of a solid on this test.
How many solutions do absolute value problems normally have?
Absolute value problems normally have two solutions, depending on whether the value inside the absolute value signs is positive or negative.
What does absolute value represent on a number line?
Absolute value represents the distance of a value from zero on the number line.
What do you call an arc that is exactly half the circumference of its circle?
An arc that is exactly half the circumference of its circle is called a semicircle.
What is an integer?
An integer is a positive whole number, a negative whole number, or zero.
Which of the following is not an integer: √ ̅1, √ ̅4, √ ̅7, or √ ̅9?
An integer is defined as a whole number (without decimal or fraction parts). Of the listed radicals, the only number that cannot be simplified to a whole number is the square root of 7.
What is a cubed number?
Any number raised to the third power. For example: 4^3 = (4)(4)(4) = 64. 64 is the cube of 4
In a triangle, the base has length 4 and the height has length 3. What is the area of the triangle, in square units?
Area = ½ × base × height = ½ × 4 × 3 = 6
What is the formula for the area of a triangle?
Area = ½ × base × height.
What is the area of a circle with a radius of 4?
Area = πr², so Area = π(4)² = 16π
What formula is used to find an average (arithmetic mean)?
Average = (Sum of terms) / (Number of terms)
What is the average speed formula?
Average Speed = (Total Distance) / (Total time)
What does the acronym BDSS mean?
BDSS is an easy way to remember the Triangle Inequality Theorem: Each side of a triangle is Bigger than the Difference (of the other side lengths) and Smaller than the Sum (of the other side lengths).
Find the perimeter of square EFGH where side EH has a length of 3
Because all four sides of a square are the same length, Perimeter = 4 × (side). If the length of one side of a square is 3, the perimeter is 4 × 3 = 12.
Triangle ABC is inscribed in circle O, with segment BC passing through the circle's center at point O. If AB = AC = 3, what is the circumference of circle O?
Because the arc of length 3 pi corresponds to a 60° inscribed angle, you know that 3π is one-sixth of the total circumference (as 60° is one-sixth of the total 360° in the circle). This gives you a circumference of 18π Since the circumference formula is 2πr, a circumference of 18 tells you that the radius is 9. Using the formula for the area of a circle (πr²), you get an area of 81π.
A circle has an inscribed angle of 60°, with a corresponding arc of length 3π What is the area of the circle?
Because the arc of length 3π corresponds to a 60° inscribed angle, you know that 3π is one-sixth of the total circumference (as 60° is one-sixth of the total 360° in the circle). This gives you a circumference of 18π. Since the circumference formula is 2πr, a circumference of 18 tells you that the radius is 9. Using the formula for the area of a circle (Area = πr²), you get an area of 81π.
A customer wants to buy three different fruits. If there are seven different fruits available at the market, how many different ways can he choose his fruit?
Because the order of the fruits chosen does not matter, this is a combination question. The combinations formula is: n! / (k!(n - k)!), where n = the number of fruits available, and k is the number of fruits being picked. In this case, n = 7 and k = 3. 7! / (3! (7 - 3)!) = (7 × 6 × 5 × 4!) / (3! 4!) = (7 × 6 × 5) / (3 × 2 × 1) = 7 × 5 = 35 There are 35 ways to choose 3 different fruits from 7 options.
Restaurant A has 5 appetizers, 20 main courses, and 4 desserts. If a meal consists of 1 appetizer, 1 main course, and 1 dessert, how many different meals can be ordered at Restaurant A?
Because you're only choosing one of each item, the number of possible outcomes from each set is the number of items in that set. There are 5 possible appetizers, 20 possible main courses, and 4 possible desserts. The number of different meals that can be ordered is (5)(20)(4) = 400.
What are consecutive numbers?
Consecutive numbers are numbers of a certain type, following one another without interruption. Numbers may be consecutive in ascending or descending order. This test prefers to test consecutive integers (e.g. -2, -1, 0, 1, 2, 3, ...), but you may encounter other types of consecutive numbers. For example, -4, -2, 0, 2, 4, 6, ... is a series of consecutive even numbers. The sequence -3, 0, 3, 6, 9, ... is a series of consecutive multiples of 3. The sequence 2, 3, 5, 7, 11, ... is a sequence of consecutive prime numbers. Most consecutive sequences contain evenly spaced terms, although certain sequences, such as consecutive primes or consecutive perfect squares, are not evenly spaced.
|x - 3| = 17 What are the possible values of x?
Consider both the positive and negative interpretations of what is contained within the absolute value signs. Here, you have that |x - 3| = 17. This could mean either x - 3 = 17 or x - 3 = -17. Dolve both equations as normal. If x - 3 = 17, then x = 20. If x - 3 = -17, then x = -14. Thus, the two possible values for x are 20 and -14.
What is coordinate geometry?
Coordinate geometry is a system in which the location of a point in a plane is shown by an x-coordinate for the horizontal position and a y-coordinate for the vertical position. A point is described by its x- and y-coordinates: (x, y).
A rectangular solid has rectangle ABCD as its top face and rectangle EFGH Vertex A is directly above vertex E. Vertex B is directly above vertex F. Vertex C is directly above vertex G. Vertex D is directly above vertex H. If edge EH has length 3, edge HG has length 4, and edge CG has length 5, what is the length of a line connecting vertices A and G?
Diagonal AC is the hypotenuse of triangle ABC; this a 3:4:5 triangle, so AC = 5. Now look at triangle ACG: ACG happens to be another special triangle, so you don't need to use the Pythagorean theorem. Because the lengths of both legs of the right triangle are equal, ACG is a 45°-45°-90° triangle, so AG = 5 times the square root of 2.
In a word problem, what are some keywords and phrases indicating a divide symbol?
Divide, quotient, divided by, unto, per, ratio, out of
What are equilateral triangles?
Equilateral triangles are triangles with three equal sides, which means they also have three equal 60° angles.
What formula is used in all probability questions?
Every probability question involves the formula: Probability = (Desired outcomes)/(Total possible outcomes)
True or False: 1 is a prime number.
False. A prime number is divisible only by "1 and itself". Put another way, a prime number needs to be divisible by exactly two numbers, by the number "1" and by the number "itself". For example, the prime 7 is divisible by 1, since 7 ÷ 1 = 7. And 7 is divisible by itself, since 7 ÷ 7 = 1. But 7 has no other factors (i.e. no other number divides 7), so 7 is prime. However, in the case of the number 1, "1" and "itself" are the same number! Thus, the number 1 is not divisible by exactly two numbers, so 1 is not prime. It's also worth remembering that the smallest prime is 2.
True or False: An easy shortcut to average-speed problems is to simply take the average of the given speeds.
False. Averaging the given speeds is, in fact, a common wrong-answer trap; test takers who make this error are failing to account for the different amounts of time spent at each speed.
True or False: Oddly shaped figures on the test often require obscure formulas to solve.
False. Oddly shaped figures on the test can almost always be divided into smaller, easily recognizable figures—most often right triangles and rectangles.
How many distinct three-digit numbers can be formed with the digits 1, 3, and 5 using each only once?
For a permutation question in which all items in the group are used in each arrangement (i.e., there are no subgroups), use the arrangement formula: n! In this case, there are three distinct digits, so n = 3. The number of arrangements is 3! = 3 × 2 × 1 = 6.
What is the combined work formula for three or more workers?
For three or more workers, the combined work formula is: 1/T = 1/A + 1/B + ... + 1/n. T is the total time it will take all the workers working together to complete one task, and A, B, ... n are the amounts of time it takes each of the individual workers to complete one task working alone. Note that solving this formula results in 1/T, the reciprocal of the total time.
How do you find the volume of a uniform rectangular solid?
How do you find the volume of a uniform rectangular solid?
If the coordinates of point A are (3, 4), and the coordinates of point B are (6, 8), what is the distance between points A and B?
If two points have different x-coordinates and different y-coordinates, draw a right triangle, calculate the leg lengths, and use the Pythagorean theorem to find the distance between the two points. Points A (3, 4) and B (6, 8) are graphed in the xy-coordinate plane. Point C has the same x-coordinate as point B and the same y-coordinate as Point A. The three points are connected by line segments to create a triangle where angle ACB is a right angle. So AC = 6 - 3 = 3, and BC = 8 - 4 = 4. If you recognize these as the legs of a 3:4:5 right triangle, you'll know immediately that the distance between points A and B must be 5.
Do x and y have identical or different signs if xy > 0? If xy < 0?
If xy > 0, then x and y have identical signs. If xy < 0, then x and y have different signs.
What is the formula for permutations?
If you're asked to find the number of ways to arrange a smaller group that's being drawn from a larger group, use the permutation formula: n! / (n - k)! where n is the number of items in the group as a whole and k is the number of items in each subgroup formed.
The average annual rainfall in Boynton for 1976-1979 was 26 inches per year. Boynton received 24 inches of rain in 1976, 30 inches in 1977, and 19 inches in 1978. How many inches of rainfall did Boynton receive in 1979?
If you're given the average, the total number of terms, and all but one of the actual numbers, you can find the missing number. You know that total rainfall equals 24 + 30 + 19 + (inches of rain in 1979). You know that the average rainfall was 26 inches per year for 4 years. So, plug these numbers into the average formula to find that. Here, let x be the number of inches in 1979: Sum of terms = (Average)(Number of terms) 24 + 30 + 19 + x = (26)(4) 73 + x = 104 x = 31
The ratio of domestic sales revenue to foreign sales revenue of a certain product is 3:5. What fraction of the total sales revenue comes from domestic sales?
In a part-to-part ratio, the total number of parts in the whole is determined by adding all the parts together. In this example, the parts are domestic sales revenue and foreign sales revenue; the ratio of 3:5 means there are 3 + 5 = 8 parts in the total revenue. The fraction of the total sales revenue that comes from domestic revenue is therefore 3 of the 8 parts, or three eighths of the total revenue.
In the formula y = mx + b, what do m and b represent? What does this entire equation represent?
In coordinate geometry, the slope-intercept equation of a line is expressed using the formula y = mx + b, where m is the slope of the line and b is the y-intercept, which is the point where the line crosses the y-axis.
At what angle does a line tangent to a circle meet that circle's radius at the point of tangency?
Lines tangent to circles and those circles' radii meet at a right angle. A line that is tangent to a circle meets the circle at exactly one point, called the point of tangency. If you draw a radius from the center of the circle to the point of tangency, the radius will always meet the tangent line at a 90° angle. Another way of saying this is that the tangent line and the radius will always be perpendicular.
Does multiplying or dividing by 1 change the number?
No
The total weight of the licorice sticks in a jar is 30 ounces. If the average weight of each licorice stick is 2 ounces, how many licorice sticks are there in the jar?
Number of terms = Sum of terms / Average = 30 / 2 = 15
How can you rapidly eliminate answer choices on average speed questions?
On average speed questions, you can rapidly eliminate answer choices by recognizing which way the speeds are weighted. If a problem involves more time spent at a fast speed than at a slower speed, the average speed will be higher than the average of the two speeds, and vice versa.
What is the formula for percent change?
Percent change = ((New value - original value) / original value) × 100% In the case of a percent increase, the percent change from the original value to the new value will always be exactly 100% less than the new value as a percentage of the original value. On the test, beware of answers exactly 100% apart.
What are the median, mode, and range of the following set of numbers? {1, 3, 1, 17, -4}
Put the numbers in order: -4, 1, 1, 3, 17. The median is the middle of the list, 1. The mode is the number that shows up the most, 1. The range is the biggest number minus the smallest number: 17 - (-4) = 21.
If 350 widgets cost $20, how much will 1,400 widgets cost at the same rate?
Set up a proportion using the fraction Number of Widgets over Cost: 350 widgets / $20 = 1400 / x Solving algebraically, you get 350x = (1,400)(20), which simplifies to: x = (1400)(20) / 350 = (140)(20) / 35 = 4(20) = $80 So, 1,400 widgets will cost $80 at the given rate.
How do you find the volume of a cube? How do you find the surface area of a cube?
Since a cube is a rectangular solid for which l = w = h, the formula for its volume can be stated in terms of any edge: Volume of a cube = l⋅w⋅h = (edge)(edge)(edge) = e³ Surface area of a cube = sum of areas of faces = 6e² where e is the length of any edge.
The average individual test score for a group of girls is 30. The average individual test score for a group of boys is 24. If there are twice as many boys as girls, what is the overall average?
Since we're only given a ratio of boys and girls, the easiest thing to do is to assume there are exactly three students: 2 boys and 1 girl. Average = (1 × 30 + 2 × 24) / 3 = 78 / 3 = 26
What is the combination formula?
Some combination questions use numbers that make quick, noncomputational solving difficult. In these cases, use the combination formula: n! / (k!(n - k)!), where n is the number of items in the group as a whole and k is the number of items in each subgroup formed.
What is a standard deviation?
Standard deviation is a measure of the dispersion of a group of numbers. The standard deviation is found using a formula based on the distances of the numbers from the average.
How is standard deviation tested on the exam?
Standard deviation measures the dispersal, or spread, of numbers around the average. You will almost never be required to calculate standard deviation on this test, but you do need to have a strong conceptual understanding of how standard deviation works. To calculate the standard deviation of a set of numbers: Find the mean. Find the differences between the mean and each of the numbers. Square each of the differences. Find the average of the squared differences. Take the square root of the average.
What is the least common multiple of 6 and 8?
Start by finding the prime factors of 6 and 8: 6 = (2)(3) 8 = (2)(2)(2) You see 2 appears as a factor three times in the prime factorization of 8, while 3 appears as only a single factor of 6. So the least common multiple of 6 and 8 will be (2)(2)(2)(3), or 24. Note that the least common multiple of two integers is smaller than their product if they have any factors in common. For instance, the product of 6 and 8 is 48, but their least common multiple is only 24.
June pays an average price of $14.50 for six articles of clothing. What is the total price of all six articles?
Sum of terms = (Average)(Number of terms) = $14.50(6) = $87
If a triangle has all three sides inscribed in a circle, and one of its sides is the diameter of the circle, what is the angle opposite the diameter?
The angle opposite the diameter will be a right angle. If line AB is a diameter of circle O, and AB is also the hypotenuse of right triangle ABC, where C is on the circumference of the circle, then angle ACB is a right angle.
What is the sum of the integers 105 through 225 inclusive?
The average of the smallest and largest terms in this set of consecutive integers is (105 + 225) ÷ 2 = 330 ÷ 2 = 165 The number of terms in this set is 225 - 105 + 1 = 121. The sum of the integers is thus 165 × 121, or 19,965.
What is the combined work formula for two workers?
The combined work formula is T = AB / (A+B). T is the total time it will take to complete one task if the two workers work together, and A and B are the amounts of time it takes each worker to complete the task alone.
In a circle with center O, points A, B, and C lie on the circumference. Radii of length 6 connect the center O to points A and C. Angle AOC is equal to 60 degrees and it subtends the minor arc ABC. In circle O above, what is the length of arc ABC?
The formula for circumference is C = 2πr; therefore, if r = 6, C = 2 × π × 6 = 12π. Since angle AOC measures 60°, arc ABC is 60°/360°, or one sixth of the circumference. Thus, the length of arc ABC is (1/6) × 12π, or 2π.
What are the formulas for percent increase and percent decrease?
The formulas for percent increase and percent decrease are alternative ways of understanding and applying the percent change formula: Percent increase = (Amount of increase)/(Original whole) × 100% Percent decrease = (Amount of decrease)/(Original whole) × 100%
What are the mean, median, and mode of the following set of numbers? {5, 8, 23, 47, 47, 62}
The mean of a set is the set's average. This set's mean is (5 + 8 + 23 + 47 + 47 + 62) ÷ 6 = 192 ÷ 6 = 32. The median of a set is the set's middle number when arranged in ascending or descending order. When a set has an even number of integers, the median is the average of the two middle numbers. This set's median is (23 + 47) ÷ 2 = 35 The mode of a set is the most frequently occurring number of the set. This set's mode is 47.
Define median, mode, and range.
The median is the middle number, or the average of the two middle numbers, of a list of numbers arranged in ascending or descending order. The mode is the number that appears most often on the list. The range is the positive difference between the largest number and smallest number on the list.
The price of an antique is reduced by 20 percent, and then this price is reduced by 10 percent. If the antique originally cost $200, what is its final price?
The most common mistake in this kind of problem is to reduce the original price by a total of 30%. This mistake occurs if you take the 20% price reduction and then add the 10% price reduction, i.e. if you assume 20% + 10% = 30%. This is not correct because the first price reduction of 20% changes the original price, then the second price reduction of 10% is applied to the NEW price! To get the correct answer, first find the new price after the 20% price reduction. You can find it by calculating either [$200 - (20% of $200)] = $160 or [80% ($200)] = $160. That new price then has to be reduced by 10% more. So calculate either [$160 - (10% of $160)] = $144 or [90% ($160)] = $144.
What are the different ways in which probability can be expressed?
The probability formula is expressed as a ratio of the number of desired outcomes to the total number of possible outcomes. So because probability is technically a ratio, it is usually expressed as a fraction, for example, "The probability of event A occurring is one-fourth." But like all fractions, this means probability can also be expressed as a decimal (e.g. 0.25) or as a percentage (e.g. 25%) Finally, probability can also be expressed in words, for example, "the probability of event A occurring is 1 in 4."
What are the maximum and minimum probabilities of a given event occurring?
The probability of any event occurring cannot exceed 1 (a probability of 1 represents a 100% chance of an event occurring) and cannot be less than 0 (a probability of 0 represents a 0% chance of an event occurring).
If 2 students are chosen at random, without replacement, from a class with 5 girls and 5 boys, what is the probability that both students chosen will be girls?
The probability that the first student chosen will be a girl is 5/10 = 1/2. After choosing a girl, there will be 4 girls and 5 boys remaining (9 total students). So, the probability that the second student chosen will be a girl is 4/9. Thus the probability that both students chosen will be girls is: (1/2)(4/9) = (1 × 4)/(2 × 9) = 4/18 = 2/9.
If the ratio of sparrows to the total number of birds in an aviary is 3:7, and there are 12 sparrows in the aviary, how many total birds does the aviary contain, not including sparrows?
The ratio of sparrows to total birds is 3:7, and there are 12 sparrows. Since 3 × 4 = 12, this means that each number in the ratio is multiplied by 4 to yield the actual number of birds. Therefore, there are 7 × 4 = 28 total birds. Subtract the 12 sparrows from the 28 total birds to find that there are a total of 16 birds in the aviary that aren't sparrows.
What is the slope of a line?
The slope of a line is the measure of a line's incline on the coordinate plane. It is measured by Δy/Δx where Δ means "change" or difference. Slope can also be expressed as rise over run using the equation slope = (y₂ - y₁) / (x₂ - x₁).
What is the sum of the differences of each term on a list from the average of that list?
The sum of the differences of each term on a list from the average of that list is 0.
What is the probability of throwing a 2, a 5, and a 3, in no particular order, on a fair six-sided die with the numbers 1 through 6 on its faces?
There are 3 dependent trials, each with 6 possible outcomes—one for each side of the die. The probability of the first trial is 3/6. (3 desired outcomes—throwing either a 2, 5, or 3— out of 6 possible outcomes). The probability of the second trial is 2/6 (the 2 remaining desired outcomes out of 6 possible outcomes). The probability of the third trial is 1/6, as only one desired outcome remains. Thus, the probability of the desired outcomes occurring in no particular order is (3/6)(2/6)(1/6) = (3 × 2 × 1)(6 × 6 × 6) = 6/216 = 1/36.
What is the probability of throwing a 2, a 5, and a 3, in that order, on a fair six-sided die with the numbers 1 through 6 on its faces?
There are 3 independent trials with 6 possible outcomes—one for each side of the die. The probability of throwing any one particular number is 1/6. Thus, the probability of the desired outcome on three consecutive trials is: (1/6)(1/6)(1/6) = (1 × 1 × 1)/(6 × 6 × 6) = 1/216.
A bucket has 50 red marbles, 25 green marbles, 15 blue marbles, and 10 orange marbles. If a marble is picked at random, what is the probability of picking a marble that is neither red nor orange?
There are a total of 50 + 25 + 15 + 10 = 100 marbles in the bucket. Red and orange marbles make up 60 of these, so the remaining marbles would amount to 40. Probability = (Desired outcomes) / (Total possible outcomes) = 40 / 100 = 2/5
What is the probability of throwing a five on a fair six-sided die?
There is 1 favorable outcome—throwing a 5. There are 6 possible outcomes—one for each side of the die. So the probability is 1/6 or, in words, the probability is one in six.
What are the factors of 36?
Thirty-six has nine distinct factors: 1, 2, 3, 4, 6, 9, 12, 18, and 36. You can group these factors in pairs: (1)(36) = (2)(18) = (3)(12) = (4)(9) = (6)(6)
A price rises by 10 percent one year and by 20 percent the next. What's the total percent increase?
This is a percent problems with an unspecified starting value. (I.e. the problem doesn't tell us what the original price is!) When faced with such a problem, choose 100 as the starting value. Here, for example, assume the starting price is $100. Year one: $100 + (10% of $100) = $100 + $10 = $110 Year two: $110 + (20% of $110) = $110 + $22 = $132 From $100 to $132—that's a 32 percent increase.
Ben spends $30 of his annual gardening budget on seed. If his total annual gardening budget is $150, what percentage of his budget does he spend on seed?
This problem specifies the whole ($150) and the part ($30) and asks for the percentage. Plugging those numbers into the percent formula gives you: Percent = (30/150)(100%) = (1/5)(100%) = 20% Ben spends 20% of his annual gardening budget on seed.
How do you find the least common multiple (LCM) of two or more numbers?
To find the LCM of two or more numbers, take the prime factorization of each number, then find the product of each prime factor raised to the highest power in each of the numbers. For 60 and 96: 60 = 2² × 3 × 5 96 = 2⁵ × 3 LCM = 2⁵ × 3 × 5 = 480
A bus travels from Station B to Station A at 50 mph, and returns from Station A to Station B at 30 mph along the same path. What is the average speed of the bus in mph?
To find the average speed, you cannot simply average the two speeds; rather, you must calculate the weighted average of the two legs of the journey. To do this you must find the total distance traveled, and divide that by the total time traveled. You are not given a distance here, so your best approach is to pick a number for the distance. Use 150 miles because 150 is a multiple of both 30 and 50. This would give you a total distance traveled of 300 miles (150 miles in each direction). The time from B to A would be 150 miles divided by 50 mph, or 3 hours. The time from A to B would be 150 miles divided by 30 mph, or 5 hours. This gives you a total time of 8 hours. The average speed of the bus would then be 300 miles divided by 8 hours, or 37.5 mph.
Two years ago, 450 seniors graduated from Inman High School. Last year, 600 seniors graduated. By what percentage did the number of graduating seniors increase?
To find the difference, just subtract the original from the new. Note that the "original" is the base from which change occurs. It may or may not be the first number mentioned in the problem. The original is the figure from the earlier time (two years ago): 450. The increase is 600 - 450, or 150. So the percentage increase is (150/450)(100%) = 33 ⅓ percent.
If the price of a $120 dress is increased by 25 percent, what is the new selling price?
To find the new whole, you'll first have to find the amount of increase. The original whole is $120, and the percent increase is 25%. Let x be the amount of increase. Then: (x/120)(100%) = 25%. (x/120) = (25/100) (x/120) = (1/4) x = 30 The amount of increase is $30, so the new selling price is $120 + $30, or $150.
How do you determine the probability of something not happening?
To find the probability that an event does not happen, subtract the probability that the event happens from 1: 1 - (Desired outcomes)/(Total possible outcomes)
True or False: In a list of consecutive integers, the mean equals the median. True. For example, for the list {0, 2, 4, 6, 8} the mean and median are both 4. What must you multiply to find the sum of the terms in a consecutive sequence?
To find the sum of the terms in a consecutive sequence (or any sequence of evenly spaced terms), multiply the average of the smallest and largest numbers in the sequence by the number of terms in the sequence.
How do you find the volume of a cylinder? How do you find the surface area of a cylinder?
To find the volume or surface area of a cylinder, you need two pieces of information—the height of the cylinder and the radius of its base. Volume of a cylinder = (area of base)(height) = πr²h Lateral surface area of a cylinder = (circumference of base)(height) = 2πrh Total surface area of a cylinder = (areas of circular ends) + (lateral surface area) = 2πr² + 2πrh
The area of a square circumscribed about a circle is 36. What is the circumference of the circle?
To get the circumference, you need the diameter or radius. In this case, because the circle fits exactly inside the square, the circle's diameter is equal in length to the square's edge. The area of a square is given by it's side length squared. Here, the square's area is 36, so the square's side length is √(36) = 6. That means the circle's diameter is also 6. And circumference = πd = 6π.
The ratio of a to b is 7:3. The ratio of b to c is 2:5. What is the ratio of a to c?
To solve combined ratios, multiply one or both ratios by whatever you need to in order to get the terms they have in common to match. That is, find the least common multiple of the common term. Multiply each member of a:b by 2 and multiply each member of b:c by 3 and you get a:b = 14:6 and b:c = 6:15. Now that the b's match, you can just take a and c and say a:c = 14:15.
If the ratio of loaves of bread to bagels in a bakery is 1:3, and there are 60 bagels, how many loaves of bread are there?
Translate the ratio into algebra: The ratio of loaves of bread to bagels can be expressed as 1x:3x, where x is the constant by which the ratio is multiplied. The question states that there are 60 bagels. Therefore, 3x = 60. Solve for x by dividing both sides by 3; you get x = 20. Now, use this value of x to solve for the number of loaves of bread: 1x = 1(20) = 20 loaves of bread.
True or False: All prime numbers are odd.
True or False: All prime numbers are odd.
True or False: In a list of consecutive integers, the mean equals the median.
True. For example, for the list {0, 2, 4, 6, 8} the mean and median are both 4.
True or False: Many complex multiplication and division problems can be simplified by reducing the terms to their prime factors.
True. On the test, test takers are rarely expected to do complex arithmetic. Most scary-looking division and multiplication will end up reducing to a simpler form once the prime factors of the terms are identified and canceled out.
True or False: When an integer is added to, subtracted from, or multiplied by another integer, the result must be an integer.
True. When only integers are involved in addition, subtraction, and multiplication, the resulting values will also be integers.
Michael's average score after four tests is 80. If he scores 100 on the fifth test, what's his new average?
Use the sum of the terms of the old average to help you find the new average when a number is added or deleted. Find the original sum from the original average: Original sum = 4 × 80 = 320 Add the fifth score to make the new sum: New sum = 320 + 100 = 420 Find the new average from the new sum: New average = 420 / 5 = 84
Does a prime number multiplied by another prime number always equal an odd number?
Usually this is true, because all prime numbers greater than 2 are odd. But 2 is also a prime number. (It is the smallest prime and the only even prime.) Thus, a prime number multiplied by 2 would be even. For example 14 = 2 × 7 is the product of primes, and yet 14 is even.
A pair of intersecting lines makes an angle of 37°. What is the measurement of its vertical angle?
Vertical angles (angles formed by two intersecting lines that share a vertex but no common side) are always equal. Thus the vertical angle of a 37° angle would be 37°.
Triangle ABC is inscribed in a circle with center O. Side AC of the triangle is a diameter of the circle. Side AB has length 5 and side BC has length 12. What is the diameter of the circle ?
When a triangle is inscribed in a circle in such a way that one side of the triangle coincides with the diameter of the circle, the triangle is a right triangle. AC is a diameter of circle O, because it passes through center point O. So triangle ABC fits the description given of a right triangle. Moreover, triangle ABC is a special 5:12:13 right triangle, with a hypotenuse of 13. Therefore, the length of diameter AC is 13.
A fair coin is flipped twice. What is the probability of it landing with the heads side facing up on both flips?
When events are independent, that is, the chance of an event occurring does not depend on the result of other event(s), the probability that several events all occur is the product of the probability of each event occurring individually. Multiply the probability for each flip: (1/2)(1/2) = 1/4 So the probability that both tosses will land heads-up is one in four.
The average of 63, 64, 85, and x is 80. What is the value of x?
When finding an average, the combined distance of the numbers above the average from the mean must be balanced with the combined distance of the numbers below the average from the mean. Think of each value in terms of its position relative to the average, which is 80: 63 is 17 less than 80 64 is 16 less than 80 85 is 5 greater than 80 So these three terms are a total of 17 + 16 - 5, or 28, less than the average. Therefore, x must be 28 greater than the average to restore the balance at 80. Therefore, x = 28 + 80 = 108.
When two objects are traveling in opposite directions, how do you find their relative speed (the speed at which they move closer to or farther away from each other)? How do you find the relative speed of two objects traveling in the same direction?
When two objects are traveling in opposite directions, add their speeds to find their relative speed. When two objects are traveling in the same direction, subtract their speeds to find their relative speed.
he average (arithmetic mean) of 24 numbers is 15. What is the sum?
When using the average to find a sum, use this formula: Sum of terms = (Average)(Number of terms) Sum = 15 × 24 = 360
When you start at the origin (the place where the x- and y-axes cross) of the coordinate plane and move in the directions listed, what happens to x or y ?
When you start at the origin of the coordinate plane and move: to the right → x is positive to the left → x is negative up → y is positive down → y is negative
What are permutations?
Within any group of items or people, there are multiple arrangements, or permutations, possible. For instance, within a group of three items A, B, and C, there are six permutations: ABC, ACB, BAC, BCA, CAB, and CBA. Permutations differ from combinations in that permutations are ordered—that is, the same combination of entities chosen in a different order counts as a separate permutation. Often, permutations questions ask for the number of differently ordered subgroups that can be drawn from a larger group.
How many parts must you know to solve for all three parts of a three-part equation (such as the percent formula, speed formula, or average formula)?
You can solve for all three parts of a three-part equation (such as the percent formula, speed formula, or average formula) if you know the values for any two parts.
What is the length of the hypotenuse of an isosceles right triangle with legs of length 4?
You can use the Pythagorean theorem to find the hypotenuse, but it's quicker to use the special right triangle ratios. In an isosceles right triangle, the ratio of leg to leg to hypotenuse is x : x : x√(2). Because the length of a leg is 4, the length of the hypotenuse must be 4√(2).
When is it possible to determine the average without knowing the exact number of terms?
You can use the weighted average formula if you have averages for proportions of the population, even if you don't have data specific to the individual members of the population. Weighted average = (Avg. of A)(Percent that are A) + (Avg. of B)(Percent that are B) + ... + (Avg. of N)(Percent that are N) Note that to use the weighted average formula, you must have averages for all portions adding up to 100 percent of the whole.
The ratio of x to y is 5:4. The ratio of y to z is 1:2. What is the ratio of x to z?
You want the y's in the two ratios to equal each other, because then you can combine the x:y ratio and the y:z ratio to form the x:y:z ratio that you need to answer this question. To make the y's equal, you can multiply the second ratio by 4. When you do so, you must perform the multiplication on both components of the ratio. Because a ratio is a constant proportion, it can be multiplied or divided by any number without losing its meaning, as long as the multiplication and division are applied to all the components of the ratio. In this case, you find that the new ratio for y to z is 4:8. You can combine this with the first ratio to find a new x to y to z ratio of 5:4:8. Therefore, the ratio of x to z is 5:8.
What is the product of zero and any number?
Zero
What are the three classic quadratics?
a^2 + 2ab + b^2 = (a + b)^2 a^2 - 2ab + b^2 = (a - b)^2 a^2 - b^2 = (a-b)(a+b)
What is the percent formula?
percent = (part)/(whole) × 100%
How many hours will it take to fill a 500-liter tank at a rate of 2 liters per minute?
plug the numbers into the rate formula: Time = Quantity / Rate Time = 500 / 2 Time = 250 minutes Now convert 250 minutes to hours: 250 minutes / 60 minutes per hour = 4 1/6 hours, or 4 hours 10 minutes, to fill the tank.
Factor the quadratic expression x^2 - 3x + 2
x^2 - 3x + 2 = (x-1)(x-2)
Change 16 over 20 into an equivalent fraction with a denominator of 10
16/20 = (16/2)/(20/2) = 8/10
What is the name for any four-sided figure in which both pairs of opposite sides are parallel? What is the formula for the figure's area?
Any four-sided figure in which both pairs of opposite sides are parallel is a parallelogram. Parallelograms include squares and rectangles. The area of a parallelogram is its base times its height.
True or False: The value inside an absolute value sign must be positive.
False. The final result of an absolute value must be non-negative (i.e. positive or zero). But numbers or variables inside the absolute value sign can be positive, negative, or zero. Watch out for this common Test Day trap.
In a word problem, what are some keywords and phrases indicating a subtract symbol?
subtract, difference, decreased by, less than, minus, reduced
Find the area of rectangle ABCD, where side AD has a length of 2 and side CD has a length of 5.
Area of a rectangle = (length)(width) The area of a 5-by-2 rectangle is 2 × 5 = 10.
Four thirds divided by four ninths equals
(4/3) / (4/9) =(4/3)*(9/4) =(4*9)/(3*4) =36/12 =3
Ten ninths times three fourths times eight fifteenths equals
(9/10)(3/4)(8/15) =(9*3*8)/(10*4*15) = 216/600 =9/25
express (a-b^2) divided by (b-a) in terms of x, if a = 2x and b = 3
(a-b^2)/(b-a) = (2x - 3^2)/(3-2x) = (2x - 9)/(3-2x)
What is the ratio of the lengths of the sides of a 45°-45°-90° triangle?
A 45°-45°-90° triangle has sides in the ratio of x to x to x times the square root of 2. This type of special right triangle occurs frequently on the test.
What are the steps for isolating a variable?
1. Eliminate any fractions by multiplying both sides and/or eliminate radicals by squaring both sides 2. Put all terms with the variable you're solving on one side 3. Combine like terms 4. Factor out the desired variable 5. Divide by everything else, leaving the desired variable by itself
What is the sum of the two non-right interior angles of a right triangle?
All triangles have interior angles that sum to exactly 180°. A right angle measures 90°, so the sum of the other two interior angles of a right triangle is 180° - 90° = 90°.
What are the steps when adding or subtracting fractions?
1: Convert to common denominator 2: add or subtract the numerators 3: simplify
If the length of the hypotenuse (longest side) of a right triangle is 26 and one of the legs (short sides) has a length of 10, what is the length of the other leg?
24. 10 = 5 × 2 and 26 = 13 × 2, so this right triangle is a 5:12:13 right triangle and the last side should be 12 × 2 = 24. Remember, the special right triangles are ratios; learn to recognize them even when they are multiplied by various constants.
Factor 9x^2 - 1
9x^2 - 1 = (3x + 1)(3x - 1)
What is a polygon?
A polygon is a closed figure whose sides are straight line segments. Families or classes of polygons are named according to the number of sides: A triangle has three sides; a quadrilateral has four sides; a pentagon has five sides; and a hexagon has six sides. Triangles and quadrilaterals are by far the most important polygons on the test; other polygons appear only occasionally.
What are the characteristics of a trapezoid?
A trapezoid is a quadrilateral with exactly one pair of parallel sides. To find the area of a trapezoid, you can use the formula ½ (base₁ + base₂) × height where base₁ and base₂ are the lengths of the parallel sides. Or you can subdivide the trapezoid into rectangles and triangles, find their individual areas, then add these areas.
What is a triangle? What is a right triangle?
A triangle is a polygon with three straight sides and three interior angles. A right triangle is a triangle with one interior angle of 90° (a right angle).
Which is greater: 5/6 or 7/9?
Find common denominator 5/6 = 14/18, 7/9 = 15/18 therefore 7/9 is greater
Translate the square root of the expression 2 and one-half multiplied by the square root of 10 equals
First, translate into math: √(2 1/2)√(10) = ? Convert the mixed number 2 1/2 to a proper fraction: √(2 1/2)√(10) = √(5/2)√(10) Combine: √(5/2)√(10) = √(5/2 × 10) Finally, treat the integer 10 as a fraction 10/1, multiply fractions as normal and simplify: √(5/2 × 10) = √((5/2)(10/1)) = √( (5 × 10) / (2 × 1) ) = √(50/2) = √(25) = 5.
What do fractional exponent relate to?
Fractional exponents relate to roots. Ex: x^(1/2) = Sqrt(x) x^(1/3) = Cbrt(x) X^(2/3) = Cbrt(x^2)
Given the lengths of two sides of a triangle, how do you find the minimum and maximum lengths for the third side?
If you know the lengths of two sides of a triangle, you know that the third side is between their positive difference and their sum. For example, the lengths of two of the sides of a triangle are 7 and 3. What is the range of possible lengths for the third side? The third side is greater than the difference (7 - 3 = 4) and less than the sum (7 + 3 = 10). In other words, 4 < side < 10.
What are the radius and diameter of a circle?
In a circle with center O, a line segment pssing through point O connects points A and B on the circmference of the circle. Another line segment connects point O and point C, which is located on the circumference of the circle. A diameter is a line segment that connects two points on the circle and passes through the center of the circle. AB is a diameter of circle O above. A radius (plural: radii) is a line segment that connects the center of the circle with any point on the circle. The radius of a circle is one-half the length of the diameter. In circle O above, OA, OB, and OC are radii.
How do you find the area of a right triangle?
In a right triangle, think of one leg as the base and the other as the height. The area is one-half the product of the legs.
In inequalities, what must you know in order to multiply or divide by variables?
In inequalities, you cannot multiply or divide by variables unless you know whether they are positive or negative. Remember, positives preserve the direction of the inequality sign but negatives reverse it; if you don't know the sign of a variable, you won't know whether the inequality sign should stay the same or be reversed.
What are isosceles triangles?
Isosceles triangles are triangles that have two equal sides. The angles opposite these sides, or base angles, are also equal.
What are parallel lines? What are perpendicular lines?
Parallel lines are lines with the same slope. If continued infinitely on the same plane, they will never meet. Perpendicular lines are lines that meet at 90° angles. Their slopes are negative reciprocals of each other.
Find the perimeter of rectangle WXYZ, where side WZ has a length of 2 and side YZ has a length of 5
Perimeter of a rectangle = 2(length + width) The perimeter of a 5-by-2 rectangle is 2(5 + 2) = 14.
Of addition, subtraction, multiplication and division which two can the radicals be combined or split? With which two can the radicals not be combined or split?
Radicals can be combined or split when the operations involved are multiplication and division Ex: Sqrt(4)*Sqrt(9) = 2 * 3 = 6 = Sqrt(36) = Sqrt(4*9) Radicals cannot be combined or split when the operations involved are addition and subtraction Sqrt(4) + Sqrt(9) = 2 + 3 = 5
What extra caution is needed when considering the solution to an inequality?
Remember that the solution to an inequality describes a range of values, not merely a single value. A number line is a great tool for visualizing such a range; for instance, this line represents all the values of x > 3: the number line shows values from -2 on the left to 6 on the right. the number 3 is circled with an open circle and has an arrow line extending to the right to the end of the number line
Does raising a fraction between 0 and 1 to an integer than 1 produce a result that is smaller or larger than the original fraction?
Smaller than the original Ex: (1/2)^2 = 1/(2+2) = 1/4
Solve for x: (3 - x) / 4 > 2
Starting with: (3 - x) / 4 > 2, Multiply both sides by 4: (3 - x) > 8. Because you're multiplying by a positive number, you do not need to change the direction of the inequality sign! Subtract 3 from both sides: -x > 5 Now multiply both sides by -1 so that x is positive: x < -5. Be careful with that last step! Because you're multiplying both sides by a negative number, you have to switch the direction of the inequality sign!
What are the two ways to solve a system of linear equations
Substitution : Isolate one variable in one equations, then substitute for all instances of that variable in the other equations Combination: Add or Subtract whole equations . Use this when it will allow you to eliminate a variable or get directly to the equation you want.
How do you subtract a negative number?
Subtracting a negative is the same as adding a positive
A pair of intersecting lines makes an angle of 37°. What is the measurement of the adjacent (supplementary) angle?
Supplementary angles always add up to 180°. The supplementary angle of a 37° angle would thus be 143°.
Let x be defined by the equation x = x^2/(1 - x^2). Evaluate (1/2)*
Symbolism problems usually require nothing more than substitution. Read the question stem carefully for a definition of the symbols and for any examples of how to use them. Then, follow the given model, substituting the numbers that are in the question stem. In this case: (1/2)* = (1/2)^2 / (1- (1/2)^2) =(1/4) / (1 - 1/4) =(1/4) * (4/3) =(1/3)
What is the Pythagorean theorem?
The Pythagorean theorem states that for a right triangle (a triangle that contains one 90° angle), a² + b² = c², where a and b are the lengths of the shorter two sides of the right triangle, known as legs, and c is the length of the longest side, or hypotenuse.
Parallel lines L1 and L2 are both intersected by transversal L3. An obtuse angle formed by the intersection of L1 and L3 measures x degrees. An acute angle formed by the intersection of L2 and L3 measures 70 degrees. What is the value of x?
The angle marked x corresponds to the angle adjacent to and to the left of the 70° angle on line L1. Therefore, it must be supplementary to the 70° angle. Because a straight line equals 180°, 70° + x° = 180°, and x equals 110°.
What is the circumference of a circle? What is the formula for circumference?
The circumference of a circle is the distance around the circle. The formula for circumference is C = 2πr where r is the radius of the circle. The formula can also be expressed as C = πd, where d is the diameter of the circle (as the diameter is twice the length of the radius).
What is a cube root?
The cube root of x is the number that, multiplied by itself 3 times (i.e., cubed), gives you x. Both positive and negative numbers have one and only one cube root, denoted by the symbol a radical symbol with the number 3 in the index , and the cube root of a number is always the same sign as the number itself. For example, (-5) × (-5) × (-5) = -125, so the cube root of negative 125 equals negative 5.
What happens when both sides of an inequality are multiplied or divided by a negative number?
The direction of the inequality sign is reversed.
Describe the distributive law of multiplication
The distributive law of multiplication allows you to "distribute" a factor over numbers that are added or subtracted, by multiplying that factor by each number in the group
What is the formula for the area of a circle?
The formula for the area of a circle is A = πr², where r is the radius of the circle.
What is the height of a triangle?
The height of a triangle is the perpendicular distance from a vertex to the side opposite the vertex. The height may fall inside or outside the triangle, or it may coincide with one of the sides. In triangle ABC, a a line segment extends from vertex A to point D, which is located on side BC. Line segment AD is perpendicular to side BC In triangle LKJ, angle LKJ is a right angle. In triangle EGF, side GF is extended to point H. A line segment connecting point E to point H is perpendicular to GH. In the triangles described above, AD, LK, and EH are heights.
What do the interior angles of a triangle sum up to? For every additional side in a polygon, how many degrees must you add to the sum of the interior angles? For example, what will the interior angles total in a five-sided figure?
The interior angles of a triangle sum up to 180°. For every side beyond three, add another 180°. For example, a five-sided figure will have interior angles totaling 180° + 2(180°) = 540°.
Rather than being a single value, the solution to an inequality is almost always what?
The solution to an inequality is almost always a range of possible values, rather than a single value. For example: a number line shows values from -4 on the left to 4 on the right. The number -1 is circled with an open circle and a line is drawn to the right stopping at the number 4 which is circled and filled in The shaded portion of the number line above shows the set of all numbers greater than -1, up to and including 4; this range would be expressed algebraically by the inequality -1 < x ≤ 4.
What is the sum of the angles of a polygon with 7 sides?
The sum of the angles of a polygon with n sides is equal to 180°(n - 2). For a polygon with 7 sides, you have: 180°(7 - 2) = 180°(5) = 900°
What is the formula for finding the degree measure of one angle in a regular polygon?
The total degree measure of the angles in a polygon is 180° × (n - 2), where n is the number of sides in the polygon. Therefore, the degree measure of each angle in a regular polygon is (180° × (n - 2)) / n
Angle BAC of triangle ABC is a right angle. Is side BC longer or shorter than side AB?
This question may seem abstract, until you draw a picture of a right triangle, labeling the vertex with the 90° angle as point A. Line segment BC has to be longer than line segment AB, since BC is opposite the triangle's largest angle. The hypotenuse (the side of a right triangle opposite the right angle) is always the longest of a triangle's three sides.
How many linear equations are needed to solve for each of three variables?
To solve for the individual values of multiple variables in a system of equations, you need at least as many linear equations as you have variable, in this case 3
Rewrite 7 - 3x > 2 in its simplest form.
Treat this inequality much like an equation—adding, subtracting, multiplying, and dividing both sides by the same thing. Just remember to reverse the inequality sign if you multiply or divide by a negative quantity. Starting with: 7 - 3x > 2, Subtract 7 from each side: -3x > -5 Divide both sides by -3 and reverse the inequality sign: x < 5/3
In a word problem, what are some keywords and phrases indicating a equal symbol?
equal, is, was, will be, result, total, equal to, has, costs, the same as, as much as
What is the least common denominator of 2/15 and 3/10?
the least common denominator of these two fraction will be the least common multiple of 15 and 10 5*3 = 15, 5*2 - 10 The least common multiple is (5)(3)(2), or 30. That makes least common denominator 3/10 or 2/15. Expressing these fraction using the LCD, we get 2/15 = 4/30 and 3/10 = 9/30
If the number inside the radical is a multiple of a perfect square, what will you accomplish by factoring out the perfect square?
the number inside the radical is a multiple of a perfect square, the expression can be simplified by factoring out the perfect square. For example: √(72) = √(36 × 2) = √(36)√(2) = 6√(2).
