Algebra Flashcards
Re-expressing Fractions:
Re-expressing Fractions:
If something costs 1/3 less than a specific amount then another way of expressing it is it costs 2/3 of that specific amount.
Reverse FOIL
Reverse FOIL
x^2 + 7x +6
same as: (x+1)(x+6)
Because between 7x and 6 there is a + you know that the two sings in parenthesis have to be the same. The sign before 7x tells you then which sign to use. If the sign between 7x and 6 was negative you would know that the signs in the new equation have to be different in both parenthesis.
Check by multiplying to see if it equals 7x:
1x + 6x = 7x
Combining equations:
Combining equations:
e.g. 4x + y = 8 and y - 3x = 7
Look for similar expressions in both equations! Then you can combine:
4x + y = 8 \+-3x + y = 7 ------------- x+2y=15
Combining Expression About Remainders:
If question says:
n/3 = x + 1 and n/7 = x + 1 the two statements are equivalent to: x/21 = x + 1 because 3 and 7 have no factors in common.
Rules of Positive and Negative Multiplication:
Rules of Positive and Negative Multiplication:
Positive x Positive = Positive
Negative x Negative = Positive
Negative x Positive = Negative
(x – 3)2 = always positive
(x + 3)2 = always positive
(x + 3)3 = could be positive or negative depending on x
(x – 3)3 = could be positive or negative depending on x
Rules of Odd and Even in Equations:
Rules of Odd and Even in Equations:
(odd x odd) = odd (even x even) = even (odd x even) = even (odd + odd) = even (odd + even) = odd (even + even) = even
You also need to know this for Number Properties questions so know this cold.
There are no odd and even rules for division, mainly because there is no guarantee that the result will be an integer. For example, if 3z = 6, then z is the even integer 2. However, if 3z = 2, then z = 2/3, which is not an integer at all.
Square vs. Cube:
Square vs. Cube:
if x^3 = x, then x = -1, 0, or 1
if x^2 = x, then x = 0 or 1
Rules of Square Roots:
Every positive square has two square roots, a positive and a negative one.
√-36 and √36 = -6^2 and 6^2
Square Roots and Perfect Squares:
If Square root of a variable is an integer it means that the variable is a perfect square, like 16, 25, or 49.
Simplifying Equations 1:
Simplifying Equations:
How to get rid of a square root:
E.g. 2c = √a/b
Rise to square on both sides to get rid (2c)2 = (√a/b)2
4c2 = a/b^2
Simplifying Equations 2:
Simplifying Equations:
How to get rid of fractions:
E.g. 4c = a/b
To solve for a =cross-multiply by b 4cb = a
Then divide by 4c to solve for b b = a/4c
Multiplying a Sum Equation:
Multiplying a sum equation:
If s = u - 10
Then s x 5 = (u-10) x 5
Don’t forget that the multiplication applies to the entire sum equation. You can’t do u – 10 x 5.
Units Digits in Multiples:
Units Digits in Multiples:
Multiples of 5 have units digits of: 0 or 5
Cubes of Numbers:
Cubes of Numbers:
Cubes of numbers with units digits of 0 or 5 keep the same units digit. E.g. cube of 10 is 1000, cube of 5 is 125)
Word Translation into Math:
Word Translation into Math:
equals, is, was, will be, has, costs, adds up to…: =
times, of, multiplied by, product of, twice, by: x
divided by, per out of, each, ratio: /
plus, added to, and, sum combined: +
minus, subtract from, smaller than, less than,
fewer, decreased by, difference between: -
a number, how much, how many, what x, n, etc.
Algebraic Vocabulary:
Algebraic Vocabulary:
- A TERM is a numerical constant or product of numerical constant and one or more variables.
E.g. 5, 3x, 4x^3y - An ALGEBRAIC EXPRESSION is combination of one or more terms, separated by + or -.
E.g. 4ab + 5cd - A COEFFICIENT is the numerical constant in a term like:
E.g. 3 in 3xy
E.g. 1 in just x - A CONSTANT TERM is a number without any variables:
E.g. just 3 - A MONOMIAL is an expression with just term:
E.g. 3xy - A BINOMIAL is an expression with two terms:
E.g. 3xy + 2ax - A TRINOMIAL is an expression with three terms:
E.g. 3xy + 2ax + x^3 - A POLYNOMIAL is general name for expressions with more than one term.
Order of Operations (PEMDAS):
Order of Operations (PEMDAS):
P = Parentheses E = Exponents M = Multiplication (in order from left to right) D = Division (in order from left to right) A = Addition (in order from left to right) S = Subtraction (in order from left to right)
Laws of Arithmetic Operations:
Laws of Arithmetic Operations:
COMMUTATIVE LAW:
2x - 5y = -5y + 2x
5a * 3b = 3b * 5a = 15ab
ASSOCIATIVE LAW:
2x - 3x + 5y + 2y = (2x -3x) + (5y + 2y) = -x + 7y
(-2x) (1/2 x) (3y) (-2y) = (-x^2) (-6y^2) = 6x^2y^2
Process of simplifying expressions by subtracting or adding together terms with same variable component is called COMBINING LIKE TERMS.
DISTRIBUTIVE LAW:
3a(2b - 5c) = (3a x 2b) - (3a x 5c) = 6ab - 15ac
For Product of two binomials apply distributive law twice:
(y+5) (y-2)
= y(y-2) + 5(y-2)
= y^2 - 2y + 5y -10
= y^2 +3y -10
So multiply the First terms first, then the Outer terms, then the Inner terms, and finally the Last terms ⇒ FOIL
Factoring Algebraic Expressions:
Factoring Algebraic Expressions:
FACTORING A POLYNOMIAL: express polynomial as product of two or more simpler expressions.
For that you can use a COMMON MONOMIAL FACTOR:
if there is a monomial factor common to every term in the polynomial, it can be factored out by using the distributive law:
2a + 6ac = 2a (1 + 3c)
because 2a is great common factor of 2a and 6ac
Remember: Making Problems look more complicated than they are by distributing a common factor is a classic GMAT trick. ⇒ Whenever algebra looks scary, check whether common factors could be factored out.
Linear/First Degree Equations:
Linear/First Degree Equations:
Equations in which all variables are raised to the first power, i.e. there are no squares or cubes.
Equations with Fractions:
Equations with Fractions:
GMAT loves to make algebra problems look harder than they need to be by using fractions. ⇒ Whenever you see a fraction in an algebraic question, always get rid of the fraction as first step.
Getting Rid of Negative Terms in Fractions:
Getting Rid of Negative Terms in Fractions:
if -VR/VT - P
Then multiply both the numerator and denominator of the fraction by -1
That gives you: VR/-VT + P
Which is the same as: VR/P - VT
Getting Rid of Fractions in Equations:
Getting Rid of Fractions in Equations:
When getting rid of fractions in equations, be careful to transfer all negative signs correctly across parentheses.
E.g. x - [2 - x^2/x] = y/x
multiply both sides by x to get rid of fractions:
= x*x - x [2 - x^2]/x = y
now if you get rid of the x in the second expression on the left you still will have a -1 there:
= x^2 -1 [2 - x^2] = y
now get rid of parentheses by multiplying -1 by every term in it.
= x^2 - 2 + x^2 = y
= 2x^2 - 2 = Y
Factoring Quadratic Expressions:
Factoring Quadratic Expressions:
When you are asked to solve for a variable that is squared, the most efficient solution is typically to factor the equation into two binomials. Factoring quadratic expressions mean using reverse FOIL to make an quadratic expression simpler. Essentially you’re factoring a polynomial into a product of two binomials. The product of the first term in each binomial must equal the first term in the polynomial. The product of the last term in each binomial must equal the last term of the polynomial. The sum of the remaining products must equal the second term of the polynomial.
How to use reverse FOIL:
- write down (x ) (x )
You now have to fill in the missing term in both parentheses. - Find the missing term in both parentheses. Remember that the product of the two missing terms will be the last term in the polynomial and the sum of the two missing terms will be the coefficient of the second term of the polynomial.E.g. x^2 -3x +2 = (x ) (x )
You know that the product of the terms you fill in has to be two and the sum -3. Through trial and error try to come up with the right answer. four constants would multiply to make 2 either 2 and 1 or -2 and -1. 2 and 1 would add up to +3 so that couldn’t be it. but -2 + -1 would add up to -3 so the answer is:
x^2 -3x +2 = (x - 2) (x - 1)
Rules for signs: If the coefficient of the constant (the last term) is negative, then you binomials will have different signs (one + and one -). If the coefficient is positive, then your binomials will both have the same sign as the coefficient in the middle term (two + or two -).
Classic Quadratics:
Classic Quadratics:
There are some patterns called “classic quadratics” which you can factor more quickly by recognizing the pattern than using reverse FOIL. The three patterns are:
1. Difference of Two Perfect Squares 2. Polynomials of the Form a^2 + 2ab + b^2 3. Polynomials of the Form a^2 - 2ab + b^2
The GMAT uses all three forms over and over so practice thoroughly. Notice that all 3 forms begin and end in a perfect square:
a^2 - b^2
a^2 + 2ab + b^2
a^2 - 2ab + b^2
Classic Quadratics: Difference of Two Perfect Squares
Classic Quadratics: Difference of Two Perfect Squares
The different of two squares can be factored into the product of two binomials:
a^2 - b^2 = (a + b) (a - b) d^2 - 16 = (d + 4) (d - 4)
Classic Quadratics: Polynomials of the Form a^2 + 2ab + b^2
Classic Quadratics: Polynomials of the Form a^2 + 2ab + b^2
Any polynomial of this form is equivalent to the square of a binomial. The binomial is the sum of two terms:
a^2 + 2ab + b^2 = (a + b)^2
x^2 + 6x + 9 = (x + 3)^2
Example of a more complicated example that might come up on the GMAT:
4x^4 + 52X^2 + 169
if you recognize that 169 is 13^2 and 4x^4 is (2x^2)^2 you know you can factor this to:
(2x^2 + 13) (2x^2 + 13) or (2x^2 + 13)^2
Classic Quadratics: Polynomials of the Form a^2 - 2ab + b^2
Classic Quadratics: Polynomials of the Form a^2 - 2ab + b^2
Any polynomial of this form is equivalent to the square of a binomial. The binomial is the difference of two terms:
a^2 - 2ab + b^2 = (a - b)^2 x^2 - 10x + 25 = (x - 5)^2
Solving Quadratic Equations:
if expression X^2 - 3x + 2 is set to 0 it’s a quadratic equation. You can find the value or values for x that make the equation work by using the factored form of the equation that you get through reverse FOIL:
X^2 - 3x + 2 = 0
(x - 2) (x - 1) = 0
A product is 0 if at least one of the terms is 0. That means in this example x = 2 or x = 1. Plugging both in will show that both make the equation work so both are values of x.
REMEMBER: Quadratic equations usually have two roots, or solutions. If both factors involving a variable are the same then there is only one distinct root. E.g. (x + 2)^2
Squares from 2 to 23:
Squares from 2 to 23:
2^2 = 4 3^2 = 9 4^2 = 16 5^2 = 25 6^2 = 36 7^2 = 49 8^2 = 64 9^2 = 81 10^2 = 100 11^2 = 121 12^2 = 144 13^2 = 169 14^2 = 196 15^2 = 225 16^2 = 256 17^2 = 289 18^2 = 324 19^2 = 361 20^2 = 400 21^2 = 441 22^2 = 484 23^2 = 529
Finding all Variables in Linear Equations:
REMEMBER: If a problems has multiple variables, you need as many distinct equations as you have variables to find unique numerical values for all variables. Distinct means every equation provides new, different information.
There are two approaches to combine the equations to simplify calculations:
- Isolate one variable in one equation and plug it into the other equation.
- Add or subtract whole equations from each other to eliminate one of the variables.
Example for 2:
Find values of x and y if 4x + 3y = 27 and 3x - 6y = -21
If you look closely you can multiply the first equation to get rid of the y:
2(4x + 3y) = 2*27
= 8x + 6y = 54
Now add new and second equation:
8x + 6y = 54
\+ 3x - 6y = -21
----------------------
11x = 33
x = 3
x can now be plugged into either equation to find y.
⇒ Always look for ways in which you can add or subtract equations and change one equation (on both sides) to change the subtraction and addition in a way to solve for one variable.
Steps to take in these questions:
- Check how many distinct linear equations and how many variables. If two distinct equations and two distinct variables you can solve for the variables.
- Choose the technique. If equations look easy to simplify, substituting one equation into the other will be the most efficient approach.
- Remember that if in the question stem is says that you are looking for x then you have to simplify one equation for to solve for y first and then plug that into the other so that you have eliminated the y and can now solve for x.
E.g. after simplifying and solving for y you might have: y = x - 5 then you can plug that into the other equation to solve for x.
Special Cases in Systems of Linear Equations:
In some cases you don’t need to solve for equations to find the value of each variable but you are asked to figure out relationships between variables, ratios, difference etc. Most frequently they ask for sums, differences, averages, and ratios and most of these special cases in systems of linear equations appear in the DS section.
REMEMBER: Don’t assume that you can’t answer a question just because there are more variables than equations because by simplifying the equation you may be able to cancel out one or more variables.
How to solve:
- Use critical thinking and pattern recognition to solve problems involving special cases in systems of linear equations.
- When you are asked to solve for the sum/difference/product/quotient of variables, you may not need to solve for each variable.
- Make sure to cancel out or combine variables before you determine that there are more variables than equations.
Simplifying expressions in DS questions:
GMAT questions rarely give you algebraic statements in their most useful form. It’s a safe bet that you’ll need to simplify or re-express almost all the algebra you see on Test Day. Doing so will help steer clear of common wrong answers. In DS questions you may also see that simplifying the equation in statement results in the same equation as you were given in the other statement so that both statements are really giving you the same equation but in a concealed way.
Absolute Value:
Absolute value of a number is the number’s distance from zero on the number line. So, because 5 and -5 are the same distance from zero, both numbers have the same absolute value of 5. So, since the absolute value is a distance, it is always nonnegative.
E.g. both 3 and -3 are three units from zero, so the absolute for either one is 3.
The absolute value of a number is denoted by two vertical lines:
E.g. I-3I = 3
I3I = 3
Also: if |a|>|b| then you know that a is father away from 0 than b. That means that a couldn’t be 0 because then b couldn’t be closer to 0 than a because there is no value closer to 0 than 0.
Largests/Smallest Absolute Value Questions with Variables:
If you are asked for the largest or smallest absolute value of answer choices that contain variables, it’s efficient to pick numbers and go through every answer choice quickly.
E.g. If - 2 smaller or same as x smaller or same as 2 then which of the following has the largest possible absolute value?
3x - 1 x^2 - x 3 - x x - 3 x^2 + 1
Pick numbers that are the highest values on the range, so -2 and 2 and go through every answer choice.
Absolute value of Expressions with Variables that are Negatives:
Expressions with variables that are negatives of each other will always produce the same absolute value.
E.g. x - 3 and 3 - x are negatives of each other. No matter what you fill in for x, it will produce the same absolute value (distance from 0).
Distributive Law in Arithmetic:
You may distribute a factor among the terms being added or subtracted.
E.g. a(b + c) = ab + ac
Same in division:
E.g. 4 + 6/2 = 4/2 + 6/2
But not when sum or difference is in the denominator:
E.g. 9/4+ 5 is not the same as 9/4 + 9/5.
Factoring in Arithmetic:
Use factoring as the distributive law in its reverse form in order to simplify calculations:
E.g. 11 + 22 + 33 + 44 = 11 (1 + 2 + 3 + 4)
Equivalent Fractions in Arithmetic:
multiplying or dividing the numerator and denominator of a fraction by the same number (any number other than 0) will leave the fraction unchanged.
E.g. 1/2 = 1x2/2x2
You can use this technique to get rid of decimals in the denominator:
2/0.5 = 2 x 2/0.5 x 2 = 4/1
Canceling and Reducing Fractions in Arithmetic:
Put fractions on the GMAT in their lowest form.
Dividing Fractions in Arithmetic:
To divide one fraction by another, just multiply the first fraction by the reciprocal of the divisor (second fraction)
E.g. 4/3 / 4/9 = 4/3 x 9/4
Exponents:
In the term 3x^2, 3 is the coefficient, x is the base, and 2 is the exponent.
Remember that in 3x^2 only the x is being squared. If the whole term was to be squared you would write: (3x)^2.
Remember that in 3x^2 you raise to the power before you multiply by 3.
A number multiplied by itself twice is called square.
A number multiplied by itself three times is called cube.
Any number raised to the first power equals itself:
E.g. a^1 = a
Any number except zero that is raised to the zero power is equal 1:
E.g. a^0 = 1
0^0 is undefined.
ALSO REMEMBER:
x^r+s = (x^r) (X^s)
Multiplying with Different Exponents:
To multiply two terms with the same base, keep the base and add the exponents:
E.g. 2^2 x 2^3 = 2^5
Dividing with Exponents:
To divide two terms with same base, keep base and subtract the exponent of the denominator from the exponent of the numerator:
E.g. 4^4/4^2 = 4^4-2 = 4^2
Exponents of Exponents:
To raise a power to another power, multiply exponents:
E.g. (3^2)^3 = 3^6
Multiplying with Same Exponents:
To multiply two terms with the same exponent but different bases, multiply the bases together and keep the exponent:
E.g. (2^3) (3^3) = 6^3.
Negative Exponents:
Negative exponents indicate a reciprocal. To get an equivalent expression, take the reciprocal of the base and change the sign of the exponent:
E.g. a^-n = 1/a^n or (1/a)^n
