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283 Cards in this Set
- Front
- Back
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What are 2 good practices when doing word problems?
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i. Write down notes as you read. Especially in word problems
ii. Look for what you know and don’t know. Don’t mix them up. |
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What is a good practice when doing data sufficiency problems?
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In Data Sufficiency questions, simplify the question
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When is a good time to approximate?
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Avoid calculations and try to approximate whenever possible when answers are far apart. When answers are close together don’t approximate.
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Give shortcuts for finding the following percentages of numbers
a. 10% b. 1% c. 20% d. 5% e. 15% f. 30% g. 16% h. 9% i. 89% |
Avoid calculations when there are percents
1. 10% of x : move the decimal point 1 digit 2. 1% of x: move the decimal point 2 digits 3. 20% of x: double 10% 4. 5%: half 10% 5. 15%: 10% + 5% 6. 30%: triple 10% 7. 16%: 10% + 5% + 1% 8. 9%: 10% - 1% 9. 89% of x: x – (10% of x) – (1% of x) |
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Give shortcuts for the following calculations
a. Multiply by 5 b. Divide by 5 |
1. Mult by 5: mult by 10 and divide by 2
2. Divide by 5: divide by 10 and mult by 2 |
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What is the nearest neighbor technique for multiplication? for division?
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Multiple by nearest neighbor numbers + or – their parts: example: 12(21) = 12(20 + 1) = 240 + 12 = 252 or 39.5(13) = (40 - .5)(13) = 520 – 6.5 = 513.5
Use the nearest neighbor technique when doing division a. Example: 162/3: find the nearest multiple of 3 closest to 162, which is 150 aka 50(3). 162 is 12 away from 150 aka 3(4). Therefore you need 3(50+4) to get to 162 b. 163/3: use the same steps as above to find that 162 is 54(3) and that 163 is 1 greater than that. i.e. 163/3 is equal to 54 1/3 c. 343/7: 350 is 50(7) and 7 (1 multiple of 7) less. i.e. 7(50-1) = 7(49) |
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Are mixed numbers better to use than improper fractions?
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It's better to use improper fractions than a mixed number (a whole number and a fraction)
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Dividing by a fraction is the same as
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Dividing is the same as multiplying by the reciprocal
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What should you try to do when dealing with decimals
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Can be a source of calculation errors, so try to convert to a fraction whenever possible
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9. What are the following common decimal to fraction equivalents?
a. ½ b. 1/3 c. 2/3 d. ¼ e. ¾ f. 1/5 g. 2/5 h. 3/5 i. 4/5 j. 1/6 k. 5/6 l. 1/7 m. 1/8 n. 3/8 o. 5/8 p. 7/8 q. 1/9 r. 2/9 s. 4/9 t. 7/9 u. 8/9 |
ii. Common decimal to fraction equivalences
a. ½ = .5 b. 1/3 = .333… c. 2/3 = .666… d. ¼ = .25 e. ¾ = .75 f. 1/5 = .2 g. 2/5 = .4 h. 3/5 = .6 i. 4/5 = .8 j. 1/6 = .1666… k. 5/6 = .8333… l. 1/7 = approx .14 m. 1/8 = .125 n. 3/8 = .375 o. 5/8 =.625 p. 7/8 = .875 q. 1/9 = .111 r. 2/9 = .222 s. 4/9 = .444 t. 7/9 = .777 u. 8/9 = .888 |
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What should you do when you are dealing with a wide range of decimals? Like a very large number and a very small number?
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Convert to scientific notation when possible.
.000006 x 45000 = ? same as (6.0 x 10^-6 ) x (4.5 x 10^4) = ((6 x 4.5) x 10^(-6 + 4)) = (27 x 10^-2) |
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What is the order of operation and its acronym?
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Order of operation, PEMDAS: Parentheses, exponents, multiplication, division, addition, subtraction
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What is the difference between an expression and an equation?
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Equation vs expression: an equation has an equal sign with an expression on each side of it
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Characteristic of a linear equation?
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Has no exponents or roots, thus it will graph as a line
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How you do reduce equation when both side of an equation are fractions?
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Use cross multiplication when both sides of an equation are fractions
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How many linear equations do you need to solve for n variables?
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You need n linear equations to solve for those n variables
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What are the two ways to solve for variables when you have a system of linear of equations?
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Use combination and substitution to solve for the variables. When using combination, cancel out one of the variables by adding the expressions together (you may need to multiply one expression by an appropriate number so that one of the variables will cancel out when adding).
When using substitution, to solve for one variable, substitute the other variable. |
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What should you avoid doing in data sufficiency problems when you have a system of linear equations?
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In data sufficiency questions, you don’t need to solve the equations. Knowing that you have n distinct linear equation for n variables is enough.
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What should you look out for when you have a system of linear equations (2 or more)?
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Beware that some linear equations may look different but are exactly the same when you factor them
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What is the form for quadratic equations?
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Quadratic equations
i. Form: ax^2 +bx +c ii. Usually a is 1 or not present and the equation usually equals 0 |
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What is the method of solving for quadratic equations?
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Use the FOIL (and reverse FOIL) method to solve: First, Outer, Inner, Last
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How many equations do quadratics factor out to (in the GMAT) and what does it equal?
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Quadratic equations factor out into two separate expressions that both equal zero.
example: x^2 +6x +9 = (x+3)(x+3) = 0 |
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What are the 3 most common types of quadratic equations?
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Common quadratics:
a. X^2 – y^2 = (x+y)(x-y) b. X^2+2xy+y^2 = (x+y)(x+y) = (x+y)^2 c. X^2-2xy+y^2 = (x-y)(x-y) = (x-y)^2 |
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What must you do in an inequality when you multiply or divide by a negative number?
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If you multiply or divide by a negative number, then you must reverse the inequality.
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In an inequality, when multiplying or dividing by a variable, what must your answer show?
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If multiplying or dividing by a variable, then you must consider that the variable can be negative and your answer must show both the positive and negative case
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if you are multiplying or dividing by a variable in an inequality in a data sufficiency question, what is needed to conclude that there is enough information to solve the problem?
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In data sufficiency questions, you DON’T have enough information if the question doesn’t say whether the variable is positive or negative.
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How do you handle 2-part inequalities?
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In 2 part inequalities, do the same thing to all three parts of the inequality.
(this is pretty rare on the gmat) |
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What is another name for absolute value?
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Absolute Value
Also known as “the positive difference”. |
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How do you handle abs values in equations and inequalities?
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To find evaluate an expression, first evaluate what’s inside the absolute value signs, then turn it into a positive number.
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What should you look out for in the question stem regarding variables in absolute value?
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iii. When dealing with abs values in equations and inequalities, you must consider two possibilities. When the value inside the abs is positive, you do nothing; when the value inside the abs is negative, then you multiply the value by -1 and consider both the positive and negative case. Example z = | x-y | , two possible answers, z = x-y and z = -1(x-y) = y-x
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if you are multiplying or dividing by a variable in an inequality in a data sufficiency question, what is needed to conclude that there is enough information to solve the problem?
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In data sufficiency questions, you DON’T have enough information if the question doesn’t say whether the variable is positive or negative.
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How do you handle 2-part inequalities?
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In 2 part inequalities, do the same thing to all three parts of the inequality.
(this is pretty rare on the gmat) |
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What is another name for absolute value?
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Absolute Value
Also known as “the positive difference”. |
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How do you handle abs values in equations and inequalities?
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To find evaluate an expression, first evaluate what’s inside the absolute value signs, then turn it into a positive number.
When dealing with abs values in equations and inequalities, you must consider two possibilities. When the value inside the abs is positive, you do nothing; when the value inside the abs is negative, then you multiply the value by -1 and consider both the positive and negative case. Example z = | x-y | , two possible answers, z = x-y and z = -1(x-y) = y-x |
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What should you look out for in the question stem regarding variables in absolute value?
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Watch out for any information the question says about signs of the variables in the absolute values. Even squares of a variable will always be positive. Subtracting a negative number is the same as adding a positive.
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What exponents of a variable will always be positive?
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even exponents
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How do you convert an exponent in the denominator of a fraction?
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An exponent in the denominator of a fraction can be converted to a negative exponent that multiplies the numerator.
example: y/ x^2 = y * x ^-2 |
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How do you convert a fractional exponent? n^(y/x)
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A fractional exponent can be converted to a radical: x root (n^y ) = n^(y/x)
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How do you handle multiplying same variables with exponents? How does this apply to fractional and negative exponents?
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When multiplying same variables with exponents, you can add the exponents. ( x^2)(x^3) = x^(2+3). This applies the same way to fractional and negative exponents as well.
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How can you convert an addition of an exponent? x^(y+z) = ?
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An addition in the exponent can be converted to a multiple: x^(y+z) = (x^y)(x^z)
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How can you convert a subtraction of an exponent? x^(y-z) = ?
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A subtraction in the exponent can be converted to a fraction (divison): x^(y-z) = (x^y) / (x^z)
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How can you convert a multiplication of exponents? x^(yz) = ?
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A multiple in the exponent can be converted to an embedded exponent: (x^y)^z = x^(yz) = (x^z)^y
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How do you solve equations with different bases of exponents?
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vii. To solve equations with different bases of exponents, convert the base to be the same base, then solve. When the bases on two sides of an exponent are the same, they can be dropped, leaving just the exponents
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What can you do when the bases of an exponential on two sides of an equation are the same?
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When the bases of an exponential on two sides of an equation are the same, they can be dropped, leaving just the exponents
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How can you convert a multiple to an exponent? (xy)^2 = ?
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A multiple to an exponent can be converted to a each individual variable or number to that exponent (xy)^z = (x^z)(y^z). And the reverse as well, example: (2^5)(5^5) = (10)^5
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How do you find (or simplify) square roots of an unfamiliar number?
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i. Roots and exponents are the mirror images of each other
ii. No matter how large or complicated a number is, take the prime factorization of the number and remove one of the factors for every two that appear. (since this number is a square of another number, it will have twice the prime factors of the square root) |
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How can you convert a root of a multiple? Root(xy) = ?
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Root(xy) = root(x) (root(y))
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How can you convert a root of a fraction? Root (x/y) = ?
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Root (x/y) = root(x)/ (root(y))
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Can you convert a root of an addition or subtraction?
example: sq. root (x+y) = sq. rt (x) + sq. rt. (y) ? |
Root (x+y) is NOT equal to root(x) + root(y)
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What should you do with a root in the denominator?
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You should generally not have a root in the denominator as an answer
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What is the cube root of: 8, 27, 64, 125?
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vii. Generally you only need to know cube roots of 8, 27,64, 125 so if a question appears to be testing your knowledge of a cube root any larger than that, there’s some shortcut, or a way you can use the answer choices to determine which is correct
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What is the number of degrees on a straight line?
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The number of degrees on a line is 180
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What can you assume about diagrams in DS questions?
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In data sufficiency questions, assume that diagrams are not drawn to scale. Assume nothing
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What are the 2 characteristics of an equilateral triangle?
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Equilateral: 3 sides equal and 3 equal angles
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What are the 2 characteristics of an isosceles triangle?
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Isosceles: at least 2 sides equal and 2 equal angles
1. The two equal sides corresponds with the 2 equal angles |
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What is the sum of the interior angles of a triangle?
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Sum of the interior angles = 180
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Area of a triangle = ?
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Area = ½ (base)(height): The height and base only need to be perpendicular, it doesn’t matter if the height isn’t inside the triangle
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What are the characteristics of the height and base of a triangle?
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The height and base only need to be perpendicular, it doesn’t matter if the height isn’t inside the triangle
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What is true about the sum of two sides of a triangle?
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No side can be greater than or equal to the sum of the other two sides
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What is true about the difference of two sides of a triangle?
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No side can be less than or equal to the difference of the other two sides.
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When can you apply the Pythagorean Theorem?
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Pythagorean theorem: can be applied to right triangles, a^2+b^2=c^2, where a & b are the legs and c is the hypotenuse
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Give some common right triangle multiples
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Right triangle multiples: 3,4,5; 5,12,13; 7,24,25 and multiples of these triplets
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What are the degrees of the angles of an isosceles right triangle?
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Isosceles right triangle: 45:45:90 degrees, ratio of the length of the sides: x: x: x root(2)
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What is the ratio of the length of the sides of isosceles triangle with respect to its angles?
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Isosceles right triangle: 45:45:90 degrees, ratio of the length of the sides: x: x: x root(2)
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What are the degrees of half of an equilateral triangle?
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30:60:90 right triangle a.k.a half of a equilateral triangle: ratio of the sides x:x root(3):2x …
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What is the ratio of the length of the sides of half of an equilateral triangle with respect to its angles?
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iv. 30:60:90 right triangle a.k.a half of a equilateral triangle: ratio of the sides x:x root(3):2x …
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Perimeter of a square =?
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Square: 4s, where s is one side
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Perimeter of a rectangle =?
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Rect: 2L+ 2W, L is the length and W is the width
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Area of a square =?
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Square: s^2
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Area of a rectangle=?
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Rect: LW
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Area of a trapezoid=?
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Trapezoid: h((b1 + b2)/2 ); h-height, b1-base1, b2-base2
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What type of triangle are squares composed of?
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Squares can be divided into two isosceles right triangles (45:45:90)
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Diameter of a circle with respect to its radius =?
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Diameter = 2*radius
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Circumference of a circle=?
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Circumference = 2*pi*r = pi*d
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Area of a circle =?
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Area = pi*r^2
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Length of an arc of a circle with respect to its angle and circumference =?
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Length of arc: 2*pi*r(angle/360) = (circumference) (angle/360)
i.e. arc/circum = angle / 360 |
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Area of a sector of a circle =? with respect to its angle and area?
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Area or sector: pi*r^2 (360) = (area of circle ) (angle)
i.e. sector area / area of the circle = angle of the sector / 360 |
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What is the difference between a major arc vs. minor arc?
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Major and minor arcs: minor arc is the smaller arc and major arc is the larger one
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Central angle of a circle =
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Central angle = angle of the intercepted arc from the center of the circle
Given two points A and B, lines from them to center of the circle form the central angle. The central angle is the smaller of the two at the center. |
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Inscribed angle of a circle=?
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Given two points A and B on a circle, lines from them to a third point P on the circle form the inscribed angle
Inscribed angle = ½* angle of intercepted arc |
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Angle formed by two chords intersecting in a circle =?
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Angle formed by two chords intersecting in a circle (in degrees)= ½ * (arc1 + arc2)
The measure of the angle formed by two chords that intersect inside the circle is ½ the sum of the chords' intercepted arcs. Note: This theorem applies to the angles and arcs of any chords that intersect within the circle. It is NOT necessary for these chords to intersect at the center of the circle for this theorem to apply. |
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Angle formed outside a circle by two secants, two tangents, or secant and a tangent = ?
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The formulas for all THREE of these situations are the same:
Angle Formed Outside = 1/2 * Difference of Intercepted Arcs = ½ * (arc1 – arc2) tangent = A line that contacts an arc or circle at only one point. secant = A line that intersects a curve or circle at two points |
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What is the angle formed by a tangent and the radius?
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a line tangent to a circle will be perpendicular to the radius/diameter of that circle.
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What can you say about 2 tangent segments drawn to a circle from the same external point?
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they are congruent
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Volume of a rectangular solid=?
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Rectangle = length*height*width
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Volume of a cube=?
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Cube = (edge)^3
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Volume of a cylinder=?
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Cylinder = pi*(radius)^2*height
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Surface area of a rectangular solid =?
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Rectangular: 2(lw)+2(lh)+2(wh); l-length, w-width, h-height
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Surface area of a cylinder =?
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Cylinder: area of 2 bases + surface of the cylinder = 2*pi*r^2 + height* (circumference) = 2*pi*r^2 + 2*pi*r*h
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Where is the first quadrant in a coordinate graph and which way does it progress?
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4 quadrants: going counter-clockwise from upper right corner
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Define the slope of a line.
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Slope = rise / run aka measure of steepness
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How can you visualize a positive slope vs. a negative slope?
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Positive slopes go from bottom left to upper right; negative slopes go from upper left to lower right
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What is the slope of the x-axis and y-axis?
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X axis has a slope of 0 and Y-axis has a slope of infinity
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How do you find the slope of a line using two points?
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To find using two points: (y2-y1)/x2-x1)
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What is the equation of a line?
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1. Y = mx + b where m = slope, b = y-intercept (where x coordinate equals 0 also where the line intersects the y-axis),
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What does the x-intercept tell you?
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x-intercept: tells you the point where the line intersects the x-axis (where the y coordinate equals 0)
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What does the y-intercept tell you?
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y-intercept (where x coordinate equals 0 also where the line intersects the y-axis),
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How do you find the x or y intercept using the equation of a line?
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To find the x or y intercept, plug in zero for the appropriate coordinate in the equation of a line and solve for the remaining variable.
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How do you find the distance between two points?
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Use the Pythagorean theorem: a^2 + b^2 = c^2 and the Pythagorean triples to find the length (3,4,5; 5,12,13; 7,24,25) i.e (x1-x2)^2 + (y1-y2)^2 = distance^2
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What are characteristics of a regular polygon? (2 of them)
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Regular polygon = all interior angles are equal and all sides are equal.
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What is the sum of the interior angles of a polygon?
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Sum of interior angles in a polygon = 180(n-2) where n is the number of sides of the polygon
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Ratios can be written in what two forms?
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Part to part: x:y or x:y:z
Part to whole is more like a fraction. The first part is a subset of the total number |
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How to convert between the two forms of ratios?
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Can convert from part-to-part to part-to-whole and vice versa.
the sum of all the individual parts will make up the whole. part to part: 5/7 part to whole: 5/12 |
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Can ratios be reversed?
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Ratios can be reversed.. x:y can also means y:x
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What is the common equation needed to find ratios or values?
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Common equation needed: (ratio part) / (ratio whole) = (actual part) / (actual whole)
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How do you handle the ratio when it doubles or halves or others?
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If the ratio doubles or halves, just multiply the ratio by the corresponding multiple (i.e. 2 or ½)
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How do you handle a ratio when adding or subtracting from a part?
Example if boys to girls ratio is 2:3 , and if the number of boys increases by two then ratio becomes ? |
When adding or subtracting from a part, convert to an expression with a variable and add or subtract accordingly.
Example if boys to girls ratio is 2:3 , and if the number of boys increases by two then ratio becomes (2x+2):3x |
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What should try to do when you have percents?
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Convert to fractions whenever possible
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How do you handle percent increases or decreases?
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Convert increases or decreases to appropriate percentage multipliers, example 20% decrease of x = 80% of x, 20% increase of x = 120% of x
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How do you find the percentage change for two numbers?
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Finding percentage change: (difference)/original
1. In a percent increase the original number is the smaller number 2. In a percent decrease the original number is the larger number |
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If numbers are the same, will a percent increase be the same as a percent decrease?
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Even if actual numbers are the same in an increase vs a decrease the percentage changes will be DIFFERENT, example , number goes from 24 to 30: (30-24)/24 = 6/24 = ¼… number goes from 30 to 24: (30-24)/30 = 6/30 = 1/5
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What is the difference between a rate and a ratio?
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Rates express the relationship between two unlike things whereas ratios express the relationship between like things
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What are the 2 characteristics of average rates?
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Average rates
1. Based on Totals, i.e. (total distance) / (total time) 2. The average is weighted (example: by the amount of time in a distance problem) |
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How do you handle the rates when objects move toward or away from each other?
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Objects moving toward or away from each other
1. Objects move toward each other at the sum of their rates |
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How do you handle the rates when one objects is catching up to the other?
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One object catching up to another
1. Subtract the rates to get the “catch-up rate” |
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What is the total rate for two or more things working together to achieve the same goal (with respect to their individual rates)?
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If two things are both working to achieve the same goal at the same time, their combined rate is the sum of their respective rates: rate 1 +rate 2 = total rate
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How is the work rate equation expressed in time?
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The same equation expressed in time (which is what most GMAT work questions will provide) is : 1/ (time 1) + 1/ (time 2) = 1/(total time)
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How do you handle the work equation when you have more than two rates?
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For multiple rates: 1/(time 1) + 1/(time 2) + 1/(time 3) + etc… = 1 / (total time)
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What is the shortcut equation for when want to find out the amount of time that a combined work effort will take?
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shortcut equation: AB/(A+B) = T, where A is time 1 and B is time 2 and T is the total time
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When should you use the rate equation vs. the time equation?
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v. General rule: If the question gives you amount of time, use the shortcut
shortcut equation: AB/(A+B) = T, where A is time 1 and B is time 2 and T is the total time If it gives you rates, or you have to do some preliminary work to determine rates or times, use the original equation: rate 1 +rate 2 = total rate Also look out for the form of units the answers are in. know that rate is the reciprical of time (if you set up the units correctly) and thus rate1 + rate2 = Total rate thus 1/time1 + 1/time2 = 1/total Time thus (time1 + time2) /( time1 * time2) = 1 / total time thus total time = (time1 * time 2) / (time 1 + time 2) i.e. AB/(A+B) |
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What is the best way to handle mixture problems?
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Use a table.
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In a mixed solution of two liquids, what will the final solution be the ratio of with respect to each liquid?
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The final solution will be equal to a fraction where the numerator is the total amount of a substance in question and the denominator is the total quantity of the two solutions mixed together.
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In mixture problems, How do you handle it when the question asks for ratios or fractions? (or gives only ratios or fractions)?
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iii. Ratios or fractions: when the question asks for ratios or fractions, setup the denominator as equal to 1 (represents the total quantity of the two solutions together)
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Difference between simple interest vs. compound interest?
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When money is borrowed, interest is charged for the use of that money for a certain period of time. When the money is paid back, the principal (amount of money that was borrowed) and the interest is paid back. The amount to interest depends on the interest rate, the amount of money borrowed (principal) and the length of time that the money is borrowed.
The formula for finding simple interest is: Interest = Principal * Rate * Time. If $100 was borrowed for 2 years at a 10% interest rate, the interest would be $100*(10/100)*2 = $20. The total amount that would be due would be $100+$20=$120 Compound interest: The interest for each period is based on the principal accumulated up to that period. |
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Which grows faster simple or compound interest?
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As long as interest is accrued over more than one period,
compound interest will be greater than an equal rate of simple interest. |
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How do you handle the frequency of the compounding in a compounded interest problem?
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Frequency of compounding determines how frequently interest in added to the principal.
1. The interest rate for each period is less than the annual interest rate 2. The more frequent the compounding, the more interest there will be. |
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Define prime number.
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Any positive number that is divisible only by itself and 1
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What common numbers are NOT prime?
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1 is NOT prime and negative numbers are NOT prime
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What is the only even prime number?
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2 is the only even prime number
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List all of the prime numbers from 1-30
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2,3,5,7,11,13,17,19,23,29….
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What numbers have exactly 3 factors?
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The only numbers that have exactly 3 factors are squares of primes. A square of a prime is divisible by 1, itself, and its square root.
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How do you check if an unfamiliar number is prime?
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To check if an unfamiliar large number is prime, approximate its square root, and then try each of the prime numbers less than its approximate square root to see if they are factors.
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What is the difference between a factor and a multiple?
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Factors are less than or equal to a number, multiple is greater than or equal to the number
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X is a factor of Y if Y/X =?
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X is a factor of Y if Y/X = integer
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How do you come up with a list of all of the factors of a number?
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To come up with the list of all factors of a number:
1. Start with 1 2. Try each successive number 3. If a number turns out to be a factor of x, then its “match” is also a factor..i.e. 4 is a factor of 24.. 24/4 = 6, thus, 6 is also a factor of 24 4. Once you reach the square root of x, stop. Example: a. 1 is a factor of 24, as is 24 b. 2 is a factor of 24, as is 12 c. 3 is a factor of 24, as is 8 d. 4 is a factor of 24, as is 6 e. We are done here since the square root of 24 is less than 5 |
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How do you get the prime factorization of a number?
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Prime factorization: list of all the prime factors of a number as well as the number of times each prime factor occurs.
1. Example 24 = 2*2*2*3, thus, 24’s prime factors are 2 and 3 and that you need three 2’s and one 3. Many numbers will have the same prime factors but only one number has a certain prime factorization 2. Use factor trees to get prime factorization a. Find the easiest pair of numbers you can, that multiplies to the number itself. If either of the numbers is prime, don’t break that one down any further. (it’s a good practice to circle the prime numbers when found) b. Continue this process until the tree has branched out to all prime factors c. There’s no right or wrong number to start with |
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What are the divisibility rules for the following numbers?
a. 2 b. 3 c. 4 d. 5 e. 6 f. 8 g. 9 |
Divisibility rules
1. If x is even, its divisible by 2 2. If the sum of the digits is divisible by 3, then the number is divisible by 3 3. If the last two digits are a multiple of 4, the number is a multiple of 4 4. If a number ends in 0 or 5, its divisible by 5 5. If a number is divisible by 2 and 3, then its divisible by 6 6. Divide by 2 and then divide the result by 2, if the result is even, , then its divisible by 8 7. If the sum of the digits is divisible by 9, then the number is divisible by 9 |
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Define prime factorization.
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Prime factorization: list of all the prime factors of a number as well as the number of times each prime factor occurs.
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Can factors and multiples be negative?
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Factors and multiples are ALWAYS positive
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Can a number that is a divisible be negative?
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Divisible numbers COULD be negative
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Define LCM
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Least Common Multiple (LCM): the smallest number that is a multiple of the two numbers.
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How do you get the LCM for two (or more) numbers?
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To calculate the LCM of two numbers:
1. Find the prime factorization of each one written in prime numbers raised to certain powers 2. Find every unique prime number in either factorization 3. For each prime, the exponent you need is the largest exponent associated with that prime 4. Example: LCM of 12 and 27; 12 = (2^2)(3), 27 = (3^3); LCM = (2^x)(3^y); LCM = (2^2)(3^3) = (4)(27)=108 |
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How can you find the LCM of two multiples?
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You can use the LCM to find the LCM of two multiples: if x is a multiple of 12 and 27, then x must be a multiple of (12)(27) = 108
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When is the LCM of two numbers equal to the product of those two numbers?
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The LCM of two numbers equals the product of those two numbers only when they have no common factors. LCM of 6 & 7 = 6*7
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Can even/odd numbers be negative?
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Can be negative
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Is 0 an even number?
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0 is an even number
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How do you write an even number as a variable? What about an odd number as a variable?
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An even number can be written as 2i , and odd number can written as 2i-1
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What are the arithmetic rules for even/odd numbers?
a. Even + even = ? b. Even + odd = ? c. Odd + odd = ? d. Even * even = ? e. Even * odd = ? f. Odd * odd = ? |
Arithmetic rules
1. Even + even = even 2. Even + odd = odd 3. Odd + odd = even 4. Even * even = even 5. Even * odd = even 6. Odd * odd = odd 7. Use simple numbers to remember the above rules instead of memorizing |
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Define consecutive numbers.
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Numbers that are equally spaced (usually by 1)
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How far apart are consecutive numbers usually?
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Can be one apart or more
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How do you write consecutive multiples?
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Consecutive multiples: x, x+5, x+10, x+15, etc.. (consecutive multiples of 5)
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When are the average and median of a set of numbers equal?
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Average and median are equal in a series of consecutive numbers (regardless if the amount of numbers is odd or even)
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Median =?
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Median is the middle number in a series where the total number of items is odd. If the total number of items is even, then it is in between the two middle numbers.
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What will never exceed the largest value in a set?
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The mean can never exceed the largest value in a set.
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If there is at least one negative number in a set, what can you say about the mean in relation to the range of the set?
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The mean will always be smaller than the range if there is at least one negative number in the set.
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How do you write consecutive odds or evens?
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Consecutive odds / evens; x, x+2, x+4, etc..
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How often do multiples of primes arise in a consecutive sequence of numbers?
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In any sequence of consecutive numbers, multiples of prime numbers will arise in roughly the same order that they do among integers
a. Multiples of 3’s arise every 3 numbers, multiples of 5 arise every 5 numbers, multiples of 7, every 7 numbers |
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What is the product of two consecutive numbers?
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The product of any two consecutive integers is even. Every other number will be even, every third number will be a multiple of 3, every fifth number will be a multiple of 5, etc…
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Define average
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a.k.a. arithmetic mean, mean, average
average = (sum of terms) / (number of terms) |
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How do you use an average within an overall average to find the overall average?
"Joe bowled three games with an average score of 194. If the average of his first two scores was 196, what was his third score?" |
average-within-an-average
1. Given the average of some terms and the number of terms, you can find the sum.. That’s enough to find some new averages, or a missing term that figures into a new average. average = (sum of first two terms)+x / number of terms We know the average (194) and the number of terms (3). Using the average formula itself, we can find the sum of the fi rst two terms: 196*2 = sum of fi rst two terms = 392 Now we can plug in all of our data to the initial equation: 194*3 = 392+x From here, solve for x |
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What is a shortcut way to solve weighted average problems? Give an example
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Sets – Weighted Average
i. The weights matter but the numbers themselves don’t. You can subtract the lowest number from all the numbers to solve the weighted average. Then add the number back to the solution. Example: "If Jason paid an average price of $185 for 5 suits, and the average price of three of the suits was $189, what was the average price of the remaining suits?" lets look at the traditional setup: (189(3)+x(2) ) / 5 = 185 That equation will lead us through some time-consuming arithmetic. In this case, the smallest quantity is 185, so subtract 185 from each of the dollar amounts, leaving x as is: (4(3)+2x ) / 5 = 0 12 + 2x = 0 2x = -12 x = -6 Our answer, then, is 185 greater than our result of -6. The average price of the remaining suits is 179 |
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If a set of numbers has the median equal to the mean, does this mean that it is consecutive?
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NO! example (1,2,2,3). The mean is 2, the median is 2.
However, in a set of consecutive numbers, the median will be equal to the mean. |
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How do you find the median in an odd numbered set of numbers? What about an even numbered set?
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Median is the middle number in a series where the total number of items is odd. If the total number of items is even, then it is in between the two middle numbers.
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When does the median equal the mean in a set of numbers?
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When the terms of a set are consecutive, the median = the mean. It doesn’t matter if they are consecutive integers, consec. Odds or consec. multiples… so long as the numbers are equally spaced.
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Define mode in a set of numbers.
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Mode: the most frequently occurring term. Could have more than one mode if 2 or more numbers occur the same number of times
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Can you have more than one mode for a set?
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Could have more than one mode if 2 or more numbers occur the same number of times
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Define range in a set of numbers.
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Range: difference between the largest term and the smallest term. Useful to first put the set in ascending or descending order.
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If in a set of numbers, the median equals the mean, does this mean that the set must be consecutive?
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no.
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Define the standard deviation of a set
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Standard deviation: measure of variability (how spread out the numbers are ) it’s the square root of the average of (the difference between each number and the mean) ^ squared.
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Define the variance of a set
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Variance: is the standard deviation squared (or just don’t take the square root when you are computing the standard deviation)
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In a Venn diagram, how is each area defined?
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Overlapping sets
i. Use a Venn diagram (example circle x and circle y) 1. Intersection of both circles = x & y 2. Outside the circles = not_x & not_y 3. Part of circle x not part of intersection: x & not_y 4. Part of circle y not part of intersection: not_x & y |
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What equation do you use for overlapping sets?
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Total = Group 1 + Group 2 – Both + Neither
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What is the best way to solve for overlapping sets?
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Use a table
1. Setup columns and rows such that they represent the different groups |
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Define probability
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Probability = (number of desired outcomes) / (number of possible outcomes)
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What is the probability when two independent things both happen?
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Probability that two things both happen: multiply the individual properties
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when calculating probabilities, what word signals the both happening?
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"and"
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What is the probability of either thing happening? What word signals this?
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When probabilities are dependent, the word “or” signals addition: find the probabilities of each of the desired outcomes and add them together
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What is a good way to solve for complex probabilities?
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When the probabilities are discrete, convert the question into one with an “and”, figure out what the opposite of the desired probability is. Then subtract it from 1. Prob of desired outcome + prob of the opposite of the desired outcome = 1
Sometimes you can use brute force: it might be faster to generate all of the possibilities and count them up. |
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Difference combinations and permutations?
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Combinations vs permutations
1. If order doesn’t matter, it’s a combinations problem. If order does matter, it’s a permutations problem. |
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What is the formula for combinations?
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C = (n!)/(k!)((n-k)!) where C is the number of combinations, n is the size of the group and k is the size of the desired subgroup
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How do you handle it when you have to combine combinations?
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When you are combining combinations, simply multiply the results of the individual combinations
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What is the formula for permutations?
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Perm = (n!)/((n-k)!)
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What is the easiest way to figure out permutations?
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Use the visual method whenever possible:
____ _____ _____ etc.. fill in the blanks with the appropriate selection available after each selection. |
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0!=?
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0! = 1
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How do you handle problems with multiple permutations?
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For problems with multiple permutations, solve for each individual permutation then find the product of the results
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What should you look out for in problems dealing with symbolic algebra?
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Symbolism
i. Same as algebra, just plug in the values for the symbols ii. Can be nested, just plug n chug. iii. A question might use a symbol that stands for a traditional operator, and you need to determine the operation that the symbol stands for. Watch out for statements that appear to provide enough information but really leave multiple options. |
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Define function
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i. F(x) = some equation of x
ii. Similar to symbolism and can have symbolism in it iii. Functions can be combined and embedded in each other. Work your way out of the deepest embedded functions first.. |
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Define sequence. What are its two parts?
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sequences
i. A set of numbers generated by a function ii. Can also have each term rely on the term before it iii. Sequences can sometimes be disguised in word problems two parts of a sequence: sequence a. rule for constructing a value in the sequence b. value of a term in the sequence |
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If x/m and x/n are integers and m and n are integers, then what can you say about x?
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If x/m and x/n are integers where m and n are integers, then x is a multiple of mn
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What values do you need to consider for all unknowns?
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Unknowns can be 0,1,fractions, any positive or negative number (including fractions)
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What happens when a value is raised to an odd power?
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A value raise to an odd power will lead to the sign of the value
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What should you watch out for when dealing with even roots (square roots)?
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Equations may have more than one solutions such as square roots may be positive or negative
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Knowing any one of these 4 things of an equilateral triangle will give the others?
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If you know any one of the following dimensions of an equilateral triangle, you can find the others:
a. length of a side b. perimeter c. altitude d. area |
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Knowing any one of these 4 things of a square will give you the others.
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if you know any one of the following dimensions of a square, you can find the others:
a. perimeter b. diagonal c. area d. side |
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Knowing any one of these 4 things of a circle will give you the others.
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If you know any one of the following dimension of a circle, you can find the others:
a. radius r b. diameter 2r c. circumference 2r d. area r2 |
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In DS questions, what answers should you check first?
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Always check choices A,B or D first, then check C and E
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What should you keep in mind when doing DS questions?
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Steps for answering data sufficiency questions:
a. Read the question carefully b. Pay careful attention to any information provided in the question stem c. Consider each statement in isolation d. Eliminate choices and, if necessary, guess. e. Do as little calculations as necessary |
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When you are testing answer choices that have concrete values, which answer should you start with?
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Sometimes you can save time by testing answer choice
a. always start with choice C because the quantities are almost always arranged in order |
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When you have unknowns, what is a one possible way of doing the problem?
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Sometimes you can substitute simple numerical values for unknowns
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When doing calculations, what should you keep in mind?
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make sure your answer is in the unit that is asked for in the question
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What are the steps used to do problem solving?
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Steps for solving problem solving questions:
a. read the question carefully a1. READ THE QUESTION CAREFULLY b. before solving the problem, check the answers c. eliminate choices that are completely off the radar screen d. for complex questions, break down the problem. i. Formulate a statement of what is needed vs what is known. ii. Find the numbers you need 1. Perform the required calculations |
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How do you handle the probability of an event that could happen in more than one way?
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6. To find the probability of an event that could happen in more than one way, find the probabilities of the individual ways and ADD them.
7. In multiple scenario question, if there are significantly fewer ways for the event not to happen, consider finding the probability that it does not happen and subtract from 1. |
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What should you always do when you have to divide by a decimal?
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when dividing by a decimal, always change the decimal to an integer by moving the decimal point to the end of the divisor.
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How do solve for two quantities that vary directly?
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Direct variation
a. Two quantities vary directly if they change in the same direction and thus can be setup by a proportion. This way you can find missing terms b. You can apply the same change as that of the denominator or numerator to the missing term to determine the missing term c. Two quantities vary inversely if the change in opposite directions, whenever two quantities vary inversely, you can find a missing term by using multiplication. Multiply the first quantity by the second and set the products equal. |
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Convert the following percentages to fractions and decimals:
a. 12 ½% b. 37 ½% c. 62 ½% d. 87 ½% e. 16 2/3% f. 83 1/3% g. 33 1/3% h. 66 2/3% |
12 ½ % = 1/8 = .125
37 ½ % = 3/8 = .375 62 ½ % = 5/8 = .625 87 ½ % = 7/8 = .875 16 2/3 % = 1/6 = .16666 83 1/3 % = 5/6 = .8333333 33 1/3% = 1/3 66 2/3 % = 2/3 |
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Define discount, commission, taxes
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Verbal problems involving percent
a. Discount: usually expressed as a percent of the marked price that will be deducted from the marked price to determine the sale price b. Commission: a percentage of the value of the good sold. c. Taxes: a percent of the money spent or earned |
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When using backsolving, which answer should you start with when the answers are variables? When they are concrete values?
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Backsolving: Process of trying out answer choices
a. For answers with variables, start from the last choice and work backwards. b. For answers with concrete values, start with the middle value. This way you maybe able to tell whether it is too large or too small and move in the appropriate direction. |
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When the base of an exponent is a fraction, what happens as the exponent increases?
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When the base is a fraction between 0 and 1, the greater the exponent, the smaller the value of the term
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What is the most common source of error when doing problem solving questions?
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Not Reading accurately!! and thus not understanding what is being asked to find. Represent what you are looking for algebraically
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Best way to handle coin problems?
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Coin problems: it’s best to change the values of all monies to cents before writing an equation.
Also, try to keep monies in the same unit as the answer choices. If the answers are in dollars, change to dollars, if in cents, change to cents. This will save you a step and precious time. |
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How do you find the interest paid on a principal if you have the rate?
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Interest Problems: Amount of interest paid on an investment is found by multiplying the amount of principal invested by the rate(percent) of interest paid. Principal * Rate = Interest Income
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Distance formula =?
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Rate * Time = Distance
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What can you say about the distance when two objects are either moving away or toward each other?
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Motion in opposite directions: when two objects start at the same time and move in opposite directions, or when two objects start at points at a given distance apart and move toward each other until they meet, the total distance traveled equals the sum of the distances traveled by each object. d1 + d2 == Total Distance
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What can you say about the distance when two objects are moving in the same direction?
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Motion in the same direction: the two distance must be equal. d1= d2
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What can you say about the distance when doing a round trip?
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Round trip: rate going is usually different than rate returning. But the distances must be equal. d1=d2
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Area of a rectangle? Square? Parallelogram? Rhombus? Triangle? Equilateral triangle when you know one side? Trapezoid? Circle?
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Areas
a. Rectangle = base * height b. Parellelogram = base * height , where the height is the perpendicular distance from the base c. Rhombus = ½ *(diagnol1 * diagnol2), a rhombus is a parellogram that has all sides equal but is not a square d. Square = (side)^2 or ½ * (diagnol)2 e. Triangle = ½ * base * height, where the height is the perpendicular distance from the base f. Equilateral triangle = ¼*(side)^2*squareroot(3) g. Trapezoid = ½ * height * (base1 + base2) h. Circle = pi*r2 |
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Perimeter of a polygon? Circle?
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Perimeters
a. Polygon = sum of all sides b. Circle = pi*diameter or pi*2*radius |
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The distance that a wheel moves in one revolution is the same as?
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Distance of one revolution of a wheel = perimeter = circumference = pi*2*radius
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Inscribed angle of a circle is what with respected to its intercepted arc?
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Inscribed angle = ½* angle of intercepted arc
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What is the measure of the exterior angle of a triangle with respect to the two remote interior angles?
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The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles
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What can you say when two angles of one triangle are congruent to two angles of another triangle?
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If two angles of one triangle are congruent to two angles of a second triangle, the third angles are also congruent
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When you know the side of an equilateral triangle, what can you say about the altitude?
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an altitude in an equilateral triangle forms a 30-60-90 triangle and is therefore equal to ½ * hypotenuse * squareroot(3).
the hypotenuse in this case is the same as the side. |
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When you know the side of a square, what can you say about the diagonal?
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the diagonal in a square forms a 45-45-90 triangle and is therefore equal to a side*squareroot(2)
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What are 6 characteristics of a parallelogram?
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In a parellelogram:
i. Opposite sides are parallel ii. Opposite sides are congruent iii. Opposite angles are congruent iv. Consecutive angles are supplementary v. Diagonals bisect each other vi. Each diagonal bisects the parallelogram into two congruent triangles |
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What can you say about the diagonals of a rectangle?
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Diagonals are congruent
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What are some features of a rhombus (with regards to its diagonals)?
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The rhombus is a quadrilateral with all sides equal in length
in a rhombus, in addition to the properties for a parallelogram: i. all sides are congruent ii. diagonals are perpendicular iii. diagonals bisect the angles |
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What are 11 characteristics of a square?
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b. In a parellelogram:
i. Opposite sides are parallel ii. Opposite sides are congruent iii. Opposite angles are congruent iv. Consecutive angles are supplementary v. Diagonals bisect each other vi. Each diagonal bisects the parallelogram into two congruent triangles c. In addition to the above, In a rectangle: i. All angles are right angles ii. Diagonals are congruent d. in a rhombus, in addition to the properties for a parallelogram, above: i. all sides are congruent ii. diagonals are perpendicular iii. diagonals bisect the angles e. A square has all of the properties listed in b, c, d above |
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Apothem of a regular polygon =?
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The apothem of a regular polygon is perpendicular to a side, bisects that side, and also bisects a central angle
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Area of regular polygon in terms of its apothem and perimeter =?
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Area of a regular polygon = ½ * apothem*perimeter
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When you have similar polygons, what can you say about:
a. The corresponding angles? b. Corresponding sides? c. Corresponding linear ratios? Give example d. Ratio of their areas? e. Ratio of their volumes? |
Similar Polygons
a. Corresponding angles of similar polygons are congruent b. Corresponding sides of similar polygons are in proportion c. When figures are similar, all corresponding linear ratios are equal. The ratio of one side to its corresponding side is the same as perimeter to perimeter, apothem to apothem, altitude to altitude, etc.. d. When figures are similar, the ratio of their areas, is equal to the square of the ratio between two corresponding linear quantities e. When figures are similar, the ratio of their volumes = (ratio between two corresponding linear quantities)^3 |
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How do you find the midpoint coordinates of a line segment when you know its endpoint coordinates?
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Midpoint coordinates of a line segment = ½*(x1+x2) , ½*(y1+y2)
i.e. take the average of the two coordinates for x and y |
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How to check if a number is a multiple of 12?
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Sum of digits is multiple of 3, last two digits
multiple of 4. |
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How do you find the common factors of two numbers?
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Break down both numbers to their prime factors to
see what factors they have in common. Multiply shared prime factors to find all common factors. - What factors greater than 1 do 135 and 225 have in common? 135 = 3 x 3 x 3 x 5 225 = 3 x 3 x 5 x 5 Both share 3 x 3 x 5 in common—find all combinations of these numbers: 3 x 3 = 9; 3 x 5 = 15; 3 x 3 x 5 = 45 |
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For events E and F, what is the probability of the following:
a. Not E? b. E or F? c. E and F? |
For events E and F:
- not E = P(not E) = 1 – P(E) - E or F = P(E or F) = P(E) + P(F) – P(E and F) - E and F = P(E and F) = P(E)P(F) |
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What is the number of ways that a set of n independent things can be ordered?
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Counting the number of ways that a set of objects
can be ordered: = n! |
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The number of ways independent events can occur together can be determined by?
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The number of ways independent events can occur
together can be determined by multiplying together the number of possible outcomes for each event. |
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For two independent events A & B, which is greater?
a. A and B vs. A or B? b. A or B vs. Individual probabilities of A, B? c. Which gives more options: (A and B) vs. (A or B)? |
A and B < A or B
A or B > Individual probabilities of A, B P(A and B) = P(A) x P(B) “less options” P(A or B) = P(A) + P(B) “more options” |
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How do you find the number of distinct permutations of a set of items with indistinguishable items? Example: how many ways can the letters in TRUST be arranged?
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To find the number of distinct permutations of a set
of items with indistinguishable items, divide the factorial of the items in the set by the product of the factorials of the number of indistinguishable elements. - How many ways can the letters in TRUST be arranged? 5!/2! = 60 |
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What is the number of ways to arrange n distinct objects along a fixed circle is?
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The number of ways to arrange n distinct objects
along a fixed circle is: (n – 1)! |
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235. Order doesn’t matter in permutations or combinations?
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combinations
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236. What is the number of ways to order r objects from a set of n objects?
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Number of permutations of r objects from a set of n
objects: n! / (n – r)! |
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Can odd numbers have even factors?
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Odd numbers have only odd factors.
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What is the quadratic formula given the following equation: ax^2 + bx + c = 0?
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To find roots of quadratic equation, ax^2 + bx + c = 0:
x = [-b +- sq. root (b^2 – 4ac)] / 2a this will most likely not come up in the gmat. |
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How do you find the LCM and how do you find the Highest common factor of two numbers? Example: 60 & 72?
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Highest Common Factor (HCF), Lowest Common
Multiple (LCM) – Prime Factorization 1. Start by writing each number as product of its prime factors. 2. Write so that each new prime factor begins in same place. 3. Highest Common Factor is found by multiplying all factors appearing on BOTH lists. 60 = 2 x 2 x 3 x 5 72 = 2 x 2 x 2 x 3 x 3 HCF = 2 x 2 x 3 = 12 4. Lowest common multiple found by multiplying all factors in EITHER list. 60 = 2 x 2 x 3 x 5 72 = 2 x 2 x 2 x 3 x 3 LCM = 2 x 2 x 2 x 3 x 3 x 5 = 360 |
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When you are comparing two average rates with a fixed distance, how is the time related? i.e. if Mieko’s average speeds was ¾ of Chan’s, what was her average time in relation to Chan’s average time?
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For a fixed distance, the average speed is inversely
related to the amount of time required to make the trip. - Since Mieko’s average speed was ¾ of Chan’s, her time was 4/3 as long. - rt = d - (3/4)r(4/3)t = d |
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How can you check if an unfamiliar number is prime?
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1. Pick a number n.
2. Start with the least prime number, 2. See if 2 is a factor of your number. If it is, your number is not prime. 3. If 2 is not a factor, check to see if the next prime, 3, is a factor. If it is, your number is not prime. 4. Keep trying the next prime number until you reach one that is a factor (in which case n is not prime), or you reach a prime number that is equal to or greater than the square root of n. 5. If you have not found a number less than or equal to the square root of n, you can be sure that your number is prime. |
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What is the relationship of an inscribed angle to its minor arc in a circle?
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angle of Minor arc = 2 x angle of (inscribed angle)
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|A Union B| = ?
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|A union B| = |A| + |B| - |A intersect B|
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Is Zero an even number? and is it positive or negative?
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Zero is an even integer.
Zero is NEITHER positive NOR negative |
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For two similar triangles, what is the ratio of the areas with respect to their lengths?
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The ratio of the areas of two similar triangles is the
square of the ratio of corresponding lengths. - Each side of triangle DEF is 2 times the length of corresponding triangle ABC - Triangle DEF must have 2^2, or 4, times the area of triangle ABC. |
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Difference between gross vs net?
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Gross is the total amount before any deductions
are made. Net is the amount after deductions are made. |
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How To multiply or divide two radicals, 6 root (3) x 2 root(5) = ?; 12 root (15) / 2 root (5) = ?
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To multiply or divide one radical by another, multiply or divide the
numbers outside the radical signs, then the numbers inside the radical signs. 6(sq.rt. 3) x 2(sq.rt. 5) = (6 x 2)((sq.rt. 3 x sq.rt. 5) = 12(sq.rt. 15) (12(sq.rt. 15))/(2(sq.rt. 5)) = (12/2)(sq.rt. 15/sq.rt. 5) = 6(sq.rt. 3) |
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How to convert a fraction to a percent? Example 1/400 = what %?
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To make a percentage, multiply by 100%:
- 1/400 = ¼% = 0.25% |
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How to convert a fractional percent to a number? Example ½% = what number?
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To drop a percent, divide by 100%:
- ½% x 1/100% = 1/200 |
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How do you find all the divisors of a number?
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You can find all the divisors of a number by finding all
the prime factors. |
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What is the volume of a sphere with respect to its radius?
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(4/3)*π*r^3
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What is the number of integers from A to B inclusive? Example from 33 to 954 inclusive?
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Number of integers from A to B inclusive = B – A + 1
Ex. How many consecutive integers are there from 73 through 419, inclusive? 419 – 73 + 1 = 347 |
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What is a quick way to find the average of a set of CONSECUTIVE numbers? Average of all integers from 13 to 77 = ?
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The average of a set of evenly spaced CONSECUTIVE
numbers is the average of the smallest and largest numbers in the set. - Ex. What is the average of all integers from 13 to 77? (13 + 77)/2 = 90/2 = 45 |
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Which answer should you guess if you have a problem where you have to plug n chug?
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If you have to guess in a problem solving
question, go with (D) or (E). - Especially with problems that force you to use or plug in the answer choices. |
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Approximate the following: sq. root of 2 =? Sq. root of 3 = ?
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Square root of 2 = 1.4
Square root of 3 = 1.7 |
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What is a good strategy for DS questions where it presents you with an equation?
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A good data sufficiency strategy is to
rephrase the information in a question: Ex. z + z < z? => z < 0? |
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How do you find the number added or deleted to a set of numbers based on the average of that set?
Ex. The average of 5 numbers is 2. After one number is deleted, the new average is –3. What number was deleted? |
Number added: (new sum) – (original sum)
Number deleted: (original sum) – (new sum) Ex. The average of 5 numbers is 2. After one number is deleted, the new average is –3. What number was deleted? Original sum: 5 x 2 = 10 New sum: 4 x (-3) = -12 Number deleted = 10 – (-12) = 22 |
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What 3 types of things are you usually asked to to find in a DS question? And what should you do once you determine that?
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1. A specific value.
2. A range of numbers 3. Yes/No Immediately write out the DS problem type (value, range, yes/no) on your scratch paper before you begin a DS problem. |
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For the first DS question, what is a good thing to do?
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Calculate out the first DS questions to make sure they
are correct. It is important to start out the section strong. |
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What is a good strategy for a question that you don’t understand?
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Skip statements that you do not understand.
- Eliminate as much as possible. and take an educated guess at the answer. |
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On very hard DS questions, which answers are more likely?
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On harder DS questions, answer choices tend to be
more sufficient than they might appear. - DON’T CHOOSE (E) if you have to guess. - Pick between (A) or (C), if you can eliminate (B). - Historically, (A) is slightly more common as the right answer. |
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In yes/no DS questions, does an answer need to conclude affirmatively or negatively or both or neither or either?
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either
On YES/NO DS questions, if a statement answers the question conclusively in the affirmative or in the negative, then IT IS SUFFICIENT. |
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in an isosceles right triangle, what is the length of one of the sides when the hypotenuse is x?
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x/ root(2)
or (x * root (2) ) / 2 using the typical ratio for an isosceles triangle: 45:45:90 corresponds to x:x:x*root(2) just divide all of the values by root (2) to make the value of the hypotenuse equal to x. |
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in a 30-60-90 right triangle, what is the value of the other two sides when the length corresponding to the 60 degrees angle is x?
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using the typical ratio for an isosceles triangle: 30:60:90 corresponds to x:x * root (3) : 2x
just divide all of the values by root (3) to make the value of the lenght of the side that corresponds to 60 degrees equal to x. 30 degrees: x / root (3) = x * root (3) / 3 90 degrees: 2x / root (3) = 2x * root (3) / 3 |
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solve and explain:
what is the units digit of (5^2)(9^2)(4^3)? |
0
5 * 5 = 25, drop the tens digit and keep the units digit 5. 9*9 = 81, drop the tens digit and keep the units digit 1, 4*4*4 = 64, keep the unites digit, 4. 5*1*4 = 20, drop the tens digit and keep the units digit 0. answer = 0. |
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what is a quick way to compare fractions?
which is greater: 4/7 or 5/9? |
cross multiply and put the result with the corresponding numerator:
4/7 5/9 4*9 7*5 36 > 35 thus 4/7 is greater than 5/9 |
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how do you solve the following quickly?
what is 10/22 of 5/18 of 2000? |
approximate the fractions
10/22 is approx. 1/2 5/18 is approx. 1/4 1/2 of 1/4 of 2000 = approx (1/2 of 500) = approx. 250. |
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How do you find the total number of divisors of an integer?
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1. take the prime factorization of the number.
2. Add 1 to each of the exponents of the prime factors. 3. take the product of these numbers. example: 24 24 = (2^3)(3) (3+1)(1+1) = 4*2 = 8 verify: total number of divisors of 24: 1, 2, 3, 4, 6, 8, 12, 24 |
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what is the ratio of the length of the apothem of an equilateral triangle to its height?
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the apothem of an equilateral triangle will be 1/3 its height.
NOTE: this applies only to equilateral triangles |
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if m and n are integers, how do you know when m/n results in a terminating decimal?
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convert the number and denominator to prime factors and simplify the fraction. If the denominator has a values other than 2 or 5, it will result in a nonterminating decimal.
example: 3/8 = .375 3/7 = 0.428571428571... 3/5 = .6 3/9 = 1/3 = .33333..... 111/5 = 22.2 7/6 = 7/(2*3) = 1.16666666... it will not terminate since we have a 3 in the denominator 21/8 = 2.625 |
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what is the product of the slopes of 2 lines that are perpendicular to each other?
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the product of a the slope of 2 lines that are perpendicular to each other will be -1
(rise1/run1)(rise2/run2) = -1 you can potentially use that to solve for unknown coordinates. |
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If you have a set of consecutive integers, what is the one thing that you need to know to determine the standard deviation of the set?
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you only need to know the number of items in the set.
remember that standard deviation can be thought of, to first-order approximation, as the AVERAGE DISTANCE TO THE MEAN from the data points in the set. this is not the exact definition of standard deviation, but it's more than adequate for any SD problem you'll ever actually have to face on this test. |
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if you have a set of numbers with a standard deviation d, how does adding a constant to all the numbers affect the deviation?
what about multiplying a constant (that is not 1 or -1) to all the numbers? what about dividing a constant (that is not 1 or -1) to all the numbers? what about changing the sign of the elements by multiplying by -1 ? |
1) Adding or subtracting a constant from each element in the set has no effect on standard deviation. Remains the same
2) Multiplying the elements of a set with an abolsute value greater than 1 increases the standard deviation 3) Dividing the elements of a set with an abolsute vaue greater than 1 decreases the std deviation 4)Changing the signs of the element of a set or multiplying by -1 has no effect on std deviation.Remains the sam |
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Is a triangle inscribed in a circle always equilateral?
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NO! don't assume the shape of the triangle unless the problem specifically states it. A triangle inscribed in a circle only has to have its 3 points on the circle and hence can be very very small, a right triangle (if one of its sides is the diameter), or an equilateral triangle (which will have the maximum area).
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what is zero to the zero power?
0^0 ? |
0^0 (zero to the zero power) is undefined and does NOT equal 1.
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What is a quick way to find the following?
sum of the integers from 1 to n? sum of the odd integers from 1 to n? sum of the even integers from 1 to n? sum of the first n odd integers? sum of the first n even integers? |
sum of the integers from 1 to n = n*[(n+1)/2]
sum of the odd integers from 1 to n = (n+1)*[(n+1)/4] sum of the even integers from 1 to n = (n/2)*[(n/2)+1] sum of the first n odd integers = n*n or n^2 sum of the first n even integers = n*(n+1) |
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Can consecutive numbers share primes?
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NO!
the idea is this: if a number is divisible by some prime p, then the next multiple of p will be p units bigger. for instance, 75 is divisible by 5. this means that the next greatest multiple of 5 is 80, which is 5 units away. hopefully, this fact is clear. once you realize this, it follows that consecutive integers can't share ANY primes, because they're only 1 unit apart (too close together to work for any common factor except 1, which is trivially a factor of any integer at all, anywhere). |
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what is the main diagonal of a rectangular box?
What is a quick way to find the length of the main diagonal of a rectangular box if you know the 3 sides of the box? |
it is the line from one corner to the opposite corner (in 3 dimensions)
Use the deluxe pyth theorem: d^2 = x^2+y^2+z^2 |
