FDP time saver
Use benchmark numbers like 10 & derivations like: 1/(2⁹ x 5³) = 1/((2³ x 5³) 2⁶) = 1/(1000 x 64)
How do you find the multiples of "x" in "y" factorial
First, all of the multiples of "x" up to "y"
Then, count the number of multiples of "x²" up to y
Then, count the number of multiples of "x³" up to y, etc.
What is the area for similar solids with corresponding sides in ratio a:b?
To save time & improve accuracy...
REPHRASE!!! the question stem & the answer choices
Ex: If C = 2R - 12 & you are looking for C < R?, substitute:
2R - 12 < R? EQUALS: R < 12?
A VIC technique that may sometimes be an easier & faster way of solving a problem
Performing test cases
A technique to finding unique solutions on DS questions
Running test cases to determine if you can eliminate an answer choice that results in a true and false case.
An unstated characteristic that is often pivotal in solving quant problems
Overlapping Sets Problems require
Organization of information into mutually-exclusive options, with tools such as the Double-Set Matrix &/or Venn Diagram
An important final step on every problem to avoid trap answers
Make sure that you are answering the correct question. Trap answers are often the correct answer to the wrong question
What single word can change your approach to problem?
"Approximate." It means you round & simplifies the math that you can use
What word often means that you have to be "inclusive" of extremes?
"Between." Your answer will include extremes if you are asked for the range that the answer is between & you furthermore may have to test minimum/maximum scenarios
How would you graph | x + 1| ≤ 4
First, find the midpoint: x = -1. Then, find the distance from the midpoint to the extremes & be inclusive, since you have the ≤ sign
Will you always use all of the boxes in a double-set matrix?
No. Some questions may only have 3 mutually-exclusive options, etc. This means that you may not need as much information to solve the problem as you would otherwise require
Why should you reread the question stem slowly after a quick first read to find the question to solve?
Most of the information given in the question stem, such as equations, whether a number is positive, an integer, or divisible by a number, is critical to solving the problem correctly
What do you do when you see a chart/graph?
ALWAYS write out the chart/graph & REPHRASE it with variables & other data
An important skill to improve speed & accuracy
ORGANIZATION. To help, create your own tables, variables, & expressions
When setting up before/after scenarios, ORGANIZATION will help ensure that variable expressions are translated correctly
Objective on DS questions
ALWAYS try to prove the statement insufficient by showing that the statement(s) will result in multiple answers
A way to eliminate erroneous answers on PS questions
Follow through on math. Trap answers will often play on incomplete math or guesses
Another concept that saves time on DS questions
Disregarding Combo & Mismatch problems, recognize that you can ALWAYS find variables when given "that many" distinct linear equations
How can knowing the ratios of a triangle's sides help?
Not only will it help save time, but it can help you determine the angles of the triangle, etc. Ex: Recognize that a 1:2 ratio between the leg & hypotenuse of a right triangle means you have a 30-60-90 right triangle: x : x√3 : 2x
How should you go about testing cases with equations?
First, simplify the equation and set equal to zero. Then, test numbers
What fact about weighted average can help you quickly approximate an answer?
Not only will the result fall between datapoints in the equation, but it will also be closer to the larger number
How do you find the average of rates?
You must first find the total distance & total time of the various rates, then divide the total distance by the total time. If you simply average the rates you will receive the incorrect answer which is often a trap answer
A deathly mistake to avoid
Making assumptions. NEVER assume that a statement is sufficient or not sufficient. You ALWAYS have to prove
What may be an easier alternative to manipulate information rather than using percents?
Fractions, especially when multiplying or dividing
What is an important detail to be mindful of when multiplying by successive percents or fractions?
ALWAYS multiply by (1+(%increase)) or (1-(%decrease))
An important rule to remember when manipulating equations
Whatever you do on one side of an equation, you MUST distribute across the entire equation--on both sides of the equal sign.
Ex: x = 48/x + 2 does NOT result in: x² = 48 + 2.
Rather, x = 48/x + 2 EQUALS: x² = 48 + 2x
Ex: 2(B+10) = 2B + 20 ≠ 2B + 10
Whatever you do to one side of an equation you must ALWAYS do the same on the other side of the equation.
Ex: If party A works 10 hours & party B works 8 hours & they both received X, & then B gives Y to A, it would be written as: (X + Y)/10 = (X - Y)/8, & ≠ X/8
What principal can help you save time when manipulating expressions?
Put the expression in terms of the variable that you ultimately want
Should you always use the RTD chart for rates questions?
No, especially if the rate is constant. It would be faster to solve these problems by setting up ratios
What does it mean when the GMAT asks for the "positive square root" of something?
This is a clue that "0" is not an answer, since it is neither positive nor negative. Additionally, there are no negative possibilities
What information can help you solve equations faster?
Knowing if the variable is not equal to zero and/or not less than zero. There are special operations that you can perform if you know this.
Ex: √(2x/3) = x EQUALS: 2x/3 = x² EQUALS: 2/3 = x
Otherwise, ALWAYS solve as a quadratic:
√(2x/3) = x EQUALS: 2x/3 = x² EQUALS: 0 = x² - 2x/3 EQUALS: 0 = x(x-2/3) EQUALS: x = 0 & 2/3
What should you do when you are presented with two or more pieces of information in the question stem?
REPHRASE that information into a simpler form
An automatic type of rephrasing
FACTOR when distributed & DISTRIBUTE when factored
What is the formula for counting consecutive multiples/evenly spaced sets?
((Last-First)/Increment) + 1
What are some properties of evenly spaced sets?
The average of the set is the same as the average of the first & last term of the set.
Also, the sum of the terms in the set is equal to the average term of the set times the number of terms in the set
The mean & median of an evenly spaced set are equal
What can help with speed & accuracy on Prime Factorizations?
Starting with the largest recognizable factor.
Ex: 7150 = 715 x 10
Also, REMEMBER to check prime numbers up to the square root of a number when finding factors
What is the divisibility rule for 11 for all three-digit integers?
If the units digit & the hundreds digit sum to the tens digit, the number is divisible by 11.
Ex: 143 = 11 x 13
When rephrasing, how should you start each problem?
Write ALL relevant formulas to the problem that you are solving, then REPHRASE those formula(s) with the information, variables, & graphs/charts from the problem.
ALWAYS write ALL relevant formulas first, so that you do not overlook a step & select a trap wrong answer.
Rephrase by LABELING all information given in the question stem & then labeling all of the formulas relevant to the question & then solving
What should be the first thing that you think of when you see an inequality?
That you will have to test positive/negative cases
Why should you be careful when rephrasing information?
A rephrasing of the information may obscure other methods of finding the correct solution.
Ex: When rephrasing what is C to what is r, because C = 2πr, be open to other methods that provide for C without giving r
How should you start EIV questions?
By rephrasing the given information into the form of the question. Often you can find the correct solution by combining information given in the question statement regardless of how ugly the math looks.
Ex: If n = positive, integer; n/25 = integer; √n > 25; n/25 = ?
√n > 25 EQUALS: n > 25² EQUALS: n/25 > 25²/25 EQUALS: n/25 > 25
What are the steps to solving complex VIC questions?
Step 1: Name the variables & minimize the number of variables created by rephrasing some variables in terms of others where appropriate.
Step 2: State the formula that you are trying to find.
Step 3: State ALL of the given equations.
Step 4: Manipulate the equations into the formula that you are trying to solve - OR - Try to find one of the inputs for the formula that you are solving & enter that into your master formula to solve.
*Be very careful as these questions require an organized approach & the information is often haphazardly interspersed throughout the question stem & may require some rephrasing
What is a faster way of solving for "x" if (x-1)² = 400?
Outside of doing the normal quadratic, RECOGNIZE that since both sides are squares, you can take the square root of both sides & then employ positive/negative cases.
This can only be done if both sides are squares. Otherwise use quadratics
Ex: x-1=+20 & -20 EQUALS: x = -19 or +21
What are the two methods that you can solve the expression: 1 - x² ≥ 0?
1. Arithmetically, 1 - x² ≥ 0 EQUALS: 1 ≥ x². Then, perform positive/negative cases: -1 ≤ x ≤ 1
2. Quadratically, 1 - x² ≥ 0 EQUALS: (1+x)(1-x) ≥ 0 EQUALS: x ≥ -1 or -x ≥ -1 EQUALS: -1 ≤ x ≤ 1
Be familiar with both methods & which is appropriate & faster
What principle can make complex probability problems easier to solve?
1-x Principle. It may be easier to calculate the probability of not getting "x," which is "1-x," than it is to find the probability of getting "x."
Ex: If you were asked to find the probability of getting tails at least once on three coin tosses & the probability of getting heads is 1/2. Find the probability of not getting any tails which is the probability of getting heads every time: (1/2)(1/2)(1/2) = 1/8. Then subtract 1 by the not probability: 1 - 1/8 = 7/8
What advice did your elementary math teacher give you that can help minimize errors?
ALWAYS show your work, especially on questions in which you are asked to evaluate the merit of multiple equations. While the prompt may be easy, you have to avoid silly math mistakes that present themselves when you do not show your work.
Ex: As "x" increase, "1/x" decreases, however expression "1-(1/x)" actually increases; an observation that may be overlooked if not written down
What are some common squares?
0²=0; 1²=1; 2²=4; 3²=9; 4²=16; 5²=25; 6²=36; 7²=49; 8²=64; 9²=81; 10²=100; 11²=121; 12²=144; 13²=169; 14²=196; 15²=225; 16²=256; 17²=289; 18²=324; 19²=361; 20²; 400; 21²=441; 22²=484; 23²=529; 24²=576; 25²=625; 30²=900; 40²=1600; 50²=2500
When should you use Venn diagrams?
When you have overlapping sets of 3 or more categories.
What is the general formula for overlapping sets problems?
Total number of distinct items = total number of items in each category - number of overlapping items
Total = Group 1 + Group 2 - Both + Neither
How can the following be expressed algebraically: M & N are two digit positive integers with the same digits, but in reverse order
M = 10x +y
N = 10y + x
Given the expression: 3/50 = y/100, what are two ways to solve for y?
1. 3/50 = y/100 EQUALS: 6/100 = y/100 EQUALS: y = 6
2. 3/50 = y/100 EQUALS: .06 = y/100 EQUALS: y = 6
If given a rectangular box with dimensions 6 x 8 x 10 & asked what the radius is of a cylindrical tube placed in the box that maximizes the volume of the tube, how would you solve?
First, write volume formula: V = πr²h. Since we are looking for max(r), RECOGNIZE that as the box is a rectangle, the maximum radius will be limited by the size of the other side. Do NOT fall for a TRAP, by assuming it is a square box. So, if you take the largest side 8 x 10, the maximum radius would be 4. Finally, verify that radius of 4 provides the maximum volume by running test cases.
How do you change the scale of a ratio by a factor?
Apply that factor to the first number in the ratio. Think of fractions. Ex: If you double the ratio of 2:50 EQUALS: 2(2:50) EQUALS: 4:50 EQUALS: 2(2/50) EQUALS: 4/50.
If you are changing the ratio for a ratio within a ratio, consider breaking the original ratio into separate ratios & then combining them back together again. This method is less error prone than doing them all together
The key to solving Algebraic Translations questions correctly
ORGANIZATION. Start with labeling the variables correctly. Then, set the correct expressions. Then, solve.
Do NOT be intimidated by ugly math, specifically quadratics
A good rule of thumb when comparing numbers
ALWAYS adjust them so that they are expressed in the same units, i.e. hours VS cm/mins EQUALS: mins VS cm/mins.
A good rule of thumb when running test cases
It is safer & more accurate to write out all of the possibilities & then apply the other restrictions that the question requires.
Ex: If you are selecting how many different positive even divisors "n" has including "n," & n = 4p &, p = a prime number > 2 EQUALS: n = 4 & p =3 EQUALS: n = 12 EQUALS: Total factors: 1, 2, 3, 4, 6, 12 EQUALS: 4 = answer
What should you do on VIC questions that ask you to find an unknown value/quantity?
Designate a variable to represent that unknown value/quantity. This will make it easier to manipulate & prevent errors.
Ex: What % of x = y EQUALS: y = v(x)/100
Also pay VERY close attention to the verbiage on VIC questions so you translate the question correctly.
Ex: "is" EQUALS: "=", "what percent of x" EQUALS: "(?/100)x"
The key to VIC questions
Properly translating the words into algebraic expressions. The math usually is not terribly complex, but trap answers play on incorrect translations.
Something to remember about shortcuts
Even if you forget how to do a shortcut, there is often another way to solve a problem.
Ex: On average problems remember that even if you forget the residual formula you can still solve the problem by translating the information algebraically
What should you be doing in the first 15 seconds of a VIC question?
Determining what method that you are going to use to solve the problem, i.e. algebraically or pick numbers & calculate a target
How should you solve a VIC problem with only one variable stated in a variable expression & the answer choices are varying algebraic expressions?
Best method is often to Pick Numbers & Calculate a Target
What is an important rule to remember when translating unknown digit's places represented by variables into algebra?
Express each variable as a multiple of its appropriate digits place.
Ex: The following expression would be written as: "If a two digit positive integer has it digits reversed, with the resulting integer differing by 27" EQUALS: xy - 27 = yx EQUALS: 10x + y = 10y + x + 27
What should you be mindful of when flipping fractions?
You can only flip a fraction AFTER each side consists entirely of a single fraction.
Ex: 1/r = 1/x + 1/y ≠ r = x + y. Rather it EQUALS: 1/r = (x + y)/xy EQUALS: r = xy/(x + y)
On Probability problem AND means what & OR means what?
AND = MULTIPLY the probabilities.
OR = ADD the probabilities.
Intuitively, if you multiply two fractions the result will be a smaller number, showing the lower likelihood of both events occurring at the same time--AND. However, adding two fractions creates a larger number, meaning there is a greater chance of either event occurring--OR.
Ex: If probability of x = 1/4, y = 1/2, & z = 5/8, & you are looking for the probability of x & y, but not z EQUALS: (1/4)(1/2)(1-(5/8)) = (1/8)(3/8) = 3/64
What should you always do before adding or subtracting fractions?
ALWAYS find a common denominator BEFORE combining the terms
How should you solve a VIC problem that is a variable expression with one variable & you are asked what that unknown variable "could" be & the answer choices are real numbers?
Consider solving algebraically as the expression may be solved by using quadratics.
Ex: If 1/x - 1/(x+1) = 1/(x+4), then x could be EQUALS: (x+1-x)/(x(x+1)) = 1/(x+4) EQUALS: 1/(x(x+1)) = 1/(x+4) EQUALS: x²+x = x+4 EQUALS: x²-4 = 0 EQUALS: x = +2 or -2