SAT- Math
How do you know if an integer is a multiple of 5 or 10?
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.
What is the Triangle Inequality Theorem?
The Triangle Inequality Theorem says that the length of one side of a triangle must be greater than the difference and less than the sum of the lengths of the other two sides. For example, if it is given that the length of one side is 3 and the length of another side is 7, then you know that the length of the third side must be greater than 7 - 3 = 4 and less than 7 + 3 = 10.
How do you determine the absolute value of a number?
The absolute value of a number is the distance of the number from zero on the number line. Because absolute value is a distance, it is always positive. The absolute value of 7 is 7. Similarly the absolute value of -7 is 7. Every positive number is the absolute value of two numbers: itself and its opposite.
How do you calculate the total number of possibilities for several events to occur?
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.
What is the prime factorization of a number?
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 are the properties of a 30-60-90 triangle?
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.
How do you express the union and intersection of sets?
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 find the missing number in a series when you are given the average?
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.
What is the formula used to find average?
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 multiply fractions?
To multiply fractions, simply multiply the numerators and multiply the denominators. For example" 5/7 x 3/4 = 5x3/7x4 = 15/28
What are numbers that are relatively prime?
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.
How do you factor the differences of two squares?
One of the test maker's favorite expressions to factor is the difference of squares. a^2 - b^2 = (a + b)(a - b) For example, x^2 - 9 factors to (x + 3)(x - 3)
How do you add and subtract roots?
Only like radicals can be added/subtracted from one another. To combine radicals, just add or subtract their coefficients.
What are the properties of similar triangles?
Similar triangles are triangles whose corresponding angles are equal and whose corresponding sides are proportional.
What are the properties of the interior and exterior angles of a triangle?
The 3 interior angles of any triangle add up to 180 degrees. An exterior angle of a triangle is equal to the sum of the remote, that is, non-adjacent, interior angles.
How do you find the median of a set of numbers?
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 find the midpoint between two points?
The midpoint of two points of a line segment is the average of the x-coordinates of the endpoints and the average of the y-coordinates of the endpoints. If the endpoints are (x^1, y^1) and (x^2, y^2), the midpoint is: (x^1 + x^2) / 2, (y^1 + y^2) / 2 For example, the midpoint of (3,5) and (9,1) is: (3 + 9) / 2, (1 + 5) / 2 = (12 / 2, 6 / 2) = (6,3)
How do you find the sum of the interior angles of a polygon?
The sum of the measures of the interior angles of a polygon = (n - 2) x 180, where n is the number of sides. For example, the eight angles of an octagon add up to (8 - 2) x 180 = 1,080 degrees.
How do you convert from an improper fraction to a mixed number?
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 count consecutive integers?
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 divide by a fraction?
To divide by a fraction, multiply by its reciprocal.
How do you divide expressions with exponents that have a common base?
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 evaluate an algebraic expression?
To evaluate an algebraic expression, substitute the given values for the unknowns and calculate according to the correct order of operations, or PEMDAS. For example, to find the value of x^2 = 5x - 6 when x = -2, replace x with -2: (-2)^2 + 5 (-2) - 6 = 4 + (-10) - 6 = -12.
How do you find the reciprocal of a fraction?
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 multiply and divide positive and negative numbers?
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.
How do yo know if the sum, difference, or product of several numbers will be even or odd?
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 raise a power to a power?
To raise a power to a power, multiply the exponents: (x^3)^4 = x^3 x 4 = x^12
How do you simplify a fraction to lowest terms?
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?
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 simplify square roots?
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.
What is an important property of a line tangent to a circle?
When a line is tangent to a circle, or intersects the circle exactly once, the radius of the circle is perpendicular to the line at the point of contact.
What types of angles are formed when two lines intersect?
When two lines intersect, four pairs of adjacent angles are formed and two pairs of vertical angles are formed. Vertical angles are equal. These adjacent angles are supplementary because they form a line.
Let r # t = t^r - r^t for all positive integers r and t. What is the value of 4 # 3?
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: 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.
How do you add/subtract fractions?
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 multiply monomials?
To multiply monomials, multiply the coefficients and the variables separately. Be sure to add the exponents when multiplying the bases. For example: 2a x 3a = (2 x 3) (a x a) = 6a^2
How do you find the surface area of a rectangular solid?
To find the surface area, find the sum of the area of each face. If the lenght is l, the width is w, and the height is h, the formula is: Surface Area = 2lw + 2wh + 2lh
How do you find the volume of a cylinder?
Volume of a Cylinder = (pi)r^2h
If f(x) = x^1/3 + 1/3x, then what is the value of f(27)?
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
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: 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 2 + [6-x] = 10 and x > 0, what is the value of 2x?
Answer: 28 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.
If the first term of a geometric sequence is 4, and the fourth term is 108, then what is the common ratio?
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: 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.
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: 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.
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: 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
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: 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 x^2 = 7x + 18, what is the positive value of x?
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.
If the average (arithmetic mean) of a, b, and 48 is 48, what is the value of a + b?
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 g(x) = x^2 + x - 20 and a is the x-coordinate of the point at which the graph of y = g(x) crosses the y-axis, what is the value of g(a)? (A) -20 (B) -5 (C) -4 (D) 0 (E) 5
Answer: A Strategic Advice: Making a simple graph of x^2 + x - 20 on your calculator makes this problem simple. Or, you can remember that when a line crosses the y-axis, its x-value is zero. Getting to the Answer: If you graph this on your calculator, you can see that the graph crosses the y-axis at x = -20. Or, plugging in zero for your a value, you can see that 0^2 + 0 = -20.
If [x - 3] = [y - 4], and y = 2, what is one possible value of x? (A) 1 (B) 2 (C) 4 (D) 6 (E) 8
Answer: A Strategic Advice: Remember that when you solve for the inside of an absolute value, you will have two solutions- one where the inside is a positive value, and one where the inside is a negative value. Getting to the Answer: First, substitute y = 2 into the right side of the equation and simplify. [x - 3] = [2 - 4] [x - 3] = [-2] [x - 3] = 2 So, x - 3 = 2 or x - 3 = -2 If x - 3 = 2, then x = 5. If x - 3 = -2, then x = 1. 5 is not among the answer choices, so choice (A), 1, is your correct answer.
The lengths of the sides of a right triangle are consecutive odd integers, and the length of the longest side is k. Which of the following equations could be used to find k? (A) (k - 4)^2 + (k - 2)^2 = k^2 (B) k + (k - 2) = (k - 4) (C) (k - 2) + (k - 1) = k (D) (k - 2)^2 + (k - 1)^2 = k^2 (E) (k - 3)^2 + (k - 1)^2 = k^2
Answer: A Strategic Advice: Since it's a right triangle, you'll probably use the Pythagorean Theorem. If the longest side is k, and the numbers are consecutive long integers, how would you write the other two sides? Getting to the Answer: The other two sides will be 2 less and 4 less, or (k - 2) and (k - 4). Now plug into the Pythagorean Theorem (remember that the longest side is the hypotenuse): (k - 4)^2 + (k - 2)^2 = k^2
What strategy would you use to complete the problem below? If g(x) is a linear function, and g(8) = g(7) + 3, which of the following expressions could represent g(x)? (A) 3x + 2 (B) 6x + 1 (C) x/3 (D) -3x (E) x + 3
Answer: A Strategic Advice: This problem can be solved by using a form of Backsolving. That is, substitute 7 and 8 into the answer choices to determine when g(8) = g(7) + 3. If (g)x = 3x + 2 g(8) = 3(8) + 2 = 26 g(7) + 3 = (3(7) + 2) + 3 = 21 + 2 + 3 = 26 Therefore g(8) = g(7) + 3 when (g)x = 3x + 2.
How would you approach the problem below? A riverboat leaves Mildura and travels upstream to Renmark at an average speed of 6 miles per hour. It returns by the same route at an average speed of 9 miles per hour. What is the average speed for the round trip, in miles per hour? (A) 7.0 (B) 7.2 (C) 7.5 (D) 7.8 (E) 8.2
Answer: B Strategic Advice: Average speed is equal to total distance divided by total time. Since no distance from Mildura to Renmark is given, pick a distance that is divisible by 6 and 9. Find the time spent on each leg; then divide the round-trip distance by the total time. Getting to the Answer: Try 18 since it is evenly divisible by both speeds. On the first leg of the trip, traveling 18 miles at 6 miles per hour will take 3 hours. On the return trip, traveling 18 miles at 9 miles per hour will take 2 hours. So the average speed for the entire trip is (18 + 18) / (3 + 2) = 7 1/5, or 7.2 miles per hour, choice (B).
w/10 = x/2 x/3 = y/5 y/7 = z/6 If w, x, y and z in the equations above are positive integers, which of the following is true? (A) w < x < y < z (B) x < z < y < w (C) x < z < w < y (D) y < z < x < w (E) z < x < y < w
Answer: B Strategic Advice: Cross-multiply so it is easier to compare the values of the variables. Getting to the Answer: First, get rid of all the fractions, and then divide each equation by a number that isolates one variable. w = 5x x = 3/5y y = 7/6z Cross multiply each proportion. w/10 = x/2 x/3 = y/5 y/7 = z/6 5x = w So, w = 5x = 3y and since 7z = 6y, 3.5z = 3y. Therefore, w = 5x = 3z = 3.5z. This means x is the smallest, then z, then y, and w is the largest: x < z < y < w
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: 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.
The number of ice cream cones the Ice Cream Hut sells is directly proportional to the highest temperature recorded that day. On a day with a high of 88 degrees, the Ice Cream Hut sold 220 ice cream cones. How many cones did the Ice Cream Hut sell on a day with a high of 94 degrees? (A) 226 (B) 235 (C) 286 (D) 308 (E) 385
Answer: B Strategic Advice: If two values vary directly, the value increase at the same rate. Express the relationship as cones/degrees, and use a proportion to solve. Getting to the Answer: cones/degrees = 220/88 Letting x equal the number of cones sold on the 94 degree day, the proportion is 220/88 = x/94. Cross multiply and solve to get x = 235.
Number of Bus Tickets - 1 (Total Price: $1.50) 8 (Total Price: $9.95) 15 (Total Price: $17.95) What would be the least amount of money needed to purchase exactly 27 bus tickets? (A) $29.40 (B) $33.90 (C) $34.35 (D) $35.95 (E) $40.50
Answer: B Strategic Advice: In this case, it's cheapest to buy the largest quantities possible. Getting to the Answer: First, buy as many books of 15 as possible (1). Then, buy as many books of 8 as possible (1). This leaves 4 to buy at the single rate. The total cost is 1($17.95) / 1($9.95) + 4($1.50) = $33.90
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: 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 the ratio of boys to girs in a class is 5 to 3, which of the following could not be the number of students in the class? (A) 32 (B) 36 (C) 40 (D) 48 (E) 56
Answer: B Strategic Advice: The ratio 5 boys to 3 girls tells you that for every 5 boys in the class there must be 3 girls in the class. So the total number of students in the class must be a multiple of 8, since the smallest possible total is 5 + 3, or 8. Getting to the Answer: Since 36 is not divisible by 8, 36 cannot be the total number of students.
Number of Bus Tickets - 1 (Total Price: $1.50) 8 (Total Price: $9.95) 15 (Total Price: $17.95) Using the prices listed above, which is the closest approximation of the cost per ticket when one purchases a book of 8 tickets? (A) $1.20 (B) $1.25 (C) $1.30 (D) $1.35 (E) $1.50
Answer: B Strategic Advice: This is straightforward division. Getting to the Answer: 8 tickets cost $9.95, so they cost 9.95 / 8 = 1.24; (B) is closest.
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: B 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
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: 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.
If the average (arithmetic mean) of t and s is 25 and the average of r and s is 40, what is the value of r - t? (A) 65 (B) 32.5 (C) 30 (D) 15 (E) It cannot be determined from the information given.
Answer: C Strategic Advice: If the average of r and s is 4, then you know that r + s = 80. If the average of t and s is 25, t + s = 50. Getting to the Answer: Subtract the equation t + s = 50 from the equation r + s = 80: t + s = 80 / -(t + s = 50) = r - t = 30
If f(x) = 1/x^2, what is the domain of f(x)? (A) all real numbers (B) all positive numbers (C) all real numbers except zero (D) all integers (E) all integers except zero
Answer: C 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).
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: 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.
If [-3x - 7] = 5, x = (A) -2/3 (B) -4 (C) -2/3 or - 4 (D) 2/3 or - 4 (E) 4
Answer: C 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
A buffet is set up to have the same number of apples and oranges. After 6 oranges and no apples are eaten, there are three times as many apples as oranges. How many apples are on the buffet? (A) 3 (B) 6 (C) 9 (D) 12 (E) 15
Answer: C Strategic Advice: Say you started with x of each. Write an equation for the amounts after the oranges were eaten. Getting to the Answer: We were left with x - 6 oranges and 3 times this the number of apples (which is still x): 3(x - 6) = x 3x - 18 = x -18 = -2x 9 = x You could also try Backsolving to see which answer gives you the same number back when you subtract 6 and multiply 3.
In the xy-plane, at what point does the graph of the equation x + 3y = 9 cross the y-axis? (A) (-3, 0) (B) (0, -3) (C) (-, 3) (D) (0, 9) (E) (9, 0)
Answer: C Strategic Advice: Since every point on the y-axis has an x-coordinate of zero, when a line crosses the y-axis, its x-coordinate is zero. This already eliminates (A) and (E). Getting to the Answer: Substitute x = 0 into the equation to find the y-coordinate when it crosses the y-axis. x + 3y = 9 0 + 3y = 9 3y = 9 y = 3 So the line crosses the y-axis at (0, 3), making (C) correct.
What strategy would you use to solve the problem below? The price of a refrigerator was first decreased by 20 percent and then the new price was increased by 15 percent. The final price was what percent of the initial price? (A) 87% (B) 90% (C) 92% (D) 94% (E) 97%
Answer: C Strategic Advice: Since you are not given the price of the refrigerator, picking numbers is a good strategy. When dealing with percents it is often easiest to start with $100. Getting to the Answer: If the fridge started at $100, then it dropped by 20% of $$100 = 2 x 100 = $20, so the price changed to $80. Then it increased by 15% of $80 = .15(80) = $12, so the final price was $92. This price is 92/100 = 92% of the original price.
A 60-second clock by a pool allows swimmers to time their laps via a second hand that moves around from 1 to 60. When the second hand is pointing at 49, where is the tail of the second hand pointing? (A) 3 (B) 11 (C) 19 (D) 23 (E) 29
Answer: C Strategic Advice: Sketch the clock and look for the pattern. Getting to the Answer: When the second hand points to 15, the tail points to 45 = 15 + 30. In general, the tail will always point to 30 less or 30 more than the head (depending on whether the head points to a number between 1 and 30 or a number between 31 and 60). So if the head points to 49, the tail points to 49 - 30 = 19.
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: 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.
In a weight loss competition, Fabian noticed that he is both the 14th lightest in his group and the 14th heaviest in his group. If everyone in his group has a different weight, how many people are in the group? (A) 25 (B) 26 (C) 27 (D) 28 (E) 29
Answer: C Strategic Advice: There are 13 people above him, 13 below him, and he is in the middle. In other words, his weight is the median of the group. Getting to the Answer: There are 13 + 1 + 13 = 27 people.
36 percent of 18 is 18 percent of what number? (A) 9 (B) 24 (C) 36 (D) 40 (E) 48
Answer: C Strategic Advice: This question involves a principle that appears frequently on the SAT: a percent of b = b percent of a. Getting to the Answer: 36% of 18 = ? 6.48 = 18x 6.48/.18 = x 36 = x
Which of the following lines is parallel to the line y = 4x - 6? I. y = x - 6 II. y = 6x - 4 III. y = 4x + 6 (A) I only (B) II only (C) III only (D) I and II only (E) II and III only
Answer: C Strategic Advice: Two lines are parallel if their slopes are the same. When a line is written in the form y = mx + b, m equals the slope of the line. Getting to the Answer: Only III has the same slopes as the line given in the question stem, so (C) is correct.
n and p are inversely proportional, If p = 100 when n = 2, what is n when p = 2? (A) 25 (B) 50 (C) 100 (D) 200 (E) 400
Answer: C 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: 100(2) = pn Getting to the Answer: 100 = n If p = 2, pn = 200, and n = 100. Answer (C) is correct.
If x is an integer and 2 is the remainder when 3x - 2 is divided by 4, then what is a possible value of x? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5
Answer: D Strategic Advice: Backsolving this one is the way to go. Getting to the Answer: (A) x = 1, (3x - 2) / 4 = 1/4 = 0 r 1 (B) x = 2, (3x - 2) / 4 = 4/4 = 1 r 0 (C) x = 3, (3x - 2) / 4 = 7/4 = 1 r 3 (D) x = 4, (3x - 2) / 4 = 10/4 = 2 r 2 (E) x = 5, (3x - 2) / 4 = 13/4 = 3 r 1 Only (D) has a remainder of 2.
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: 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? 1. Choose a number between 0.25 and 0.75. 2. Multiply the number from the previous line by 100. 3. Divide the number found in the previous step by 5. 4. Determine the greatest integer less than or equal to the number obtained from the previous step. 5. Print the resulting number. Which of the following could be a number printed in step 5 after steps 1 through 4 are performed? (A) 0 (B) 4 (C) 11.4 (D) 12 (E) 17
Answer: D Strategic Advice: Find the minimum and maximum values that could be printed out based on the range of the initial number in step 1. Getting to the Answer: Step 1: Number is between .25 and .75 Step 2: New number is between 25 and 75 Step 3: New number is between 5 and 15 Step 4: New number is an integer between 5 and 15 The only answer choice that works is (D).
If [2x - 9] =33, x = (A) -21 (B) -12 (C) -12 or -21 (D) -12 or 21 (E) 12
Answer: D Strategic Advice: If [2x - 9] = 33, then 2x - 9 can equal 33 or -33. Getting to the Answer: 2x = -24 or 2x = 42 x = -12 or 21
A subway car passes 3 stations every 10 minutes. At this rate, how many stations will it pass in 1 hour? (A) 2 (B) 12 (C) 15 (D) 18 (E) 30
Answer: D Strategic Advice: Remember to pay attention to your units. There are 60 minutes in an hour. Getting to the Answer: The subway will pass 6 times as many stations in 60 minutes, as it passes in 10 minutes. In ten minutes, it passes 3 stations; in 60 minutes, it must pass 6 x 3 or 18 stations.
If x > 1 and a/b = 1 - 1/x, then b/a = (A) x (B) x -1 (C) (x -1) / x (D) x / (x -1O (E) 1/x
Answer: D Strategic Advice: Since b/a is the reciprocal of a/b, b/a must be the reciprocal of 1 - 1/x as well. Getting to the Answer: Combine the terms in 1 - 1/x and then find its reciprocal. a/b = 1 -1/x = x/x - 1/x = (x - 1) / x Therefore, b/a = x / (x - 1)
If f(x) = x^2 + 5 and g(x) = x + 7, what are the values of x where f(x) = g(x)? (A) -2 and 1 (B) -2 and 2 (C) -1 and 1 (D) -1 and 2 (E) 5 and 7
Answer: D Strategic Advice: The easiest way to find the solution is to graph the two functions (on a graphing calculator if possible) and find their intersection points. Alternatively you can set f(x) = g(x) and solve for x. Getting to the Answer: The two points of interesection are (-1, 6) and (2, 9). Since we are only looking for x-values, (D) is your answer.
If Set A consists of all the positive integers from 10 through 20, inclsive, and Set B consists of all the odd integers between 2 and 14, which of the following numbers is in the intersection of sets A and B? (A) 3 (B) 9 (C) 10 (D) 11 (E) 14
Answer: D Strategic Advice: The intersection of two sets consists of only the elements that are in both sets. Getting to the Answer: There are only two integers that are odd integers between 2 and 14 and are also among the integers between 2 and 14 and are also among the integers 10 through 20, inclusive: 11 and 13. Choice (D) is 11, and 13 is not among the answer choices.
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: D 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.
What strategy would you use to complete the problem below? What is the value of x if (x + 1)/(x - 3) - (x + 2)/(x - 4) = 0? (A) -2 (B) -1 (C) 0 (D) 1 (E) 2
Answer: D 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 starting with (C), the middle value. If the evaluation is that x is less than 0, try (D); if it is more than 0, try (B). This way you can narrow down your choices to find the correct answer. Getting to the Answer: (C) (x + 1)/(x - 3) - (x + 2)(x - 4) = (0 + 1)/(0 - 3) - (0 + 2)/(0 - 4) = -1/3 + 1/2 = not 0 (D) (x + 1)/(x - 3) - (x + 2)/(x - 4) = (1 + 1)/(1 - 3) - (1 + 2)/(1 -4) = -1 + 1 = 0 (D) is your answer.
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: D 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: 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
Five runners run in a race. The runners who come in first, second, and third place will win gold, silver, and bronze medals respectively. How many possible outcomes for gold, silver, and bronze medal winners are there? (A) 5 (B) 10 (C) 15 (D) 30 (E) 60
Answer: E Strategic Advice: Any of the 5 runners could come in first place, leaving 4 runners who could come in second place, leaving 3 runners who could come in third place. Getting to the Answer: This leaves a total of 5 x 4 x 3 = 60 possible outcomes for gold, silver, and bronze medal winners.
If f(x) = [6x^2 + 3x], what is f(-3)? (A) -45 (B) 12 (C) 18 (D) 30 (E) 45
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 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: E 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
When the number p is multiplied by 5, the result is the same as when 7 is added to p. What is the value of 4p? (A) 5/7 (B) 1 (C) 7/5 (D) 5 (E) 7
Answer: E Strategic Advice: It's actually much easier to solve for 4p than for just p. Translate into algebra first, then manipulate the equation until you get 4p. Getting to the Answer: 5p = p + 7 4p = 7
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: 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 do you find the area of a circle?
Area of a Circle = (pi)r^2 The area of a circle with a radius of 4 is (pi)(4)^2 = 16(pi)
How do you find the circumference of a circle?
Circumference = 2(pi)r
What was the initial amount of water in a barrel, in liters, if there are now x liters, y liters were lost, and 6 liters were added? (A) x + y - 6 (B) y - x + 6 (C) x + y + 6 (D) y - x - 6 (E) x - y + 6
Strategic Advice: Working out your own equations is the best tactic but you could also try picking numbers for x and y or backsolving by thinking through each answer choice. Getting to the Answer: We are trying to find the original amount of water after some has been removed and some has been added. Working backwards in time, start with the current amount, x, and undo the removal of y liters by adding y to x, and undo the addition of 6 liters by subtracting 6 from x + y. Your equation will be x + y - 6 or choice (A).
How do you identify which number of a fraction is the part and which number is the whole?
The key to solving most fractions and percents story problems is to identify the part and the whole. Usually, you'll find the part associated with the verb is/are and the whole associated with the word of. For example, take the following sentence: "In a group of 260 boys, half of the boys are blonds." 1/2 of 260 boys are blonds 1/2 of 260 = # of blond boys 130 = # of blond boys
How do you find the mode of a set of numbers?
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 add a positive number to a negative number?
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 convert from a mixed number to an improper fraction?
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.
How do you find the volume of a rectangular solid?
Volume of a Rectangular Solid = lwh For example, the volume of a 4-by-5-by-6 box is: 4 x 5 x 6 = 120 A cube is a rectangular solid with length, width, and height all equal. If e is the lenght of an edge of a cube, the volume formula is: Volume of a Cube = e^3
How do you solve a linear equation?
To solve a linear equation, do whatever is necessary to both sides to isolate the variable. For example, to solve the equation 5x - 12 = -2x + 9, first get all the x's on one side by adding 2x to both sides: 7x - 12 = 9. Then, add 12 to both sides: 7x = 12. Then, divide both sides by 7; x = 3.
How do you solve a proportion?
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 quadratic equation?
To solve a quadratic equation, put it in the "ax^2 + bx + c = 0" form, factor the left side (if you can), and set each factor equal to 0 separately to get the two solutions. For example, to solve x^2 + 12 = 7x, first rewrite it as x^2 - 7x + 12 = 0. Then factor on the left side: (x - 3)(x - 4) = 0 x - 3 = 0 or x - 4 = 0 x = 3 or 4 If the equation is not factorable, it can be solved by using the quadratic formula.
How do you solve a problem involving rates?
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
How do you find the area of a sector?
A sector is a fraction of the area of a circle. If n is the degree measure of the sector's central angle, then the formula is: Area of a Sector = (n/360)((pi)r^2)
How do you find the length of an arc in a circle?
An arc is a fraction of the circumference. If n is the degree measure of the arc's central angle, then the formula is: Length of an arc = (n/360)(2(pi)r)
What is the slope of the line perpendicular to the line with linear equation 4x + 2y = 12?
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.
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: 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.
What is the positive difference between the answers to the equation 35 + x^2 = 12x?
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.
What is the greatest of three consecutive odd integers whose sum is 75?
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.
What value of x is not in the domain of the function? f(x) = x - 2/ x - 3?
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
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: 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 value of x^2 - y^2, if x - y = 5 and x + y = 7?
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.
If 3^3x = 9^3 + 2, what is the value of x?
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
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: 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
A sporting goods store ordered 30 more white tennis balls than yellow tennis balls, making the ratio of white balls to yellow balls 6:5. How many tennis balls did the store order? (A) 120 (B) 150 (C) 180 (D) 300 (E) 330
Answer: E Strategic Advice: Let the number of yellow balls received be x. Then the number of white balls received is 30 more than this, or x + 30. Getting to the Answer: So # of white balls / # of yellow balls = 6/3 = (x + 30) / x Cross multiply: 6x = 5(x + 30) Solve for x: 6x = 5x + 150 x = 150 So they ordered 150 yellow balls and (150 + 30) = 180 white tennis balls. The question asks for the total number of the tennis balls in the order: 150 white + 180 yellow = 330 total.
The price of the newspaper rises from 5 cents to 15 cents. What is the percent increase in price? (A) 50% (B) 75% (C) 100% (D) 150% (E) 200%
Answer: E Strategic Advice: Remember that the original whole is the price before the increase. The amount of increase is the difference beween the increase price and the original price. Getting to the Answer: So the amount of increase is 15 cents - 5 cents = 10 cents. % increase = 10/5 x 100 = 2 x 100 = 200%
What strategy would you use to complete the problem below? From 1990 through 2000, the population of City Q increased by 20 percent. From 2000 through 2010, the population increased by 30 percent. What was the combined percent increase for the period 1990-2010? (A) 25% (B) 26% (C) 36% (D) 50% (E) 56%
Answer: E Strategic Advice: Since you are not given the population of City Q in 1990, Picking Numbers is a good strategy. When dealing with percents it is often easiest to start with 100. Getting to the Answer: If the 1990 population was 100, then a 20 percent increase would put the 2000 population at 120. Now, to figure out the 30% increase, multiply 120 by 30%: x = 1.3(120) x = 156 Since the population went from 100 to 156, that's a 56% increase.
The average (arithetic mean) of 3 numbers is 45. If 2 of the numbers are 35 and 45, what is the third number? (A) 30 (B) 40 (C) 45 (D) 50 (E) 55
Answer: E Strategic Advice: The average (arithemetic mean) of a set of numbers is the sum of the numbers divided by the total amount of numbers in the set. Getting to the Answer: Let the unknown number = x. Then (x + 35 + 45) / 3 = 45. This simplifies to x + 35 + 45 = 135, or x + 80 = 135, or x = 55.
Set R contains the integers from -10 through -2, inclusive. Set S contains th absolute value of each number in Set R. How many numbers are in the intersection of sets R and S? (A) 18 (B) 16 (C) 6 (D) 4 (E) 0
Answer: E Strategic Advice: The intersection of two sets consist of only the elements that are in both sets. Getting to the Answer: Since all the elements in Set R are negative and all the elements in Set S are positive, no number can be in both sets.
How do you factor a quadratic polynomial (or perform FOIL "in reverse")?
To factor a quadratic expression, think about what binomials you could use FOIL on to get that quadratic expression. For example, to factor x^2 - 5x + 6, think about what First terms will produce x^2, what Last terms will simplify to +6, and what Outer and Inner terms will simplify to -5x. When the leading coefficient is 1, you are looking for factors of the last term that sum to the middle term. Some common sense-- and a little trial and error-- lead you to the fact that -2 x -3 = 6 and -2 + -3 = -5, and thus the factors (x - 2)(x - 3).
How do you find the least common multiple (LCM) of two or more numbers?
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.
If 4^x^2 - 10x + 25 = 1, what is the value of x?
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
What are scalene, isosceles, and equilateral triangles?
Scalene triangles are triangles in which no sides, and no angles, are equal. Isosceles triangles are triangles that have at least two equal sides. Not only are two sides equal, but the angles opposite for equal sides, called base angles, are also equal. Equilateral triangles are triangles in which all three sides are equal. Since all the sides are equal, all the angles are also equal. All three angles in an equilateral triangle measure 60 degrees, regardless of the lengths of the sides.
How do you simplify an algebraic fraction?
Simplifying an algebraic fraction is a lot like simplifying a numerical fraction. The idea is to cancel factors common to the numerator and denominator. Thus, simplifying an algebraic fraction begins with factoring. For example, to simplify x^2 - x - 12 / x^2 - 9, first factor the numerator and denominator: x^2 - x - 12 / x^2 - 9 = (x - 4)(x + 3) / (x - 3)(x +3)... Canceling the factor x + 3 from the numerator and denominator leaves you with (x - 4) / (x - 3)
How do you add and subtract monomials?
To add or subtract monomials, first make sure that the terms to be combined have exactly the same variable(s). This makes them like terms. Then, to combine like terms, keep the variable part unchanged while adding or subtracting the coefficients. For example: 2a + 3a = (2 + 3)a = 5a
How do you add or subtract polynomials?
To add or subtract polynomials, combine like terms, which are terms with exactly the same variable(s). For example: (3x^2 + 5x - 7) - (x^2 + 12) = (3x^2 - x^2) + 5x + (-7 - 12) = 2x^2 + 5x - 19
How do you convert from a fraction to a decimal, and from a decimal to a fraction?
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.
When given a series of percent increases and decreases, how do you determine your ending value?
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 set up a ratio?
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.
How do you find the distance between two points on a coordinate graph?
To find the distance between coordinated points, use the distance formula.
How do you find the greatest common factor (GCF) of two or more numbers?
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.
If given an equation of a line, how do you find its y-intercept?
To find the y-intercept, you can either put the equation into y = mx +b (slope-intercept form)-- in which case b, the constant term, is the y-intercept-- or you can just plug x = 0 into the equation and solve for y. For example, take 3x - y = 5. Solving for y, if you put the equation in terms of y = mx + b form, you arrive at y = 3x - 5. The constant term, b, is -5 so the y-intercept is -5. Using the other method, substitute 0 for x. 3(0) - y = 5 so y = -5. Again, the y-intercept is -5.
How do you increase or decrease a number by a certain percentage?
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.
How do you multiply binomials?
To multiply binomials, use FOIL.. For example, to multiply (y + 3) by (y +4), first multiply the First terms: y x y = y^2. Next the Outer terms: y x 4 = 4y. Then the Inner terms: 3 x 4 = 12. Then, combine like terms: y^2 + 4y + 3y + 12 = y^2 + 7y + 12.
What does solving "in terms of" mean?
To solve an equation for one variable in terms of another means to isolate the one variable on one side of the equation, leaving an expression containing the other variable on the other side of the equation. For example, to solve the equation 3x - 10y = -5x + 6y for x in terms of y, isolate x: 3x - 10y = -5x + 6y 3x + 5x = 6y + 10y 8x = 16y x = 2y
How do you solve an inequality?
To solve an inequality, do whatever is necessary to both sides to isolate the variable. Just remember that when you multiply or divide both sides by a negative number, you must reverse the inequality symbol. For example, to solve -5x + 7 < -3, subtract 7 from both sides to get: -5x < -10. Now divide both sides by -5, remembering to reverse the sign: x > 2.
How do you translate English expressions into algebraic expressions?
To translate from English into algebra, systematically turn phrases into algebraic expressions. This will allow you to transform an algebraic sentence into an equation. Be careful about order, especially when a problem calls for subtraction. Take the following example: "The charge for a phone call is a flat rate of 'r' cents for up to minutes and 's' cents for each minute thereafter. What is the cost, in cents, of a phone call lasting exactly 't' minutes?" The charge begins with 'r', and then something more is added, depending on the length of the call. The amount added is 's' times the number of minutes past 3 minutes. If the total number of minutes is 't', then the number of minutes past 3 is t - 3. So the charge is r + s(t - 3).
If you have an average of a series of numbers, how can you find their sum?
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.
What are the properties of a 45-45-90 triangle?
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.
What are the characteristics of a rectangle?
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.
How do you solve a radical equation?
A radical equation contains a variable under a radical. Solve radical equations by isolating the radical term and squaring both sides to remove the radical. After solving the equation, check all roots since this method often introduces extraneous roots.
What is the definition of a rational number?
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 are the characteristics of a square?
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.
What types of angles are formed when a transversal crosses parallel lines?
A transversal across parallel lines forms four equal acute angles and four equal obtuse angles.
What are the characteristics of a parallelogram?
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 is a function and how do you evaluate one?
A function is a mathematical rule that associates every input with exactly one output. Standard function notation is written f(x) and read "f of x." To evaluate the function f(x) = 2x + 3 for f(4), replace every x with 4 and simplify: f(4) = 2(4) + 3 = 11.
How do you calculate negative exponents and fractional exponents?
A negative exponent indicates a reciprocal. To arrive at an equivalent expression, take the reciprocal of the base and change the sign of the exponent. A fractional exponent indicates a root.
How do you know if an integer is a multiple of 2 or 4?
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 you know if an integer is a multiple of 3 or 9?
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.
How do you find the area of a triangle?
Area of a triangle: 1/2(base)(height) The height is the perpendicular distance between the side that's chosen as the base and the opposite vertex.
What is the average rate and how do you find it?
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.
How do you quickly find the average of a series of evenly spaced numbers?
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
How do you convert from part-to-part ratios to part-to-whole ratios?
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.
What is the definition of an irrational number?
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.)
What is the general procedure for multiplying polynomials?
If you want to multiply polynomials with more than two terms, make sure you multiply each term in the first polynomial by each term in the second. In other words, use the distributive property and then combine like terms. For example: (x^2 + 3x + 4)(x + 5) = x^2(x + 5) + 3x(x + 5) + 4(x + 5) = x^3 + 5x^2 + 3x^2 + 15x + 4x + 20 = x^3 + 8x^2 + 19x + 20 After multiplying two polynomials together, the number of terms in your expression before simplifying should equal the number of terms in one polynomial multiplied by the number of terms in the second. In the example, you should have 3 x 2 = 6 terms in the product before you combine like terms.
Describe direct variation and inverse variation.
In direct variation, y = kx, where k is a constant. In direct variation, the variable y changes directly as x does. As x gets larger, y gets larger. For example, if a unit of Currency A is worth two units of Currency B, then A = 2B. If the number of units of B were to double, the number of units of A would double, and so on for halving, tripling, etc. In inverse variation, xy = k, where x and y are variables and k is a constant. As x gets larger y gets smaller. A commonly used inverse relationship is rate x time = distance, where distance is constant. For example, imagine having to travel 24 miles. If you were to travel at 12 miles per hour, you'd need 2 hours. But if you were to travel at half your rate, you would have to double your time. This is just another way of saying that rate and time vary inversely.
What is the definition of an integer?
Integers are whole numbers, negative whole numbers, and zero. For example: -15, -4, 0, 1, and 457 are all integers.
How do you calculate the probability that an event will take place?
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 compare the values of two or more fractions?
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.
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?
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.
What is the Pythagorean Theorem?
The Pythagorean Theorem can be applied to any right triangle: (leg^1)^2 + (leg^2)^2 = (hypotenuse)^2 If c represents the hypotenuse, and one leg is 2 and the other leg is 3, then: 2^2 + 3^2 = c^2 c^2 = 4 + 9 c^2 = 13 c = (square root of) 13
How do you find the domain and range of a function?
The domain of a function is the set of values for which the function is defined. For example, the domain of f(x) = 1 / (1 - x^2) is all values of except 1 and -1, because for those values the denominator has a value of 0 and is therefore undefined. The range of a function is the set of outputs or results of the function. For example, the range of f(x) = x^2 is all numbers greater than or equal to zero, because x^2 cannot be negative.
What is the difference between a factor and a multiple?
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.
How do you multiply and divide roots?
The product of more than one square root is equal to the square root of their total product. The quotient of more than one square root is equal to the square root of their total quotient.
How can you use ratios to find the length of a missing side of a right triangle?
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.
How do you find an original value, before it was increased or decreased by a certain percentage?
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.
If given an equation of a line, how do you find its slope?
To find the slope of a line from an equation, put the equation into the slope-intercept form: y = mx + b. The slope is m, the coefficient of x. For example, to find the slope of the equation 3x + 2y = 4, solve for y: 3x + 2y = 4 2y = -3x + 4 y = -3/2x + 2 The slope is the coefficient of x, -3/2.
How do you multiply expressions with exponents that have a common base?
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
Explain the memory device "PEMDAS"
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).
What is the formula used to find percent?
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 solve a system of equations?
You can solve for two variables only if you have two distinct equations with different slopes. Two forms of the same equation will not be adequate. Combine the equations so that one of the variables cancels out. For example, to solve the two equations 4x + 3y = 8 and x + y = 3, multiply both sides of the second equation by -3 to get -3x - 3y = -9. Now, add the two equations. The 3y and the -3y cancel out, leaving x = -1. Plug that back into either one of the original equations and you'll find that y = 4.
How do you find the slope of a line, when given two points on the line?
slope = change in y / change in x = rise / run
