SAT MATH EXAM 2025/2026 | OFFICIAL EXAM MATERIAL WITH STUDY GUIDE AND PRACTICE TEST | VERIFIED QUESTIONS AND ANSWERS FROM THE REAL EXAM | UPDATED EDITION | PROVEN SUCCESS RATE

SAT MATH EXAM 2025/2026 | OFFICIAL EXAM MATERIAL WITH STUDY GUIDE AND PRACTICE TEST | VERIFIED QUESTIONS AND ANSWERS FROM THE REAL EXAM | UPDATED EDITION | PROVEN SUCCESS RATE 1) Strategic Advice: Know the difference between domain and range. The domain of a function is the set of values for which the function is defined. The range of a function is the set of outputs, or results, of the funtion. Getting to the Answers: You know that x cannot be zero because, if it were, the function would be undefined. This consideration eliminates all choices except (c) and (E). The value x = 1/2 satisfies the function, so eliminate (E), since 1/2 is not an integer. This leaves only (C). Achmed finds that by wearing different combinations of the jackets, shirts, and pairs of trousers that he owns, he can make up 90 different outfits. If he owns 5 jackets and 3 pairs of trousers, how many shirts does he own? (A) 3 (B) 6 (C) 12 (D) 18 (E) 30 - ANSWER Answer: B 2) Strategic Advice: Use the Fundamental Counting Principle: if there are m ways one event can happen and n ways a second event can happen, then there are m x n ways for the 2 events to happen. Getting to the Answer: The Fundamental Counting Principle is applied for the 3 events. Let s be the number of shirts Achmed owns: 5 x 3 x s = 90 15s = 90 s = 6 A class of 40 students is to be divided into smaller groups. If each group is to contain 3, 4, or 5 people, what is the largest number of groups possible? (A) 8 (B) 10 (C) 12 (D) 13 (E) 14 - ANSWER Answer: D 3) Strategic Advice: We will get the maximum number of groups by making each group as small as possible. Getting to the Answer: Each group must have at least 3 people, so divide 40 by 3 to find the number of 3-person groups: 40/3 = 13 with a remainder of 1. So we have 13 groups with 1 person left over. Since each group must have at least 3 people, we must throw the extra student in with one of the other groups. So we have 12 groups with 3 students each, and 1 group with 4 students, for a total of 13 groups. What is the value of a if ab + ac = 21 and b + c = 7? (A) -3 (B) -1 (C) 0 (D) 1 (E) 3 - ANSWER Answer: E Strategic Advice: Always look for ways to factor complicated- looking equations to get them into simpler form. Getting to the Answer: To first use the equation we need to factor out of a: ab + ac = 21 a(b + c) = 21 Since b + c = 7, substitute 7 for b + c: a(7) = 21 Solve for a: a = 21/7 = 3 If m @ n is defined by the equation m @ n = m^2 - n + 1 for all nonzero m and n, then 3 @ 1 = (A) -3 (B) 3 (C) 5 (D) 8 (E) 9 - ANSWER Answer: E 4) Strategic Advice: Here we have a symbolism problem, involving a symbol, @, that doesn't exist in mathematics. All you need to do is follow the directions given in the definition of this symbol. Getting to the Answer: To find the value of 3 @ 1, plug 3 and 1 into the formula given for m @ n, substituting 3 for m and 1 for n. Then the equation becomes: 3 @ 1 = 3^2 -1 + 1 = 9 - 1 + 1 = 9 5) Set J is the set of all positive even integers and Set K is the set of all numbers between -2 and 2, inclusive. Which of the following represents the intersection of J and K? (A) all integers (B) all positive integers (C) all positive even integers (D) {2} (E) {0, 2} - ANSWER Answer: D 6) Strategic Advice: The intersection of two sets contains only the numbers that reside in both sets. Rember, 0 is neither positive nor negative. Getting to the Answer: Set J is {2, 4, 6, ...} and Set K is all values from -2 to 2, including the endpoints themselves. The only number that is both a positive even integer and in the range is 2. If [-3x - 7] = 5, x = (A) -2/3 (B) -4 (C) -2/3 or - 4 (D) 2/3 or - 4 (E) 4 - ANSWER Answer: C 7) mode - ANSWER Most frequently occurring number 8) straight angle - ANSWER An angle that is 180 degrees 9) parallel lines - ANSWER lines in the same plane that never intersect 10) alternate interior angles - ANSWER Angles that lie within a pair of lines and on opposite side of a transversal. 11) alternate exterior angles - ANSWER Angles that lay outside the parallel lines and are on opposite sides of the transversal; They are congruent. 12) corresponding angles - ANSWER Angles formed by a transversal cutting through 2 or more lines that are in the same relative position. 13) same side interior angles - ANSWER two interior angles on the same side of the transversal 14) measures of the 3 angles in a triangle - ANSWER should add up to 180 15) measures of the 2 acute angles in a right triangle - ANSWER should add up to 90 16) how do you solve distance problems - ANSWER D= √(x₁-x₂)²+(y₁+y₂)² 17) how do you find the distance between 2 points? - ANSWER c² = a² + b² 18) counting principal - ANSWER uses multiplication to find the number of possible ways two of more events can occur 19) exterior angle of a triangle - ANSWER An angle that forms a linear pair with an angle of the triangle 20) Pythagorean theorem - ANSWER a²+b²=c² 21) properties of a 45-45-90 triangle - ANSWER 22) properties of a 30-60-90 triangle - ANSWER 23) triangle inequality theorem - ANSWER The sum of the lengths of any two sides of a triangle is greater than the length of the third side. 24) formula for the area of a CIRCLE - ANSWER A=πr² 25) special formula for the area of a triangle - ANSWER a= 1/2 bh 26) similar triangles (and their properties) - ANSWER Similar triangles have the same shape: corresponding angles are equal and corresponding sides are proportional 27) what is true about the areas of similar triangles? - ANSWER A correlation coefficient shows the relationship between two sets of numbers. 28) exterior angle of a polygon - ANSWER an angle formed by a side and an extension of an adjacent side 29) properties of a parallelogram - ANSWER opposite angles are congruent 30) properties of a rectangle - ANSWER all properties of a parallelogram, all angles are right, diagonals are congruent 31) properties of a square - ANSWER The measure of each angle is 90 32) formulas for the area of a parallelogram, rectangle, and a square - ANSWER length x width 33) what is pi? - ANSWER 3.14 34) formula for the circumference of a circle - ANSWER C = 2πr 35) how do you find the length of an arc of a circle? - ANSWER 36) how do you find the area of a sector of a circle - ANSWER 37) what is the relationship between a radius of a circle and a line that is tangent to the circle? - ANSWER A tangent line is always perpendicular to the radius 38) formula for the volume of a rectangular solid and a cube - ANSWER Cube: V=side³ Rectangular solid: l*w*h 39) the formula for the surface area of a rectangular solid and a cube - ANSWER 2(area of base) + (base perimeter)(height of prism) 40) formula for the volume of a PYRAMID or CONE - ANSWER V=1/3BH 41) how do you find the x-intercepts and the y-intercepts of a graph? - ANSWER Substitute them for zero 42) midpoint formula - ANSWER (X1 + X2)/2 and (Y1 + Y2)/2 43) distance formula - ANSWER d = √[( x₂ - x₁)² + (y₂ - y₁)²] 44) slope of a line - ANSWER y=mx+b 45) what is true about the slopes of parallel lines? - ANSWER They are equal 46) what is true about the slopes of perpendicular lines? - ANSWER They are opposite reciprocals (product of negative 1). 47) equation of a line - ANSWER y=mx+b 48) how do you find the probability that an event will occur? - ANSWER Number of favorable outcomes/total number of possible outcomes 49) what is the probability of an event that cannot occur? - ANSWER 0 50) venn diagram - ANSWER A diagram that uses circles to display elements of different sets. Overlapping circles show common elements. 51) function - ANSWER a rule relating an input value to an output value. Each input value can only be paired with one output value. 52) if you are given the graph y=f(x), what does f(a) represent - ANSWER that represents the y value when x is equal to a 53) what are the relationships between the graphs of y=f(x) and the graphs of y=f(x)+a, y=f(x)-a, y=f(x+a), y=f(x-a) - ANSWER it moves the graph up and down and left and right 54) what proportion do you use to solve percent problems? - ANSWER x/100 order of operations - ANSWER PEMDAS greatest common factor - ANSWER The largest factor that two or more numbers have in common. least common multiple - ANSWER the smallest number that two or more numbers will divide into evenly. arithmetic sequence - ANSWER a sequence in which each term is found by adding the same number to the previous term geometric sequence - ANSWER a sequence in which each term is found by multiplying the previous term by the same number range - ANSWER Distance between highest and lowest scores in a set of data. mean - ANSWER Average vertical angles - ANSWER A pair of opposite congruent angles formed by intersecting lines equation of a circle - ANSWER (x-h)²+(y-k)²=r² sine of an angle - ANSWER The ratio of the length of the side opposite the angle to the length of the hypotenuse. cosine of an angle - ANSWER Adjacent/Hypotenuse tangent of an angle - ANSWER The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. system of equations - ANSWER A set of equations with the same variables. slope formula - ANSWER (y₂- y₁) / (x₂- x₁) Strategic Advice: Remember that equations involving absolute values often have two solutions since the value inside the bars can be positive or negative. Getting to the Answer: If [-3x - 7] = 5, then -3x - 7 = 5 or -3x - 7 = - 5. Solve for x: -3x = 12 or - 3x = 2 x = -4 or x = -2/3 If f(x) = [6x^2 + 3x], what is f(-3)? (A) -45 (B) 12 (C) 18 (D) 30 (E) 45 - ANSWER Answer: E Strategic Advice: Don't be intimidated by functions- they are often the easiest problems to solve if they involve straight substitution. Getting to the Answer: Plug in -3 wherever you see x: f(x) = [6x^2 + 3x] f(-3) = [6(-3)^2 + 3(-3)] f(-3) = [54 - 9] f(-3) = 45 If k is a positive integer less than 13, which of the following is a possible pair of integral solutions for the equation x^2 + 8x + k = 0? (A) -4, -2 (B) -3, -5 (C) -2, -6 (D) -4, 12 (E) 1, 7 - ANSWER Answer: C Strategic Advice: The 8x in the equation represents the sum of the products of the outer and inner terms of the equation in factored form, (x + _)(x + _). The k represents the product of the last terms. Use Backsolving to determine which of the answer choices provides a middle term with coefficient 8 and a constant term less than 13. Remember, if (x - m) is a factor then m is a solution. Getting to the Answer: Since the constant term (k) is a positive and the middle term (8x) is positive, we know that x^2 + 8x + k = 0 factors to (x + _)(x + _) = 0. Therefore, both solutions must be negative. This eliminates answer choices (D) and (E). Substitute the remaining answer choices into the factored form to determine which of the products will be of form x^2 + 8x + k = 0 where k is a positive integer less than 13. Only (C) has a middle term of 8x and a constant term less than 13. What strategy would you use to complete the problem below? When x is divided by 5, the remainder is 4. When x is divided by 9, the remainder is 0. Which of the following is a possible value for x? (A) 24 (B) 45 (C) 59 (D) 109 (E) 144 - ANSWER Answer: E Strategy: Backsolving Strategic Advice: There are numerical answer choices and the question asks for the value of a variable. Therefore, this problem can be solved by Backsolving. The solution is the answer choice that is divisible by 9 and whose remainder is 4 when divided by 5. Getting to the Answer: Since x leaves no remainder when divided by 9, it is divisible by 9. Answer choices (A), (C), and (D) can be eliminated. Of the remainding answer choices, only (E) has a remainder of 4 when divided by 5. How would you approach the problem below? The average of 20, 70, and x is 40. If the average of 20, 70, x, and y is 50, then y = (A) 100 (B) 80 (C) 70 (D) 60 (E) 30 - ANSWER Answer: B Strategic Advice: Number of terms x average= sum of the terms. Getting to the Answer: For the first group, 3 x 40 = 120, so the sum of 20, 70, and x is 120. For the second group, 4 x 50 = 200, so 20 + 70 + x + y = 200. Since the sum of the first 3 terms is 120, 120 + y = 200, and y = 80. If f(x) = 7, what is the range of f(x)? (A) all real numbers (B) 7 (C) all real numbers other than zero (D) all integers (E) all positive numbers - ANSWER Answer: B Strategic Advice: Distinguish domain from range. THe domain of a function is the set of values for which the function is defined. The range of a function is the set of numbers that constitute the values of the function. Getting to the Answer: For all values of this function, f(x) will always equal 7. So the range is 7. If x + y = 11, y + z = 14, and x + z = 13, what is the value of x + y + z? (A) 16 (B) 17 (C) 18 (D) 19 (E) 20 - ANSWER Answer: D Strategic Advice: Combine the equations to see if you can isolate any variables, or if you can factor anything out to make an equation that looks familiar. Getting to the Answer: If you add the 3 equations together, you find that 2x + 2y + 2z = 38. Dividing both sides by 2 shows that x + y + z = 19, answer choice (D). How would you approach the problem below? If a^2 - a = 72, and b and n are integers such that b^n = a, which of the following cannot be a value for b? (A) -8 (B) -2 (C) 2 (D) 3 (E) 9 - ANSWER Answer: C Strategic Advice: Find the possible values by factoring: b^n = aso b must be a root of one of the values of a. Use Backsolving to find the choice that is NOT a root of one of the values of a. If a^2 - a = 72, then a^2 - a = 72 = 0. Factoring this quadratic equation, you get (a - 9)(a - 8) = 0. So a = 9 or -8. Getting to the Answer: (A) (-8)^1 = -8, so this can be a value for b. (B) (-2)^3 = -8, so this can be a value for b. (C) 2^3 = 8, not -8, so this answer cannot be a value for b. (D) 3^2 = 9, so this can be a value for b. (E) 9^1 = 9, so this can be a value for b. (C) is the only answer choice that cannot be a value for b. What strategy would you use to complete the problem below? If n doesn't equal 0, then which of the following must be true? I. n^2 n II. 2n n III. n + 1 n (A) I only (B) II only (C) III only (D) I and III only (E) I, II, and III - ANSWER Answer: C Strategic Advice: Picking Numbers is a good strategy since this problem is associated with variable expressions. When picking numbers for variables, it is important to consider positive and negative numbers and fractions. Getting to the Answer: Examine each of the three statements for accuracy. I. This statement does not hold true for fractions. When you square a fraction, the resulting fraction is a smaller number. For n = 1/2, (1/2)^2 = 1/4. II. Although this statement holds true for positive numbers, if you multiply a negative number by two, your result is smaller than the original number. For n = -3, -3 x 2 = -6. III. Adding 1 to any number results in a larger number. Therefore, only statement III must be true. (C) is correct. How do you convert from part-to-part ratios to part-to-whole ratios? - ANSWER If the parts add up to the whole, a part-to-part ratio can be turned into two part-to-whole ratios by putting each number in the original ratio over the sum of the numbers. For example, if the ratio of males to females is 1 to 2, then the males-to-people ratio is 1/ (1+2) = 1/3 and the females-to-people ratio is 2/ (1+2) = 2/3 In other words, 2/3 of all the people are female. How do you solve a proportion? - ANSWER To solve a proportion, cross multiply. For example: x/5 = 3/4 4x = 3 x 5 x = 15/4 = 3.75 How do you solve a problem involving rates? - ANSWER To solve a rate problem, use the units to keep things straight. For example, if asked: If snow is falling at the rate of one foot every four hours, how many inches of snow will fall in seven hours? Your setup would be: 1 foot/ 4 hours = x inches/ 7 hours 12 inches/ 4 hours = x inches/ 7 hours 4x = 12 x 7 x = 21 What is the average rate and how do you find it? - ANSWER Average rate is not simply the average of the rates. Average A per B = Total A/ Total B, so Average Speed = Total distance/ Total time For example, to find the average speed for 120 miles at 40 mph and 120 miles at 60 mph, don't just average the two speeds. First, figure out the total distance and the total time. The total distance is 120 + 120 = 240 miles. The times are three hours for the first leg and two hours for the second leg, or five hours total. The average speed, then, is 240/5 = 48 miles per hour. What is the formula used to find average? - ANSWER To find the average of a set of numbers, add them up and divide by how many there are. Average = Sum of the terms / Number of the terms How do you quickly find the average of a series of evenly spaced numbers? - ANSWER For evenly spaced numbers, the average is the middle value. To find the average of evenly spaced numbers, just average the smallest and the largest. For example, the average of all the integers from 13 through 77 is the same as the average of 13 and 77. 13 + 77/ 2 = 90/ 2 = 45 If you have an average of a series of numbers, how can you find their sum? - ANSWER Use the formula: Sum = Average x Number of terms If the average of 10 numbers is 50, then they add up to 10 x 50, or 500. How do you find the missing number in a series when you are given the average? - ANSWER To find a missing number when you're given the average, use the sum. Sum = Average x Number of Terms For example, if the average of 4 numbers is 7, then the sum of those 4 numbers is 4 x 7, or 28. Suppose that 3 of the numbers are 3, 5, and 8. These 3 numbers add up to 16 of that 28, which leaves 12 for the fourth number. How do you find the median of a set of numbers? - ANSWER The median of a set of numbers is the value that falls in the middle of the set. For example, if you have 5 test scores, and they are 88, 86, 57, 94, and 73, you must first list the scores in increasing or decreasing order: 57, 73, 86, 88, 94. The median is the middle number, or 86. If there is an even number of values in a set (6 test scores, for instance), simply take the average of the two middle numbers. How do you calculate the total number of possibilities for several events to occur? - ANSWER The fundamental counting principle: If there are m ways one event can happen and n ways a second event can happen, then there are m x n ways for the 2 events to happen. For example, with 5 shirts and 7 pairs of pants to choose from, you can have 5 x 7 = 35 different outfits. How do you find the mode of a set of numbers? - ANSWER The mode of a set of numbers is the value that appears most often. For example, if your test scores were 88, 57, 68, 85, 99, 93, 93, 84, and 81, the mode of the scores would be 93 because it appear more often than any other score. If there is a tie for the most common value in a set, the set has more than one mode. How do you multiply expressions with exponents that have a common base? - ANSWER To multiply exponential terms with the same base, add the exponents and keep the same base: x^3 x x^4 = x^3+4 = x^7 How do you calculate the probability that an event will take place? - ANSWER Number of Favorable Outcomes/ Total Possible Outcomes ^ Probability = For example, if you have 12 shirts in a drawer and 9 of them are white, the probability of picking a white shirt at random is 9/ 12 = 3/ 4. This probability can also be expressed as .75 or 75%. How do you divide expressions with exponents that have a common base? - ANSWER To divide exponential terms with the same base, subtract the exponents and keep the same base: y^13 / y^8 = y^13-8 = y^5 How do you raise a power to a power? - ANSWER To raise a power to a power, multiply the exponents: (x^3)^4 = x^3 x 4 = x^12 How do you simplify square roots? - ANSWER To simplify a square root, factor out the perfect squares under the radical. Take the square root of the perfect square and put the result in front. How do you add and subtract roots? - ANSWER Only like radicals can be added/subtracted from one another. To combine radicals, just add or subtract their coefficients. How can you use ratios to find the length of a missing side of a right triangle? - ANSWER There are special right triangles with consistent side-length ratios and by memorizing these ratios you can avoid using the Pythagorean Theorem to find a missing side length. For example, say one leg of a right triangle is 30 and the hypotenuse is 50. This is 10 times 3-4-5. Therefore, the other leg is 40. When these ratios are integers, the triangles are known as Pythagorean Triples. The Pythagorean Triples ratios you will want to know for the SAT are 3:4:5 and 5:12:13. You can use these two ratios to find the missing side of a triangle. For example, if one leg is 36 and the hypotenuse is 39, this is 3 times 5-12-13. The other leg is 15. What are the properties of a 30-60-90 triangle? - ANSWER The sides of a 30-60-90 triangle are in a ratio of x: s (square root of) 3: 2x. You don't need the Pythagorean Theorem. x is the shorter leg, opposite the smallest 30 degree angle; x(square root of) 3 is the longer leg, opposite the 60 degree angle; 2x is the longest side (the hypotenuse), opposite the right angle. What are the properties of a 45-45-90 triangle? - ANSWER You don't need the Pythagorean Theorem to solve a 45-45-90 triangle. The sides of a 45-45-90 triangle are in a ratio of x:x:x(square root of)2. x and x are the two equal sides. The hypotenuse is x(square root of)2. How do you know if an integer is a multiple of 3 or 9? - ANSWER An integer is divisible by 3 if the sum of its digits is divisible by 3. An integer is divisible by 9 if the sum of its digits is divisible by 9. For example, the sum of the digits in 957 is 21, which is divisible by 3 but not by 9. Therefore, 957 is divisible by 3 but not by 9. What is the definition of an integer? - ANSWER Integers are whole numbers, negative whole numbers, and zero. For example: -15, -4, 0, 1, and 457 are all integers. What is the definition of a rational number? - ANSWER A rational number is a number than can be expressed as a ratio of two integers. Terminating and repeating decimals are rational numbers because they can also be expressed as the ratio of two integers. What is the definition of an irrational number? - ANSWER Irrational numbers are real numbers-- they have locations on the number line-- but they can't be expressed as a fraction. they are non-repeating, non-terminating decimals. For the purposes of the SAT, the most important irrational numbers are (square root of)2, (square root of)3, and pi. (Note that 3.14 is only an approximation of pi, pi is a non-repeating, non-terminating decimal.) How do you add a positive number to a negative number? - ANSWER To add a positive and a negative, first ignore the signs and find the positive difference between the numbers. Then attach the sign of the original number with the larger absolute value. How do you multiply and divide positive and negative numbers? - ANSWER To multiply and/or divide positives and negatives, treat the number parts as usual, and attach a negative sign if there is an odd number of negatives. Explain the memory device "PEMDAS" - ANSWER When performing multiple operations, you must remember to perform them in the correct order: PEMDAS, Parentheses first, then Exponents, then Multiplication and Division (left to right), and then Addition and Subtraction (left to right). How do you count consecutive integers? - ANSWER To count consecutive integers, subtract the smallest from the largest and add 1. For example, to count the integers from 13 through 31, subtract 31 - 13 = 18. Then add 1: 18 + 1 = 19. How do you express the union and intersection of sets? - ANSWER The union of Set A and Set B, is the set of elements that are in either or both Set A and Set B. How do you set up a ratio? - ANSWER To find a ratio, put the number associated with the word "of" in the numerator and the quantity associated with the word "to" in the denominator and simplify. For example, the ratio of 20 oranges to 12 apples is 20/12, which reduces to 5/3. When given a series of percent increases and decreases, how do you determine your ending value? - ANSWER To determine the combined effect of multiple percent increases and/or decreases, start with 100 and see what happens. Take the following problem: A price went up 10% one year, and the new price went up 20% the next year. What was the combined percent increase? First year: 100 + (10% of 100) = 110 Second year: 110 + (20% of 110) = 132 The price increased by $132 - $100 = $32 % increase = change in price / original price x 100 = 32% That's a combined increase of 32 percent. How do you find an original value, before it was increased or decreased by a certain percentage? - ANSWER To find the original whole before a percent increase or decrease, set up an equation. Add/subtract the given percentage to/from 100. Then, convert the percentage into a decimal. Finally, equate the given value with the product of the decimal and a variable representing the original value. For example: If asked: After a 5 percent increase, the population was 59,346. What was the population before the increase? Your setup would be: 1.05x = 59,346. How do you increase or decrease a number by a certain percentage? - ANSWER To increase a number by a percent, add the percent to 100 percent, convert to a decimal, multiply. For example, to increase 40 by 25 percent, add 25 percent to 100 percent, which equals 125%. Then, convert 125 percent to 1.25, and multiply by 40: 1.25 x 40= 50. To decrease a number by a percent, subtract the percent from 100 percent, convert to a decimal, and multiply. For example, to decrease 50 by 20 percent, subtract 20 percent from 100 percent, which equals 80%. Then, convert 80 percent to 0.80, and multipy by 50:0.80 x 50 = 40. What is the formula used to find percent? - ANSWER Whether you need to find the part, the whole, or the percent, use the same formula: Percent = Part/Whole x 100 For example: If asked: What is 12 percent of 25? Your setup would be: 12 = part/25 x 100 If asked: 15 is 3 percent of what number? Your setup would be: 3 - 15/ whole x 100 If asked: 45 is what percent of 9? Your setup would be: Percent = 45/9 x 100 How do you convert from a fraction to a decimal, and from a decimal to a fraction? - ANSWER To convert a fraction to a decimal, divide the numerator by the denominator. For example, to convert 5/8, divide 8 into 5, yielding .625. To convert a decimal to a fraction, set the decimal over 1 and multiply the numerator and denominator by 10 raised to the number of digits to the right of the decimal point. For example: to convert .625 to a fraction, you would multiply .625/1 by 10^3/10^3 or 1,000/1,000. Then, simplify: 625/1,000 = 5/8. How do you compare the values of two or more fractions? - ANSWER One way to compare fractions is to express them with a common denominator. For example, 3/4 = 21/28 and 5/7 = 20/28. So 3/4 is greater than 5/7. Another method is to convert both fractions to decimals. For examples, 3/4 converts to .75, and 5/7 converts to approximately .714, so 3/4 is greater than 5/7. On the SAT, you are allowed to use a calculator to convert fractions to decimals. How do you find the reciprocal of a fraction? - ANSWER To find the reciprocal of a fraction, switch the numerator and the denominator. For example, the reciprocal of 3/7 is 7/3. The reciprocal of 5 is 1/5. The product of a number and its reciprocal is 1. How do you convert from an improper fraction to a mixed number? - ANSWER To convert an improper fraction to a mixed number, divide the numerator by the denominator to get a whole number quotient with a remainder. The quotient becomes the whole number part of the mixed number, and the remainder becomes the new numerator (with the same denominator) of the fractional part. For example, to convert 108/5, first divide 5 into 108, which yields 21 with a remainder of 3. Therefore, 108/5 = 21 3/5. How do you convert from a mixed number to an improper fraction? - ANSWER To convert a mixed number to an improper fraction, multiply the whole number part by the denominator, then add the numerator. The result is the new numerator over the same denominator. For example, to convert 7 1/3 , first multiply 7 by 3, then add 1, to get the new numerator of 22. Put that over the same denominator, 3, to get 22/3. function notation making equation undefined - ANSWER -if have lots of a series of numbers make it a variable (like x-5) -solve polynomial functions - ANSWER -function with variables raised to power higher than 1 polynomial factors - ANSWER if divide polynomial by equation and do not get remainder then is a factor g(x-1)= 3x^2+5x-7 solve - ANSWER make all xs equal x + 1 and then solve commutative property - ANSWER numbers can swap places and still provide the same result -addition/multiplication associative property - ANSWER different numbers groups will provide the same math result -when have multiplication/addition when subtracting an expression in parentheses - ANSWER distribute the negative sign to the numbers/variables in the parenthesis first absolute value - ANSWER distance a number is from 0 4/5 x= - ANSWER 4x/5 quadrants location - ANSWER left to right: 2 then 1 on top 3 then 4 on bottom perpendicular symbol - ANSWER parallel symbol - ANSWER area of a triangle - ANSWER A= 1/2bh parallelograms area - ANSWER A=bh circumfrence - ANSWER a circle's perimeter C=2pir area of a circle - ANSWER A=pi x r^2 axis of symmetrey - ANSWER splits a shape into 2 identical parts congruence - ANSWER -angles, lines, and shapes can be congruent (the same) -indicated by hash marks, everything with the same number of them is congruent similarity - ANSWER a shape has identical angles and proportional sides key words for inequalities - ANSWER -"describe all possible values of x" -"include the entire set of solutions for x" -range of values how to solve inequalities - ANSWER isolate x on one side inequalities in graphs in one dimension - ANSWER -in one dimension, inequalities are graphed on a number line with a shaded region -open dot indicates that have signs or -if x greater than goes right, if x less than goes left inequalities in 2 dimensions - ANSWER -solid lines for if not or because line itself is included in set boundary lines - ANSWER the lines on the graph of inequalities half planes - ANSWER the shaded region on the graph system of inequalities - ANSWER multiple equations can be combined to create a system -answer is shaded region where the shadings overlap -check shading by plugging coordinates in and seeing if they work -can do elimination or draw a graph adding inequalities - ANSWER only if have the same symbol regression equation - ANSWER equation of the line of best fit -found in top left corner of graph line of best fit - ANSWER is drawn through the data points to describe the relationship between 2 variables -have half points above and half below -gives slope (rate of change) domain - ANSWER the set of inputs, which corresponds to the x-values of the data points when plotted on graph range - ANSWER set of y-values on graph outlier - ANSWER data point that does not follow the same overall trend as the other points growth and decay linear formula - ANSWER y= kx+ xo k= rate of change growth and decay exponential equation - ANSWER y= xo (1+r)^x r= rate of change scatterplots - ANSWER -make sure to only use relevant points quadratic model - ANSWER -u-shaped -have parabola -have vertex (min/max value) -open upward when x^2= positive, opens downward when negative coefficient correlation coefficient - ANSWER -r -indicates how well a regression equation fits the data -closer r is to 1 for positive better is, closer to -1 for negative better is distance - ANSWER equals rate times time when have circle question - ANSWER draw a diagram 1 pound equals - ANSWER 16 ounces 1 yard equals - ANSWER 3 feet "strong" relationship - ANSWER good relationship but not perfect if have a series of points - ANSWER make a table can do soh cah toa - ANSWER on the 90 angle if you want to divide an exponential equation (like from years to months) you must divide - ANSWER the x, not the rate of change g(f(2)) solve - ANSWER put 2 in for f and find what x equals -put that number in for x in the g(x) equation function notation instead of f(g(x)) - ANSWER f circle g if create an equality but don't see it in the answers - ANSWER you may need to just isolate one variable the coefficient should always be - ANSWER the rate per unit (3 followers per post, 0.2 followers per repost) unit conversion questions - ANSWER use what you learned in chemistry infinite solutions - ANSWER 2+ equations have the same slope and y-intercept -if have infinite solutions the equations are dependent dependent equations - ANSWER one equation can be manipulated algebraically to get the other one (independent equations are unable to do this) to get rid of decimals - ANSWER multiply the equation ratio - ANSWER a comparison of 1 quantity to another -you can compare a part to another part or a part to a group 1 cat : 2 dogs; 1 cat: 3 pets -if you have a:b and b:c you can get a:c biweekly - ANSWER divide by 26 Common Pythagorean triplets - ANSWER 3 4 5 and 5 12 13 45-45-90 - ANSWER 30-60-90 - ANSWER radius - ANSWER The distance from the center of a circle to its edge chord - ANSWER A line segment that connects two points on a circle diameter - ANSWER A chord that passes through the center of a circle. The diameter is always the longest chord a circle can have and is twice the length of the radius. arc - ANSWER is part of a circle's circumference. Both chords and radii can cut a circle into arcs. minor vs major arcs - ANSWER If only two arcs are present, the smaller arc is called the minor arc, and the larger one is the major arc. central angle - ANSWER When radii cut a circle into multiple (but not necessarily equal) pieces, the angle at the center of the circle contained by the radii is the central angle sectors - ANSWER Radii splitting a circle into pieces can also create sectors, which are parts of the circle's area. geometry equation - ANSWER arc length/c= central angle/360= sector area/circle area inscribed angle - ANSWER An angle whose vertex is on the edge of the circle is called an inscribed angle tangent line - ANSWER A tangent line touches a circle at exactly one point and is perpendicular to a circle's radius at the point of contact. percent formula - ANSWER part/whole times 100% percent increase or decrease - ANSWER amount of increase or decrease/original times 100% 2 way table - ANSWER table that contains data on 2 variables -can be used to make comparisons and determine whether relationships exist btwn variables (bivariate data) if trying to find percent of majority - ANSWER 100- percent minority mean - ANSWER -also called average -the sum of the values divide by the number of values median - ANSWER the value that is in the middle of the set when the values are arranged in ascending order (least to greatest) mode - ANSWER the value that occurs most frequently -a set of data can have multiple modes standard deviation - ANSWER a measure of how far a typical data point is from the mean -a low standard deviation means most values in the set are fairly close to the mean, a high standard deviation means there is much more spread in the data set margin of error - ANSWER -a description of the max expected difference between a true value for a data pooling and a random sampling from the data pool -a lower margin of error is achieved by increasing the size of the data pool measures of central tendency - ANSWER mean, median, mode -can be used to represent a typical value in the data set shape of data - ANSWER -symmetric, or skewed (asymmetric) -have a head (many data points are clustered) and a tail (where number of data points slowly decreases to zero) skewed to the left - ANSWER skewed to the right - ANSWER symmetric data - ANSWER -if data evenly spaced, the mean and median will be the same -you can find the mean by taking the mean of the highest and lowest values ((12+2)/7) probability - ANSWER a fraction or decimal comparing the number of desired outcomes to the number of total possible outcomes desired outcomes/total outcomes probability of a series of events - ANSWER -multiply the probability of the first event by that of the second "with replacement" vs "without replacement" - ANSWER -with replacement- the item chosen is returned to the original group, denominator stays the same -without replacement-the item will not be returned, number of possible outcomes will change to reflect that negative correlation - ANSWER as 1 increases another decreases if have a^(2/6) - ANSWER simplify the fraction then convert when looking at a graph/word problem - ANSWER underline the units equation for exponential lines - ANSWER y= a (b)^x x-intercept - ANSWER x, 0 y-intercept - ANSWER 0, y when asked to find percent increase - ANSWER you can't add/subtract percent, use the number 100 graph of function - ANSWER -if touches then parenthesis is squared, if crosses then is not g(x-3) compared to g(x) - ANSWER the graph of g(x-3) is 3 more to the right undefined part on graph - ANSWER has a circle (not filled in) f(x)= - ANSWER y y intercept is f(0) function equation - ANSWER f(x)= kx+f(0) is linear g(x)=g(0)(1+r)^x piecewise functions - ANSWER -a function that is defined literally by multiple pieces -open dot= not included -have different rules based on what the x value is -= means 1 point greater than or less than and equal to means closed dot, opposite means open dot transformation - ANSWER occurs when a change is made to the function's equation or graph -include translations (moving graph up/down, left/right), reflections (flips about an axis or other line), and expansions/compressions (stretching or squashing horizontally or vertically) f(x+a) - ANSWER moves f(x) left a units f(x)+a - ANSWER f(x) moves up a units f(x-a) - ANSWER f(x) moves right a units f(x)-a - ANSWER f(x) moves down a units -f(x) - ANSWER f(x) is reflected over the x axis f(-x) - ANSWER f(x) is reflected over the y axis af(x) and a is fraction - ANSWER undergoes vertical compression (fatter) f(ax) and a is a fraction - ANSWER undergoes horizontal expansion af(x) and a is greater than 1 - ANSWER undergoes vertical expansion f(ax) and a is greater than 1 - ANSWER undergoes horizontal compression quadratics standard form - ANSWER ax^2+bx+c discriminant - ANSWER (b^2-4ac) -if positive, equation has 2 distinct real solutions, if 0 then 1 real solution, if negative there are no solutions factored form - ANSWER y= a(x-m)(x-n) x intercepts are m and n -vertex is halfway between m and n vertex form - ANSWER y= a(x-h)^2 + k h, k is vertex -axis of symmetry x=h -min/max is k finding x coordinate vertex - ANSWER h= -b/2a How do you divide by a fraction? - ANSWER To divide by a fraction, multiply by its reciprocal. How do you multiply fractions? - ANSWER To multiply fractions, simply multiply the numerators and multiply the denominators. For example" 5/7 x 3/4 = 5x3/7x4 = 15/28 How do you add/subtract fractions? - ANSWER To add or subtract fractions, first rewrite the fractions with a common denominator, then add or subtract the numerators. For example: 2/15 + 3/10 = 4/30 + 9/30 = (4 + 9) / 30 = 13/30 How do you simplify a fraction to lowest terms? - ANSWER To reduce a fraction to lowest terms, divide the numerator and denominator by their greatest common factor. For example, to write 28/36 in lowest terms, first find the GCF of the numerator and denominator. In this case, it's 4. Divide the numerator and denominator by 4 to reduce the fraction to 7/9. 28/36 = 7x4 / 9x4 = 7/9 What is the definition of a remainder? - ANSWER To remainder is the whole number left over after divisions. The remainder is always smaller than the number you are dividing by. For example, 487 is 2 more than 485, which is a multiple of 5, so when 487 is divided by 5, the remainder will be 2. How do you know if an integer is a multiple of 5 or 10? - ANSWER An integer is divisible by 5 if the last digit is 5 or 0. An integer is divisible by 10 if the last digit is 0. The last digit of 665 is 5, so 665 is divisible by 5 and therefore a multiple of 5 but not a multiple of 10. How do you know if an integer is a multiple of 2 or 4? - ANSWER An integer is divisible by 2 if the last digit is even. An integer is divisible by 4 if the last two digits are divisible by 4. The last digit of 562 is 2, which is even, so 562 is a multiple of 2. The last two digits are 62, which is not divisible by 4, so 562 is not a multiple of 4. The integer 512, however, is divisible by four because the last two digits form 12, which is divisible by 4. How do yo know if the sum, difference, or product of several numbers will be even or odd? - ANSWER To predict whether a sum, difference, or product will be even or odd, just take simple numbers like 1 and 2 and see what happens. There are rules -- "odd times even is even," for example -- but there's no need to memorize them. What happens with one set of numbers generally happens with all similar sets. How do you find the greatest common factor (GCF) of two or more numbers? - ANSWER To find the greatest common factor of two or more numbers, multiply all the prime factors they have in common. For example: 36 = 2 x 2 x 3 x 3, and 48 = 2 x 2 x 2 x 2 x 3. What they have in common is two 2s and one 3, so the GCF is 2 x 2 x 3 = 12. How do you find the least common multiple (LCM) of two or more numbers? - ANSWER To find the least common multiple, make a list of the multiples of the larger number until you find one that's also a multiple of the smaller. To find the LCM of 12 and 15, begin by taking the multiples of 15 : 15 is not divisible by 12; 30 is not; nor is 45. But the next multiple of 15, 60, is divisible by 12, so it's the LCM. What are numbers that are relatively prime? - ANSWER Two relatively prime numbers are integers that have no common positive factors other than 1. To determine whether two integers are relatively prime, compare their prime factorizations. For example: 35 = 5 x 7, and 54 = 2 x 3 x 3 x 3. They have no prime factors in common, so 35 and 54 are relatively prime. What is the prime factorization of a number? - ANSWER The prime factorization of a number is the expression of the number as the product of its prime factors. The easiest way to determine a number's prime factorization is to figure out a pair of factors of the number and then determine their factors, continuing the process until you are left with only prime numbers. Those primes will be the prime factorization. What is the difference between a factor and a multiple? - ANSWER The factors of integer n are the integers that divide into n with no remainder. The multiples of n are the integers that n divides into with no remainder. Factors and multiples are related in that if A is a factor of B then B is a multiple of A. For example, 6 is a factor of 12, and 24 is a multiple of 12. 12 is both a factor and a multiple of itself, since 12 x 1 = 12 and 12 / 1 = 12. What are the characteristics of a rectangle? - ANSWER A rectangle is a four-sided figure with four right angles. Opposite sides are equal. Diagonals are equal. The perimeter of a rectangle is equal to the sum of the lengths of the four sides, which is equivalent to 2(length + width). The area of a rectangle = length x width. The perimeter of a 7-by-3 rectangle is 2(7 + 3) = 20. The area of a 7-by-3 rectangle is 7 x 3 = 21. What are the characteristics of a parallelogram? - ANSWER A parallelogram has two pairs of parallel sides. Opposite sides are equal. Opposite angles are equal. Any two consecutive angles sum to 180 degrees. The area of a parallelogram = base x height. What are the characteristics of a square? - ANSWER A square is a rectangle with four equal sides. The perimeter of a square is equal to four times the length of one side. The area of a square is equal to the length of the side squared. If the side of a square equals 5 units, its perimeter is 20 units and its area is 5^2 = 25 square units. If f(x) = x^1/3 + 1/3x, then what is the value of f(27)? - ANSWER Answer: 12 Strategic Advice: First, substitute for the variable. Then evaluate the expression using the correct order of operations. Getting to the Answer: After substituting, the expression becomes: 27^1/3 + 1/3 (27) Recall that 27^1/3 is the same as the cube root of 27, which is equal to 3,and 1/3(27) = 9. 3 + 9 = 12 If 4^x^2 - 10x + 25 = 1, what is the value of x? - ANSWER Answer: 5 Strategic Advice: Don't be intimidated by the complex exponent. Remember that any non-zero base to the power of zero is equal to one. Getting to the Answer: Set the exponent equal to zero and solve for x. x^2 - 10x + 25 = 0 (x - 5)(x - 5) = 0 x - 5 = 0 x = 5 If the average (arithmetic mean) of a, b, and 48 is 48, what is the value of a + b? - ANSWER Answer: 96 Strategic Advice: Notice that this question did not ask for the value of a or b, but for the sum of a and b. The average of a set of numbers is the sum of the numbers divided by the number of terms. Getting to the Answer: Since the average of the three numbers if 48, then a + b + 48 / 3 = 48 or a + b + 48 = 144 Thus, a + b = 96. If x^2 = 7x + 18, what is the positive value of x? - ANSWER Answer: 9 Strategic Advice: To solve a quadratic equation, set the equation equal to zero and solve for the factors of the expression. Getting to the Answer: First, subtract 7x and 18 from both sides of the equation. Then, factor the trinomial into two factors. Set each factor equal to zero to find the values of x. x^2 = 7x + 18 x^2 - 7x - 18 = 0 (x - 9)(x + 2) = 0 x - 9 = 0 or x + 2 = 0 x = 9 or - 2 The positive value of x is 9. Cole biked 20 miles in 60 minutes on Monday and 7 miles in 30 minutes on Tuesday. What is his average rate of speed in miles per hour? - ANSWER Answer: 18 Strategic Advice: Average speed is the total distance divided by the total time. Convert the units as necessary. Don't be fooled; average rate is not the average of the two rates. Getting to the Answer: Since the problem asks for the rate in miles per hour, convert from minutes to hours: 60 minutes = 1 hour, 30 minutes = 1/2 hour. Average miles per hour = Total miles / Total hours Average miles per hour = 20 + 7 / 1 + 1/2 = 27 / 1 1/2 = 18 If the first term of a geometric sequence is 4, and the fourth term is 108, then what is the common ratio? - ANSWER Answer: 3 Strategic Advice: ???????? :) Each time a class of students is divided into groups of 2, 3, or 5, there is one student left over. What is the minimum number of students in the class? - ANSWER Answer: 31 Strategic Advice: First, use the least common multiple of 2, 3, and 5 to find the least number of students if no students were left over. Then, add one ( for the one student that was left over) to that result. Getting to the Answer: The least common multiple of 2, 3, and 5 is 30. Therefore, the least number of students in the class is 30 + 1 = 31. Let r # t = t^r - r^t for all positive integers r and t. What is the value of 4 # 3? - ANSWER Answer: 17 Strategic Advice: Recall that symbol questions are really just asking you to substitute for variables, and then evaluate using the correct order of operations. Getting to the Answer: Substitute r = 4 and t = 3 and evaluate the exponents before subtracting the expression. 4 # 3 = 3^4 - 4^3 = 81 - 64 = 17 What is the length of a leg of an isosceles right triangle whose hypotenuse measures 24(square root of)2? - ANSWER Answer: 24 Strategic Advice: This question appears to have some information missing. However, the fact that the triangle is isosceles makes the problem possible to solve. What strategy would you use to complete the problem below? If the sale price of a jacket is marked down 30% each week for a total of three weeks, what percent of the original price is the cost of the jacket after the three week sale period? - ANSWER Strategy: Picking Numbers Answer: 34.3 Strategic Advice: Since you are not given the price of the jacket, picking numbers is a good strategy. When dealing with percents it is often easiest to start with $100. Getting to the Answer: Suppose the original cost is $100. Then, the cost after the first week will be 100 - 0.30(100) = 100 - 30 = $70. The cost after the second week will be 70 - 0.30(70) = 70 - 21 = $49. The cost after the third week will be 49 - 0.30(49) = 49 - 14.70 = $34.30. Because you are comparing $34.30 with the original cost of $100, the sale price will e 34.3 percent of the original cost of the jacket. If 2 + [6-x] = 10 and x 0, what is the value of 2x? - ANSWER Answer: 28 Exponential Growth - ANSWER y= A(1+r)^x Exponential Decay - ANSWER y=A(1-r)^x Histogram Average - ANSWER (# on y axis x # on x axis)+ (# on y axis x # on x axis) /total number given Difference in Histograms - ANSWER -subtract y axis' from another and then divide by total number Bar Graph Average - ANSWER y axis+ y axis/ total number of x variables Line Graph Average - ANSWER highest point- lowest point/ number of variables on x axis Value of Stock Increase on a Scatter Plot - ANSWER percent increase formula Yearly or Point to Point on a Scatter Plot (Average Increase) - ANSWER slope Biggest Difference in Scatter Plot - ANSWER predicted- actual Unbiased - ANSWER everyone has an equal chance of participation Biased - ANSWER not everyone has an equal chance, and is catered to a specific group Average or the Mean - ANSWER sum of all the values/ how many numbers there are Finding an Unknown Number When an Average is Given - ANSWER sum of all the values = average x n Weighted Average - ANSWER (average x number) + (average x number) / total number of all the values in the set Median - ANSWER the middle number - if its an odd amount of numbers then the middle value is the median -if it is an even amount of numbers then the last two numbers left in the middle you find the average of Mode - ANSWER Is the number that appears the most Range - ANSWER Is the difference between the largest number and the smallest number When Transforming Data Each Value is Increased/Decreased by the Same Amount Then - ANSWER The Mean, Median, Mode are also increased or decreased by that number When Transforming Data and Each Value is Multiplied/ Divided by the Same Amount Then - ANSWER The Mean, Median, Mode are also multiplied or divided by that number x^0 - ANSWER equals 1 x^-1 - ANSWER 1/x 1/x -1 - ANSWER x 1/n - ANSWER n on the outside of the radial and x or the other number on the inside x^a/n - ANSWER n goes on the outside and x goes inside raised by a number Fractional Exponents - ANSWER 1) multiply both sides of the equation by the reciprocal of the fraction 2) using exponent rules convert the right side 3) parentheses and solve Exponential Equations - ANSWER 1) have to convert the big numbers the same 2) set equations equal to each other i^0 - ANSWER 1 i^1 - ANSWER i i^2 - ANSWER -1 i^3 - ANSWER -i i^4 - ANSWER 1 Breaking Down i - ANSWER divide by 4 and the remainder is the answer Completing Perfect Square - ANSWER (b/2)^2 a0 - ANSWER parabola goes up a0 - ANSWER parabola goes down Parabola up Means Vertex is - ANSWER the minimum point Parabola down Means Vertex is - ANSWER the maximum point Equation for Line of Symmetry or Vertex - ANSWER b/-2a What are the two Equations for parabolas and how do you Find the Roots - ANSWER A) ax^2 + bx + c or b) a(x-h)+k c) opposite of H and K is the same sign Line of Symmetry and Perpendicular Bisector - ANSWER add the two corresponding points of what you are looking for and divide by 2 Modeling Projectile Motion y- axis Means - ANSWER the height Modeling Projectile Motion Vertex is - ANSWER when the ball reaches max height Modeling Projectile Motion x - axis Means - ANSWER time elapsed Modeling Projectile Motion Finding the Max Height - ANSWER using line of symmetry equation (x= b/-2a) find x then plug x in for why Modeling Projectile Motion y - axis Means - ANSWER the height Modeling Projectile Motion Vertex Also Means - ANSWER max height Modeling Projectile Motion Finding the Vertex - ANSWER 1) use the line of symmetry equation to find x (x=b/-2a) 2) plug x in the equation Modeling Projectile Motion When the ball hits the Ground - ANSWER 1) set equation equal to 0 2) pull out a GCF 3) set the forward number equal t ozero and solve 4) set the inside equation equal to zero and solve * 0 is the time from where the ball is tossed * the other number is how long it takes to hit the ground GCF in Quadratic Formula - ANSWER goes away GCF in Completing the Square - ANSWER doesn't go away y= f(-x) - ANSWER reflection or flip over the y axis y= -f(x) - ANSWER reflection or flip over x axis y= (x)+k or y=(x)-k - ANSWER goes up if positive goes down if negative *vertical shift y= f(x-h) - ANSWER slide to the right bc its the opposite *horizontal shift y=f(x+h) - ANSWER slide to the left bc its the opposite *horizontal shift [x] + 2 - ANSWER shift up [x + 2} - ANSWER shift to the left Vertical Angles - ANSWER add to 360 Corresponding Angles - ANSWER Sum of Triangles adds to - ANSWER 180 deegrees Sum of Interior Angle Measure of Polygons - ANSWER (number of sides-2) x 180 Measure of Exterior Angles in a Polygon - ANSWER 360/ number of sides Measure of Interior Angles in a Polygon - ANSWER 180 - 360/ number of sides Isosceles Triangles - ANSWER angles with equal sides have equal measure Equilateral Triangles - ANSWER all three sides have the same measure Right Triangle - ANSWER 45 45 90 Right Triangle - ANSWER 60 30 90 Right Triangle - ANSWER 4 Characteristics of a Parallelogram - ANSWER 1) quadrilateral where opposite sides are parallel 2) opposite sides are the same length 3) opposite angles have the same measure 4) diagnosis bisect each other 3 Characteristics of a Rectangle - ANSWER 1) parallelogram with 4 right angles 2) all same properties as a parallelogram 3) diagonals are the same length Rhombus - ANSWER Parallelogram where all the sides are the same - diagonals are perpendicular to the bisector Square - ANSWER Diagonals are the same length Area of a Parallelogram - ANSWER A= b x h Area of a Rectangle - ANSWER A= b x h Area of a Rhombus - ANSWER A= 1/2 (dxD) Area of Square - ANSWER a= s^2 or a= 1/2 (d)^2 Area of Triangle - ANSWER a= 1/2 bxh Area of an Equilateral Triangle - ANSWER s^2 / 4 radical 3 Area of a Hexagon - ANSWER Equals 720 degrees equilateral formula times 6 Area of Trapezoid - ANSWER quadrilateral with one parallel side a= h x 1/2 (b+B) Radius of a Circle - ANSWER segment whose endpoints are in the center half of the diameter Chord of Circle - ANSWER are any end points through the circle Diameter of Circle - ANSWER chord that runs through the center of the circle Tangent of a Circle - ANSWER circle of a line that intersects at exactly one point Arc of a Circle - ANSWER curved section of a circle number of degrees of an arc is 360 Minor Arc of a Circle - ANSWER an arc smaller than the semi circle Major Arc of a Circle - ANSWER an arc larger than the semi circle Central Angle of a Circle - ANSWER Vertex is at the center and whose sides are the radius Inscribed Angle - ANSWER Vertex is on the circle and whose sides intercept an arc of a circle Circumference of a Circle Equations - ANSWER c= πD or c-2πr Length Circle - ANSWER n/360 x π r^2 Area of a Circle - ANSWER a= π r^2 Sector of a Circle - ANSWER n/360 x π r ^2 Central Radius Formula - ANSWER (x-h)^2 + (y-k)^2 = r^2 Distance Formula of a Box - ANSWER (l^2 + w^2 + h^2) parentheses is a radical Surface Area Formula of a Box - ANSWER 2(lxw + lxh + wxh) Surface Area Formula of a Cylinder - ANSWER SA= 2π r^2 + 2π rh area of the bases times the lateral height Another Formula of a Right Cone and a Pyramid - ANSWER V= 1/3 BxH How to Find the Area of a Rectangle From a Cylinder - ANSWER Circumference x height Radian and Degree Formula - ANSWER 2π radians = 360 π radians = 180 Converting Degrees to Radians - ANSWER number of degrees x π/180 Converting Radians to Degrees - ANSWER number of radians x 180/π 30 Degrees = - ANSWER π/6 45 Degrees= - ANSWER π/4 90 Degrees= - ANSWER π/2 Complementary Angles = - ANSWER 90 degrees or π/2 Standard Position - ANSWER remember the triangle Unit Circle With Units - ANSWER Unit Circle With Pi - ANSWER (0,1) equals π/2 (1,0) equals 2π (0,-1) equals 3π/2 (-1,0) equals π cos = sin= - ANSWER cos is x sin is y Arc Length Means - ANSWER SQR triangle Travel Counter Clockwise - ANSWER + Travel Clockwise - ANSWER - Strategic Advice: In any absolute value equation, isolate the part of the equation with the absolute value, then solve. Keep in mind you are looking for the positive value of x, since x 0. Getting to the Answer: Subtract 2 from each side of the equation to get the absolute value by itself. Then set 6 - x equal to 8 and solve. 2 + [6 - x] = 10 [6 - x] = 8 6 - x = 8 or - (6 - x) = 8 -x = 2 -6 + x = 8 x = -2 x = 14 Since the positive value of x is 14, the value of 2x is 2(14) = 28. Owen picked 1/3 of the apples on a tree in the first hour he picked, and then picked 1/2 of the remaining apples the second hour. If there were 100 apples remaining on the tree after two hours, how many apples were on the tree before Owen started picking? - ANSWER Answer: 300 Strategic Advice: Remember, after picking n/m of the apples, 1 - n/m of the apples remain on the tree. Getting to the Answer: Since Owen picked half of the apples that remained in the second hour, there were 200 left after the first hour. In the first hour, Owen picked 1/3 of the total apples in the tree, so 1 - 1/3 = 2/3 of the apples remained. If x is the number of apples that were on the tree before Owen started picking, then 2/3x = 200, so x = 300. What is the greatest of three consecutive odd integers whose sum is 75? - ANSWER Answer: 27 Strategic Advice: Consecutive odd integers are odd integers in order in a row, such as 1, 3, and 5, or -11, -9, and -7. The difference between any two consecutive odd integers is 2. Translate the question into an equation that expresses the relationship between the integers. Getting to the Answer: Let x = the 1st odd integer, x + 2 = the 2nd odd integer, and x + 4 = the 3rd odd integer. x + (x + 2) + (x + 4) = 75 3x + 6 = 75 3x = 69 x = 23 So, the largest integer is (23 + 4) or 27. At a clothing outlet, two shirts and one tie cost $39. At the same store, one shirt and one tie cost $24. At this rate, what is the cost of four shirts? - ANSWER Answer: 60 Strategic Advice: Let s = cost of one shirt and t = cost of one tie, and solve the system of equations. The solution is 4 times the value of x. Getting to the Answer: Use the equations: 2s + t = 39 and s + t= 24. By subtracting the two equations, the result is s = 15. Therefore, the cost of one shirt is $15, so the cost of 4 shirts is 4 x 15 = $60. A jar contains twice as many blue marbles as green, and three times as many red marbles as blue. If the jar contains only blue, green and red marbles, and there are a total of 27 marbles in the jar, how many red marbles does it contain? - ANSWER Answer: 18 Strategic Advice: Find the color of marble that has the least amount and start from there. Let x represent the amount of marbles of that color, and express each of the other colors in terms of x. Getting to the Answer: Let x = the number of green marbles. Therefore, 2x = the number of blue marbles, and 3(2x) = 6x = the number of red marbles. The total numbers is 27, so set up the equation: x + 2x + 6x = 27 9x = 27 x = 3 There are 6(3) = 18 red marbles. If x varies directly with y, and the value of x is 12 when y is 11, what is the value of y when x is 66? - ANSWER Answer: 60.5 Strategic Advice: If two values vary directly, the values increase at the same rate. Express the relationship as x/y, and use a proportion to solve. Getting to the Answer: Set up the values as x/y, so 12/11 = 66/y. Cross-multiply to get 12y = 726. y = 60.5 What is the slope of the line perpendicular to the line with linear equation 4x + 2y = 12? - ANSWER Answer: 1/2 or .5 Strategic Advice: Use the formula y = mx + b, and remember that the value of m is the slope of the line. The slopes of two perpendicular lines are negative reciprocals of one another. Getting to the Answer: First, change the given equation to y = mx + b form. 4x + 2y = 12 2y = -4x + 12 y = -2x + 6 Since the slope of this line is -2, the negative reciprocal is 1/2. If 3^3x = 9^3 + 2, what is the value of x? - ANSWER Answer: 4 Strategic Advice: First, get the base numbers to be the same. Then, set the exponents equal to each other and solve for x. Getting to the Answer: Change the base of 9 to 3^2. The expression then becomes: 3^3x = 3^2(x + 2) Set the exponents equal. 3x = 2(x + 2) 3x = 2x + 4 x = 4 What value of x is not in the domain of the function? f(x) = x - 2/ x - 3? - ANSWER Answer: 3 Strategic Advice: Values of x that make the denominator of a fraction equal to zero-- thereby dividing by zero, which is not possible-- are not in the domain of a function. Set the denominator equal to zero and solve for x. Getting to the Answer: x - 3 = 0 x = 3 What is the positive difference between the answers to the equation 35 + x^2 = 12x? - ANSWER Answer: 2 Strategic Advice: Set the quadratic equation equal to zero with the terms in order of decreasing exponents, and solve for the roots by factoring. Don't forget to then find the difference in the roots by subtracting them. Getting to the Answer: 35 + x^2 = 12x x^2 - 12x + 35 = 0 (x - 5)(x - 7) = 0 x - 5 = 0 or x - 7 = 0 x = 5 or 7 The positive difference is 7 - 5 = 2. Let N represent the smallest positive integer that is a multiple of 6 and 8, but leaves a remainder of 2 when divided by 7. What is the value of N? - ANSWER Answer: 72 Strategic Advice: Use common multiples of 6 and 8, and divide by 7 to find a remainder of 2. Getting to the Answer: Find the common multiples of 6 and 8 first, and then locate the smallest one whose remainder is 2 when divided by 7. The least common multiple of 6 and 8 is 24. 24 / 7 = 3 with a remainder of 3, so this value does not work. Other common multiples of 6 and 8 are multiples of 24: 48, 72, 96, 120... Try 48: 48 / 7 = 6 with a remainder of 6, so this value does not work. However, 72 / 7 = 10 with a remainder of 2, so 72 is the correct value of N. If y varies inversely with x, and y is 3 when x is 10, what is the value of x when y is 6? - ANSWER Answer: 5 Strategic Advice: Variables that have an inverse relationship will always have the same product. As one value gets larger, the other gets smaller. Set up and solve the equation: 3 x 10 = 6 x X. Getting to the Answer: 3 x 10 = 6 x X 30 = 6x 5 = x What is the value of x^2 - y^2, if x - y = 5 and x + y = 7? - ANSWER Answer: 35 Strategic Advice: Don't be confused by all of the variables. Recall that (x - y)(x + y) = x^2 = y^2. Getting to the Answer: Since (x - y)(x + y) = x^2 - y^2, then x^2 - y^2 = 5 x 7 = 35. A buffet is set up to have the same number of apples and oranges. After 6 o