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Mixed Number to Improper Fraction - Multiply the denominator by the whole number, add to the numerator. Keep the denominator the same. least common denominator - The least common multiple of the denominators of two or more fractions. To determine the LCD of the fractions, list the multiples of the denominators and find the smallest number that is common to the lists. equivalent fractions - Two fractions a/b and c/d are equivalent if ad=bc Adding/subtracting fractions with same denominator - a/b + c/b = (a+c)/b or a/b +c/b = (a-c)/b Distributive property of division over addition - (a+c)/b = a/b + c/b Adding/subtracting fractions with different denominators - a/b + c/d = (ad + bc)/bd ... a/b - c/d = (ad - bc)/bd Subtracting a fraction from a whole number - A - b/c = (c x A - b)/c Multiplying fractions - a/b x c/d = ac/bd
Dividing fractions - Multiply by the reciprocal: a/b x c/d = a/b x d/c = ad/bc Two methods for simplifying fractions - Cross cancel and top-and-bottom cancel 1 over a fraction - 1/a/b = b/a How to simplify a complex fraction: Method 1 - Multiply both the numerator of the complex fraction and the denominator of the complex fraction by the LCD How to simplify a complex fraction: Method 2 - Write the numerator of the complex fraction as a single fraction, do the same to the denominator, then divide How to simplify a complex fraction: Method 3 - After doing method 2, (a/b)/(c/d) = ad/bc Bowtie method of comparing fractions - Best way to compare size of multiple fractions - Find the least common denominator and then compare the numerators
How to add or subtract decimals - vertically line up the decimal points of the numbers and then add/subtract How to multiply decimals - Does not require aligning the decimal points. arrange with right justification, multiply like normal and then count the total number of decimal places to the right of the decimal in the numbers. Then move the decimal point of the product to the left the same number of spaces Shortcut for multiplying decimals - Count the total number of decimal places from the two numbers being multiplied. That will be the number of decimal places in the answer (unless the answer ends in 0) How to divide decimals - 1. Move the decimal point of the divisor to the right until it becomes a whole number
- Move the decimal point of the dividend to the right the same number of places
- Divide the numbers by long division. For the quotient, keep the decimal in the same place as the new dividend How to compare the size of decimals - Compare them place by place, left to right. The larger decimal will have the larger number at the first place of difference
Percent means - divide by 100 x% = - x/ How to convert a fraction to percent - Multiply by 100 and add % How to convert a decimal or integer to percent - Move decimal point two places to the right and add % Convert percent to decimal - Move the decimal two places to the left and drop the % Convert a fraction to a decimal - Divide numerator by denominator using long division Converting Terminating Decimals to Fractions - If one decimal place: write the number (as if the decimal point were not there) over 10 If two decimal places: write the number (without decimal) over 100 If three decimal places: write the number (without decimal) over 1000
-. -.
- Decimal equivalents of Base 8 fractions -. -. -. -. -. -. -.
- Decimal equivalents of Base 9 fractions -. -. -. -. -. -. -. -.
(a/b)^2 =? - a^2/b^ If x is greater than or equal to 0 and y > 0, the square root of x/y =? - square root of x over the square root of y If 0 < x < 1, it must be true that... - x^2 < x < sqrt x Absolute value of a number is... - The distance of that number from 0 on the number line PEMDAS - Parentheses, Exponents, Multiplication, Division, Addition, Subtraction Steps for order of operations - 1. Operations within parentheses (or absolute value bars or radicals)
- Exponents
- Multiplication and division FROM LEFT TO RIGHT
- Addition and subtraction FROM LEFT TO RIGHT PEMDAS Time Saving Variation: Addition and Subtraction - Combine Like Terms
Commutative Property of Multiplication - ab = ba Associative Property of Multiplication - (ab) x c = a x (bc) Distributive Property of Multiplication - a(b + c) = ab + ac. (Super useful in factoring) 0! and 1! = - 1 Expanding and Contracting Factorial Notation and Canceling Factorials within Fractions - Do this to cancel/simplify tricky fractions and large numbers and expressions The substitution method to solve a system of equations - Replacing one variable with an equivalent expression containing the other variable In the substitution method, which variable should you isolate first? - Consider isolating the variable in the equation that yields the least complex expression for the variable you are isolating
Combination method - Add one equation to another equation or subtract one equation from another equation in order to eliminate one variable and solve for the other variable When to use the substitution method - When one of the equations can be easily manipulated to isolate one of the variables on one side of the equation When to use the combination method - When neither equation can be easily manipulated to isolate one of the variables on one side of the equation Fastest way to eliminate tricky fractions in an equation - Multiply each term of the entire equation by the LCD of the fractions Meaning of "what is x in terms of y?" - We are being asked to isolate x and set it equal to some expression of y When all terms in an expression have a common factor... - the common factor can be factored out When the product of two integers is 1 then... - either both are 1 or both are -
Some quadratic equations can be created from fractions by... - Multiplying the whole equation by the LCD Equation trap 1 - Don't assume that two equations are sufficient to determine the value of two variables, especially when one equation is just a multiple of the other equation. Equation trap 2 - Don't assume that one equation must be insufficient to determine the values of two variables. If there are inherent restrictions on the variables, one equation may be sufficient Equation trap 3 - Given an equation with two variables, such as x and y, and asked to solve for some combination of those variables (x+y, x-y, y/x, etc.), don't automatically assume that the question is unanswerable because there are two variables but only one equation. See whether it is possible to isolate the expression in question Equation trap 4 - Some quadratic equations have two solutions, while others have one. If the equation has two unique roots, it may not be sufficient without more info. If the equation has only one root, it may be sufficient without more info. Before factoring out the GCF... - Move all of the terms to one side before solving
You can factor by grouping when... - Two terms share a common factor and the other two terms share a common factor Goal of factoring by grouping - Make a cubic expression into a product of two binomials When can you factor a cubic expression (ax^3 + bx^2 + cx + d) by grouping? - If and only if a/b=c/d Equation trap 5 - Don't assume that the value of a variable can't be 0 Multiplying two square roots - sqrt(a) x sqrt(b) = sqrt(a x b) Catch up and pass rate questions - DDDDR (Delta Distance Divided by Delta Rate) Whole numbers - 0,1,2,3... nonnegative integers 0 raised to any positive power is... - 0
Even integer x any integer = - Even integer Odd integer x odd integer = - odd integer Even/Even = - even or odd Even/Odd = - Even Odd/Odd - Odd Positive + Positive = - positive Negative + Negative - negative To find a multiple of any quantity x... - Multiply x by any integer First 25 prime numbers - 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
How to find total number of factors of a number - If x has y number of unique prime factors, then x^n... - Has the same unique prime factors Least common multiple - The smallest multiple that two or more numbers have in common Finding the LCM - 1. Prime factorize each integer and put result in exponent form
- Of repeated prime factors, take those with largest exponent
- Take non-repeated prime factors
- Multiply together what you found in steps 2 and 3 (2^3x3x5 = 120) If two positive integers, x and y, share no prime factors, the LCM is... - xy Otherwise, the LCM is something less than xy Greatest Common Factor - The largest number that will divide evenly into all the numbers in the set
Phrases that alert us that dividing some number x by some number y will result in an integer - x is divisible by y y is a factor of x y is a divisor of x x is a multiple of y y divides into x (evenly) x is a dividend of y If y is a factor of x, and z is a factor of y, then z is... - also a factor of x x^a is divisible by x^b if and only if - a is greater than or equal to b if z is divisible by both x and y, it must also be divisible by... - the LCM of x and y When is a number divisible by 3? - if the sum of its digits can be divided evenly by 3 (216 is because 2+1+6 is 9 and divisible by 3) When is a number divisible by 4? - when the number formed by it's last 2 digits is divisible by 4 (3,028 is because 28 is divisible by 4)
When is a number divisible by 6? - if it is divisible by both 2 and 3 (318 is because it is even and the sum of 3+1+8 is divisible by 3) When is a number divisible by 7? - Tricky, so you'll have to do the division When is a number divisible by 8? - If the number is even, divide the last three digits by 8. If there is no remainder, then the number itself is divisible by 8 When is a number divisible by 9? - If the sum of the digits is divisible by 9 When is a number divisible by 11? - When the sum of the odd-numbered place digits minus the sum of the even-numbered place digits is divisible by 11 When is a number divisible by 12? - When the number is divisible by both 3 and 4 Formula for general division with a remainder - x/y = Q + r/y Ways to manipulate the general division with a remainder formula - x = Qy + r Q = (x - r) / y r = x - Qy
