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Quantitative Reasoning Equations
Adding and Subtracting Fractions - You must find a common denominator Multiplication of Fractions - Multiply the numerators, then multiply the denominators Division of Fractions - Multiply by its reciprocal How many degrees are in a triangle? - 180 How many degrees are in a rectangle? - 360 PEMDAS - Parentheses Exponents Multiplication/Division Addition/Subtraction Percent Formula - Part = Percent x Whole Percent Increase - Amount of Increase/Original Whole x 100 Percent Decrease - Amount of Decrease/Original Whole x 100
Multiple of 2 - Last digit is even Multiple of 3 - Sum of digits is a multiple of 3 Multiple of 4 - Last two digits are a multiple of 4 Multiple of 5 - Last digit is 5 or 0 Multiple of 6 - Sum of digits is a multiple of 3, and last digit is even Multiple of 7 - Some of digits is a multiple of 9 Multiple of 0 - Last digit is 0 Multiple of 12 - Sum of digits is a multiple of 3 and last two digits are a multiple of 4 Average - Average = Sum of term / # of terms Number of Integers from A to B - B-A+
Vertical angles are equal Angles along a line add up to 180 How to find an angle formed by transversal across parallel lines - All the acute angles are equal All the obtuse angles are equal An acute plus an obtuse angle equals 180 Area of a triangle - Area = 1/2(base x height) Isosceles Triangle - Two equal sides and two equal angles Equilateral Triangle - Three equal sides and three equal 60 degree angles Similar Triangle - Corresponding angles are equal and corresponding sides are proportional Pythagorean Theorem - How to find the hypotenuse or leg of a right triangle (a^2) + (b^2) = (c^2) 3:4:5 Special Right Triangle - 3:4: 45:45: 5:12:13 Special Right Triangle - 5:12: 30:60:
Perimeter of a Rectangle - Perimeter = 2(length + width) Area of a Rectangle - Area = length x width Area of a Square - Area = (Side)^ Area of a Parallelogram - Area = base x height Area of a Trapezoid - Quadrilateral having only two parallel sides Area = (average of parallel sides) x height you can also break the figure into a rectangle and triangle or two triangles Circumference of a Circle - 2(Pi)radius or (Pi)diameter Area of a Circle - Area = (Pi)r^ Distance between points on a coordinate plane - If the x or y coordinates are equal, simply add or subtract. If the coordinates are unequal, make a right triangle an use the pythagorean theorem or apply the special right triangle attributes if applicable. Slope of a Line - Rise/Run = (Change in y)/(Change in x)
The inverse of the time it would take everyone working together equals the sum elf the inverse of the times it would take each working individually Combined Ratios - Multiply one or both ratios by whatever you need to in order to get the terms they have in common to match Example: The ration of a:b is 7:3. The ratio of b:c is 2:5. What is the ratio of a:c? a:b= 2(7:3) = 14: b:c = 3(2:5) = 6: a:c = 14: The Balancing Method (dilution or mixture problem) - Make the weaker and stronger (or cheaper and more expensive, etc.) substances balance. That is (% difference between the weaker solution and the desired solution) x (amount of weaker solution) = (% difference between stronger and desired solution) x (amount of stronger solution). Make n the amount, in liters, of the weaker solution Example: How many liters (n) of a solution that is 10% alcohol by volume (weaker) must be added to 2 liters of a solution that is 50% (stronger) of alcohol by volume to create a solution that is 15% (desired) alcohol by volume? n(15-10) = 2(50-15) 5n = 70 n= Group Problem involving BOTH/NEITHER - Group 1 + Group 2 + Neither - Both = Total Group Problem involving EITHER/OR - Organize the information into a grid 0! - Zero factorial equals 1
Permutation - The number of ways to arrange elements sequentially = the number of arrangements! (factorial). Order DOES matte n(P)k = n!/(n-k)! n = the number in the larger group k = the number you're arranging Combination - If the order or arrangement of the smaller group that's being drawn from the larger group does NOT matter, you are looking for the numbers of combinations n(C)k = n!/k!(n-k)! n = the number in the larger group k = the number you're choosing Multiplying Probabilities - When all outcomes are equally likely, the basic probability formula is P = # of Desired outcomes / # Total possible outcomes Probability both events occur - P(A) x P(B) Conditional Probability - B occurs given A occurs If 2 students are chosen at random to run an errand form class with 5 girls and 5 boys, what is the probability that both students chosen will be girls?
Exponents and 0 - Zero raised to any nonzero number equals zero (Ex: 0^6 = 0) Any nonzero number raised to the exponent 0 equals 1 (Ex: 5^0 = 1) Zero raised to the 0 power is undefined Negative Powers - A number raised to the exponent -x is the reciprocal of that number raised to the exponent x Ex: n^-1 = 1/n Fractional powers - Fractional exponents relate to roots Ex: n^1/2 = square root of n Roots - You can subtract/add roots only when the parts inside of the symbol are identical To multiply/divide roots, deal with what's inside the square root and what's outside of the square root separately Ex. 2sqr(3) x 7sqr(5) = (2x7)(sqr3 x sqr5) = 14sqr(15) Solving Multiple Equations - Usually they're easy to combine! Example: 5x - 2y = -9 and 3y - 4x = 6. What is the value of x + y? so... 5x - 2y = -
x + y = -
Sequence Problem - What is the positive difference between the fifth and fourth terms in the sequence 0, 4, 18 ... who's nth term is n2(n-1)? n(5) = (5^2)(5-1) = 100 n(4) = (4^2)(4-1) = 48 Difference = 100-48 = 52 Equation of a line - y = mx + b m = the slope of the line = rise/run b = the y-intercept where the line crosses the y-axis Finding the Maximum/Minimum lengths for a side of a triangle - You know n - the lengths of two sides of a triangle, you know that the third side is somewhere between the positive difference of the sum. Ex: The length of one side of a triangle is 7. The length of another side is 3. What is the range of possible lengths for the third side? The third side is greater than the difference (7-3 = 4) and less than the sum (7 + 3 = 10) Sum of all the angles of a regular polygon - Sum of the interior angles in a polygon with n sides (n-2) Degree measure of one angle in a regular polygon with n sides - ((n-2)180)/n Ex: What is the measure of one angle of a regular pentagon? (n=5)
