GMAT and GRE Quantitative formula sheet

1. Divisibility

1.1. Definition of Divisibility

1.2. Common Divisibility Rules

1.3. Factors and Multiples

1.4. Remainders and Divisibility

Types of Divisibility Problems on the GMAT

Strategies for Solving

  1. Memorize Divisibility Rules
  2. Break Down Numbers (Use prime factorization)
  3. Test with Small Numbers
  4. Leverage LCM and GCD
  5. Eliminate Using Properties

Practice Examples

  1. Is 1,728 divisible by 8? (Yes, 728 ÷ 8 = 91)
  2. If n is divisible by 12, is it divisible by 6? (Yes)
  3. What’s the remainder when 2^10 is divided by 9? (7)
  4. How many positive integers from 1 to 20 are divisible by 3? (6)

2. Factoring

3. Multiples

4. Divisibility with Addition and Subtraction

5. Completing Powers

6. Factorial Divisibility

7. Number Properties

8. Odds and Evens

9. Sequences

10. Counting Integers

11. Negatives and Positives

12. Exponents

13. Work Rate

14. Distance/Rate

15. Ratios

16. Combinatorics

17. Probability

18. Remainders

19. Algebra

20. Statistics, Absolute Value, and Inequalities

21. Mixtures (Weighted Averages)

22. Overlapping Sets

23. Interest

24. Geometry (GRE ONLY)

—–I’ve organized the content into sections and subsections, used bullet points for lists, and bolded key terms for emphasis. This should make the notes much easier to navigate and understand.

  1. Divisibility

1. Definition of Divisibility

2. Common Divisibility Rules

3. Factors and Multiples

4. Remainders and Divisibility

Types of Divisibility Problems on the GMAT

1. Direct Divisibility Check

2. Finding Numbers with Specific Properties

3. Remainder-Based Problems

4. Prime Factorization

5. Data Sufficiency

Strategies for Solving

  1. Memorize Divisibility Rules
    • Know rules for 2, 3, 4, 5, 6, 9, 10 cold—they’re fast and reliable.
  2. Break Down Numbers
    • Use prime factorization for complex divisibility or factor counting.
  3. Test with Small Numbers
    • Plug in values to check patterns (e.g., n = 5k + 2: 2, 7, 12).
  4. Leverage LCM and GCD
    • Least Common Multiple (LCM): Smallest number divisible by all given numbers.
    • Greatest Common Divisor (GCD): Largest number dividing all given numbers.
    • Example: LCM of 6 and 8 = 24, GCD = 2.
  5. Eliminate Using Properties
    • Use parity (even/odd) or last digits to narrow answer choices.

Practice Examples

  1. Is 1,728 divisible by 8?
    • Solution: Last 3 digits = 728. 728 ÷ 8 = 91 (yes).
  2. If n is divisible by 12, is it divisible by 6?
    • Solution: 12 = 2^2 × 3, 6 = 2 × 3. All multiples of 12 (e.g., 12, 24) have at least 2 × 3, so yes.
  3. What’s the remainder when 2^10 is divided by 9?
    • Solution: Pattern of 2^n mod 9: 2, 4, 8, 7, 5, 1 (repeats every 6). 10 mod 6 = 4, so 2^10 mod 9 = 7.
  4. How many positive integers from 1 to 20 are divisible by 3?
    • Solution: Multiples of 3: 3, 6, 9, 12, 15, 18. Count = 20 ÷ 3 = 6 (integer part).
  2. Factoring 
    1. Finding factors: i.e.  Factors of 65 = 1, 65, 5,13
      1. Start with 1, then 2, then 3, and so forth if asked for all factors, no new factors higher than the square root of the number. Just use the prime factorization to find factors usually
      2. Prime factors (factor tree) = 5 and 13 
    2. Number of distinct factors = that just means normal factors
      1. Or number of different factors → make factor tree and disregard repeated numbers
        1. 27 = 3,3,3 → 1 distinct factor, 3 
        2. 18 = 2,3,3 → 2 distinct factors, 2 and 3 
    3. Greatest common factor:
      1. Factor each number and look for the greatest that they have in common 

Least Common Multiple:

Prime each number, then the LCM of the numbers contains EACH DISTINCT PRIME AND THE HIGHEST EXPONENT ON EACH PRIME THAT ANY INDIVIDUAL NUMBER HAS

  1. Multiples 
    1. Greatest and least common multiple:
      1. List the multiples OUT of each number 
    2. If K is a multiple of 24 – then it has the prime factors of 24
      1. Will have the prime factors of 24 i.e. 2,2,2,3 (and more) but AT LEAST those to be a multiple of 24 
    3. D: Formula to find how many multiples of “x” in a range
      1. Don’t start at 1 → USE FORMULA
        1. (Greatest – Least)/ x +1 → gives you how many multiples of x you have in a given range 
      2. Example: 200-300; 3x? 
        1. (300-200)/3 + 1 = 34 
      3. When the range in question starts at 1 →  DIVIDE AND ROUND DOWN 
      4. Use the formula exactly as is when the range does not start at 1 → otherwise just divide the biggest by your number 

Very handy: For any two positive integers, the product of their least common multiple (LCM) and their greatest common divisor (GCD) is equal to the product of the two numbers themselves

Eg: x and y are positive integers. If the greatest common divisor of x and 3y is 9, and the least common multiple of 3x and 9y is 81, then what is the value of 81xy?

 Ans: 3^8

  1. Divisibility with addition and subtraction 

Rules: Multiple + Multiple  = Multiple

  1. Multiple + Nonmultiple = Nonmultiple 
  2. Nonmultiple + Nonmultiple = ???

Example 1: Y= 3X+2, which of the following cannot be a divisor of Y

  1. 3
  2. 4
  3. 5
  4. 6
  5. 7
  1. Example 2: N = 20! + 17, then n is divisible by which of the following
  1. 15
  2. 17
  3. 19
  1. Example 3: Find the number that divides 17! +1 
  1. 7
  2. 11
  3. 17
  4. 18
  5. 19
  1. Finding whatever is not divisible 
  2. 1 is not divisible by anything 
  3. 1 is also not a prime number 
  1. D: Completing powers → what is missing (the variable they are telling you)
    1. Example 1: If y is the smallest positive integer such that 3,150 multiplied by y is the square of an integer, then y must be
      1. 3,150 = (2 3^2 5^2 7)y = x^2 
      2. You need a 2 and a 7 = 14 
    2. Example 2: If n is an integer and n^3 is divisible by 24, what is the largest number that must be a factor of n
      1. N^3 = 2^3 3 
      2. Cube the 3 so that it’s a perfect cube 
      3. N^3 = 2^3 3^3
      4. Take the cube 
      5. N = 2 3 = 6 
  2. D: Factorial Divisibility
    1. Example 1: If p is the product of the integers from 1-30 (inclusive) what is the greatest integer K for which 3^K is a factor of p?
      1. Asking how many 3s are there in 30!
      2. 30!/3^k
      3. How many multiples of 3? → 30/3 = 10 
      4. 30/9 = 3 
      5. 30/27 = 1 
      6. If range was bigger you’d have to keep going multiples of 3 
      7. Total = 14 
  3. D: Unknown factorial divisibility
    1. Example 1: N is a positive integer and K is the product of all integers  (this should scream: factorial!) from 1 – n inclusive. If K is a multiple of 1440, then the smallest possible value of n is
      1. Factor→ prime factorization is 2^5 3^2 5
  4. Number properties
    1. The average of consecutive integers = the median (they are the same) 
  5. Number Properties 
    1. D: Sum of an evenly spaced set
      1. [(Least Multiple + Greatest Multiple)/2] x (# of terms)
        1. # of terms → how many multiples
          1. HOW MANY MULTIPLES OF X IN A RANGE:
            1. (greatestx – leastx / x) + 1  → number in question 
            2. 2 → if evens 
            3. 3 → if looking for multiples of three 
            4. 4 → if looking for mu ltiples of four, etc  
          2. NOW TO GET SUM OF THOSE MULTIPLES
            1. (Least + greatest / 2) * # of terms 
        2. Example 1:
        3. Least + Greatest / 2 → (100 + 300 / 2) +1 =
          1. Since they are looking for even start at 100 and end at 300 
          2. (100 + 300)/2 x (300-100/2 +1) 
          3. (300-100/2 +1) → this is number of terms
    2. Divisibility by 3 or 6?
      1. n(n+1)(n-1) is always divisible by 2, 3 and 6. Or some version of that:
  1. Odds and Evens
    1. Addition and Subtraction 
      1. O+/-O = EVEN 
      2. E+/-E = EVEN 
      3. O+/- E = ODD 
      4. The sum of two primes = always even (odd + odd = even) unless 2 is on of the primes
    2. Multiplication 
      1. If ANY number is even = EVEN 
      2. NONE of integers even = ODD
      3. Odd x Odd = Odd 
  2. Sequences: 
    1. Arithmetic mean or median are equal to each other (middle number) 
    2. For all evenly spaced sequences the average equals (First + Last)/2 The mean and median of a sequence are equal to the average of the FIRST and LAST number 
  3. Counting integers: 
    1. How many integers between 14 and 765 inclusive?
      1. (Last – First + 1) → 765-14 =751 + 1 = 752 
    2. When talking about multiples
      1. (Last – First) : Increment + 1 
  4. Negatives and Positives! Careful when a question states the following:
    1. Variable that implies it’s negative or positive:
      1. x<0 – y> 0 
    2. Product of a variable is greater or less than 0
      1. Pq > 0 → pq have the same sign either positive; or negative 
    3. Expression that has negative sign and exponent
      1. (-x)^n → positive since the ^ is even
        1. Negative raised to an odd power = negative
        2. Negative raised to an even power = positive 
    4. Strategy tip: If you are told the sign of the variable, try and make a generalization about the sign of each quantity 
  5. Exponents! Memorize:
22= 432= 942= 1652= 2562= 3672= 4982= 6492 = 81
23= 833= 2743= 6453=12563= 21673= 34383= 51293 =729
24= 1634= 8144= 25654=625 
25= 3235= 24345=1024
26= 64
27= 128
28= 256
112=121122=144132=169142=196152=225162=256172=289182=324192=361
  1. Numbers >1 get BIGGER when raised to higher powers
    1. 2^1 < 2^2 < 2^3 
    2. 2    < 4     < 8 
  2. Numbers <1 get (smaller) when you raise then to higher powers
    1. (½)^1 > (½)^2 > (½)^3
    2. 1 / 2 > 1 / 4 > 1 / 8
  3. Units digit – do not do anything just look at units digit
  1. Work rate 
    1. Basic → Work = RT 
      1. THE RULE: They never tell you the rate 
        1. Solve for the rate 
        2. Then plug back in 
        3. Plug what you have to get what you want 
    2. Multiple workers → W = RT(# of workers) 
    3. Different rates → W=R1T & W=R2T 
      1. When you have to rates
      2. Combined work rate  → W2 = (R1 + R2) T 
    4. Multiple workers with a missing individual 
      1. Get the rate that you can 
      2. Make a variable for the rate that you cannot get in the combined rate formula 
      3. And move forward with the problem 
      4. “What percent of the job” → what is the work 
    1. Work rate system of equations 
  2. Distance/Rate 
    1. Distance = RT 
      1. More often than not they give you the rate
      2. Get the formula 
      3. If catching up problem →  subtract the rates 
      4. If converging distance →  add the rates 
      5. If going same direction and do not give rates → RT = RT 
      6. Average speed → total distance/ total time 
    2. One car catching up with the other → relative motion (we subtract the rates)
      1. 10:00 AM
        1. 4 H x 110 = 40 miles 
        2. 40 = 5T — rate is 15-10 = 5 
        3. T = 8 — 8 Hours from 2pm aka 10pm 
    3. Going same direction
      1. TAXI
        1. When going in the same direction – distances are equal
          1. RT = RT (we do this because they don’t give us the rates)
    4. Converging distance
      1. Add the rates 
      2. Make sure that the 
      3. When one leaves first you get distance before the other one starts 
    5. Average speed → Total distance/ Total time
      1. If they don’t give you total distance you can make it up as a multiple of the rates combined 
      2. Do D= RT for each leg – solve each for time 
      3. Add times together to get total time 
      4. Then put total distance/total time 
      5. Do not give you distances = create something easy and make it up yourself → can be a multiple of the rates 
      6. Steps:
        1. DRTs for each leg 
        2. Solve for Time 
        3. Add times together 
        4. TOTAL DISTANCE/TOTAL TIME → BOOM DONE 
  3. Ratios 
    1. Ratios: the unknown multiplier 
      1. Ratio of men to women = 3:4; there are 56 people how many are men
        1.  so make it 3x/4x 
        2. Men + Women = 56 
        3. 3x + 4x = 56 
        4. X = 8 
      2. When to use the unknown multiplier = 
        1. Total number of items is given
        2. Neither quantity in the ratio is already equal to a number or a variable expression 
        3. Three variable ratios
          1. A recipe calls for amounts of lemon juice, wine, and water, in the ratio 2: 5 : 7 – if all three combined yield 35 ml of liquid, how much was included?
            1. 2x + 5x + 7x = 35 (total) 
            2. Solve for x : 14x = 35; x = 2.5 
    2. Multiple ratios: Make a common term
      1. A box containing action figures – there are three of Clotho for every two figures of Atropos, and five figures of Clotho for every four figures of Lachesis
        1. C:A:L → to get C to be the same 
        2. 3:2 → multiply by common multiple 5→ 15: 10
        3. 5: 4 → multiply by common multiple 3 →  15: 12 
        4. Combined ratio: 15:10:12
  1. Combinatorics

1. Fundamental Counting Principle

2. Permutations (Order Matters)

3. Combinations (Order Doesn’t Matter)

4. Factorials

Types of Combinatorics Problems on the GMAT

1. Basic Arrangements

2. Partial Arrangements

3. Selections

4. Restrictions (e.g., Identical Items)

5. Data Sufficiency

Strategies for Solving

  1. Determine Order
    • Ask: Does arrangement matter (permutations) or just selection (combinations)?
  2. Break Into Steps
    • Use the counting principle for multi-step problems (e.g., choose then arrange).
  3. Handle Restrictions
    • For identical items: Divide by factorial of repeats (e.g., n! / (r1! × r2!)).
    • For “must include” or “cannot include”: Adjust total or subtract cases.
  4. Simplify Factorials
    • Cancel terms in numerator and denominator before calculating.
    • Example: C(5, 2) = (5 × 4) / (2 × 1) instead of full 5!.
  5. Test Small Cases
    • Verify with smaller numbers (e.g., arrange 2 items) to check logic.

Practice Examples

  1. How many ways to arrange 4 distinct flags in a row?
    • Solution: 4! = 4 × 3 × 2 × 1 = 24.
  2. How many 2-digit numbers can be formed using 1, 2, 3, 4 (no repeats)?
    • Solution: Tens: 4 choices, Units: 3 choices. 4 × 3 = 12.
  3. How many ways to choose 2 cards from a deck of 52?
    • Solution: C(52, 2) = (52 × 51) / (2 × 1) = 1,326.
  4. How many ways to arrange the letters in “MISS”?
    • Solution: 4 letters (M, I, S, S), 2 S’s identical. 4! / 2! = 24 / 2 = 12.

Arranging N things in a line= N!

Arranging N things in a circle= (N-1)!

Very specific! Distributing N things to r people (r not fixed): N + R – 1CR – 1

  1. Probability 

Desired Outcomes/Total Outcomes

Use combinatorics often to count numerator and denominator

When doing pure probability, consider 1 minus approach, orders, etc

Set up slots (# of days, etc), label slots, fill in probabilities for each slot, and then consider the possible orders

  1. Basic Probability Rules
    • Definition: Probability = (Number of favorable outcomes) / (Total possible outcomes).
    • Range: Probabilities are always between 0 (impossible) and 1 (certain).
    • Example: If a die is rolled, the probability of rolling a 3 is 1/6.
  2. Complementary Events
    • The probability of an event NOT happening = 1 – Probability of the event happening.
    • Example: Probability of not rolling a 3 is 1 – 1/6 = 5/6.
  3. Independent Events
    • Two events are independent if the outcome of one doesn’t affect the other.
    • Probability of both occurring = P(A) × P(B).
    • Example: Rolling a die and flipping a coin—probability of a 4 and heads = (1/6) × (1/2) = 1/12.
  4. Mutually Exclusive Events
    • Events that cannot happen at the same time.
    • Probability of either occurring = P(A) + P(B).
    • Example: Probability of rolling a 2 or a 3 on a die = 1/6 + 1/6 = 2/6 = 1/3.
  5. Dependent Events
    • The outcome of one event affects the other.
    • Probability of both occurring = P(A) × P(B|A), where P(B|A) is the probability of B given A has occurred.
    • Example: Drawing two cards without replacement—probability of two aces = (4/52) × (3/51).
  6. Combinations and Permutations in Probability
    • Combinations (order doesn’t matter): Used when selecting items for probability (e.g., choosing 2 people from 5).
    • Permutations (order matters): Less common but can appear in sequencing problems.
    • Example: Probability of picking 2 specific people from 5 for a committee = (Number of ways to choose 2 specific people) / (Total ways to choose 2 from 5) = 1 / C(5,2) = 1/10.
  7. Conditional Probability
    • Probability of an event given another event has occurred.
    • Formula: P(A|B) = P(A and B) / P(B).
    • Example: If 30% of students pass a test and 10% pass both that test and another, the probability of passing the second given they passed the first is 10%/30% = 1/3.

Common GMAT Probability Question Types

Difficulty Level

  1. Easy: Single-event probabilities (e.g., coin flips, dice rolls).
  2. Medium: Two-step problems with independent or mutually exclusive events.
  3. Hard: Multi-step problems with dependent events, conditional probability, or tricky wording requiring careful setup.
  1. Remainders

1. Definition of a Remainder

– When dividing integer *a* by integer *b*, the remainder *r* is: *a = bq + r*, where *q* is the quotient (integer) and *0 ≤ r < b*.

– **Example**: 17 ÷ 5 = 3 remainder 2, because 17 = 5 × 3 + 2.

3. Properties of Remainders

– If *a < b*, remainder is *a*. (e.g., 3 ÷ 5 = 0 remainder 3).

– Remainders cycle with the same divisor (useful for patterns).

– Negative numbers: Add divisor until non-negative. (e.g., -7 ÷ 3: -7 = 3(-3) + 2, remainder 2).

4. Common Divisors

– GMAT often uses: 2, 3, 4, 5, 6, 7, 8, 9, 10, 12.

– **Divisibility Rules**:

  – *2*: Even number.

  – *3*: Sum of digits divisible by 3.

  – *4*: Last two digits divisible by 4.

  – *5*: Ends in 0 or 5.

  – *9*: Sum of digits divisible by 9.

1. Direct Remainder Calculation

– **Example**: “What is the remainder when 123 is divided by 7?”

– **Solution**: 123 ÷ 7 = 17 remainder 4 (123 = 7 × 17 + 4).

2. Pattern Recognition

– **Example**: “What is the remainder when 2^10 is divided by 7?”

– **Solution**:

  – 2^1 = 2,

  – 2^2 = 4,

  – 2^3 = 8 = 7 × 1 + 1 (remainder 1),

  – 2^4 = 16 = 7 × 2 + 2 (remainder 2),

  – 2^5 = 32 = 7 × 4 + 4 (remainder 4),

  – 2^6 = 64 = 7 × 9 + 1 (remainder 1).

  – Pattern: 2, 4, 1 (repeats every 3). For 2^10, 10 mod 3 = 1, remainder = 2.

3. Algebraic Remainders

– **Example**: “If *n* leaves a remainder of 3 when divided by 5, what is the remainder of *n + 7* when divided by 5?”

– **Solution**: *n = 5k + 3*. Then *n + 7 = 5k + 3 + 7 = 5k + 10 = 5(k + 2)*. Remainder = 0.

4. Word Problems

– **Example**: “A number when divided by 6 leaves a remainder of 2. What is the remainder when divided by 3?”

– **Solution**: Number = 6k + 2. Divide by 3: 6k + 2 = 3(2k) + 2. Remainder = 2.

5. Data Sufficiency

– **Example**: “What is the remainder when *n* is divided by 4? (1) *n* divided by 8 has remainder 3. (2) *n* is odd.”

– **Solution**:

  – Statement 1: *n = 8k + 3* (e.g., 3, 11, 19 → remainders 3, 3, 3). Sufficient.

  – Statement 2: Odd numbers (1, 3, 5, 7 → remainders 1, 3, 1, 3). Not sufficient.

—

Strategies for Solving

1. **Use Small Numbers to Test**

   – Plug in numbers to spot patterns. (e.g., Remainder 2 when divided by 5: Test 2, 7, 12).

2. **Leverage Patterns**

   – For exponents, find remainder cycles 

3. **Break Down Large Numbers**

   – **Example**: 127 ÷ 5. Split as 125 + 2. 125 = 5 × 25, remainder = 2.

4. **Modulo Arithmetic**

   – For sums: Compute parts’ remainders, then combine. (e.g., (13 + 17) mod 5 = (3 + 2) mod 5 = 0).

5. **Time Management**

   – If stuck, estimate or eliminate using divisibility rules.

Practice Examples

1. **What is the remainder when 3^20 is divided by 10?**

   – Last digit of 3^n: 3, 9, 7, 1 (repeats every 4). 20 mod 4 = 0, last digit = 1, remainder = 1.

2. **If *x* divided by 7 has remainder 4, what’s the remainder of 3x divided by 7?**

   – *x = 7k + 4*. 3x = 3(7k + 4) = 21k + 12 = 7(3k + 1) + 5. Remainder = 5.

  1. Algebra 

Factoring Special Polynomials

4. Quadratic Equations

  1. Standard Form: ax^2 + bx + c = 0.
  2. Quadratic Formula: x = [-b ± √(b^2 – 4ac)] / (2a).
  3. When the Discriminant (b^2 – 4ac) is…
    1. > 0: Two real roots.
    2. = 0: One real root.
    3. < 0: No real roots.
  4. Sum and Product of Roots:
    1. Sum = -b/a.
    2. Product = c/a.
  5. GMAT Application: Solving for variables or analyzing roots.
    1. Example: Solve x^2 – 5x + 6 = 0.
    2. Solution: Factors are (x – 2)(x – 3) = 0, so x = 2 or 3.
  1. Statistics, Absolute Value, and Inequalities
    1. Average: 
      1. A = S/n 
      2. The average of an evenly spaced set is: First + Last / 2 
    2. Weighted Average: 
    3. Median: 
      1. Odd set: value is the middle 
      2. Even set: value is the average of two middle numbers 
    4. Standard Deviation 
      1. Definition
        1. Measures the spread of data points around the mean.
        2. Formula: SD = √[Σ(x – μ)² / n], where μ is the mean, x is each value, and n is the number of values.
        3. GMAT Focus: You never calculate it directly—focus is on visualizing  spread and applying properties, like what would increase or decrease average distance from the average.
      2. Key Properties Tested
        1. Spread: Larger SD means more spread; smaller SD means data is clustered near the mean.
        2. Uniformity: If all values are identical, SD = 0.
        3. Scaling: Multiplying all values by a constant multiplies SD by that constant; adding a constant doesn’t change SD.
        4. Example: Set {1, 3, 5} has mean 3. Deviations are -2, 0, 2; SD = √[(4 + 0 + 4) / 3] = √(8/3) ≈ 1.63. If doubled to {2, 6, 10}, SD doubles to ≈ 3.26.
      3. Comparison of Sets
        1. GMAT may ask which set has a higher/lower SD based on data distribution.
        2. Example: {1, 2, 3} (SD ≈ 0.82) vs. {0, 2, 4} (SD ≈ 1.63)—latter has greater spread.
      4. Normal Distribution (Rare)
        1. GMAT might reference the bell curve implicitly (e.g., 68-95-99.7 rule: 68% within 1 SD, 95% within 2 SDs, 99.7% within 3 SDs).
        2. Example: If mean = 50, SD = 10, 95% of data lies between 30 and 70.

Common GMAT Statistics & SD Question Types

  2. A small SD = list clustered around the average (mean) 
  3. A big SD = list spread out widely with points appearing far from the mean 
  4. How to: Look at the average spread of the cluster
    1. 5 5 5 5 = not much SD 
    2. 2 4 6 8 = moderate SD 
    3. 0 0 10 10 = big SD 
    4. The more spread out the numbers the greater the SD 
      1. Question about comparisons → ask yourself which list is more spread out from mean 
      2. Changes moving closer/further/neither from mean → ask  yourself which list is more spread out from the mean 
  5. Variance: Is the square of  the standard deviation 
  6. Normal distribution: Looks like a classic bell shape curve 
    1. Symmetric around the mean and long tails
    2. Mean and median are equal 
    3. Data is symmetric around the mean/median  
  8. Range
    1. Is the difference between the largest number and the smallest number on a list 
    2. Z – A = RANGE  
  9. Absolute Value and Inequalities 
    1. Absolute Value: 
      1. Do a positive and negative case 
    2. Inequality: 
      1. Flip sign when dividing or multiplying  by a negative 
      2. Careful dividing and multiplying by a variable IF YOU DON’T KNOW THE SIGN → YOU NEED TO KNOW THE SIGN – YOU CANNOT DO IT 
  1. Mixtures (Weighted Averages) 
    1. Steps: TUG OF WAR 
      1. Number Line 
      2. Differences 
      3. Reduce 
      4. Flip 
    3. What percent of total or what fraction of the total → you have to ADD the parts!
  1. Overlapping Sets 
    1. Two groups Steps: 
      1. Do the chart 
      2. Set it up correctly – yes and a no for each criteria along axis
        1. 4 years not 4 years
        2. Degree not Degree 
      3. Fill in the information they tell you 
      4. Rows and columns add across and add down 
    3. Formulas: 
      1. Two group formulas:
        1. Total = A+B – Both + Neither 
      2. Three group formulas: 
        1. Lay out each of the little group overlaps: 

Total = A + B + C – (sum of 2 group overlaps) + (all 3 groups) + none

OR:

Total = A + B + C – (# in exactly 2 groups) – 2(all 3 groups) + none

Interest

1. Simple Interest

2. Compound Interest

3. Key Differences

Types of Interest Problems on the GMAT

1. Basic Calculation

2. Compound Interest Growth

3. Finding Unknown Variables

4. Comparing Simple vs. Compound

5. Data Sufficiency

Strategies for Solving

  1. Identify the Type
    • Look for keywords: “simple” or “compounded” (with frequency like “annually,” “quarterly”).
  2. Convert Rates and Time
    • Change percentages to decimals (e.g., 6% = 0.06).
    • Adjust time to years if given in months (e.g., 18 months = 1.5 years).
  3. Approximate for Compound Interest
    • For small rates and short times, (1 + R/n)^(nT) ≈ 1 + R × T (like simple), then adjust slightly higher.
    • Example: 5% annual for 2 years ≈ 10% (simple) + a bit more (≈ 10.25%).
  4. Break Down Compounding
    • Calculate step-by-step for each period if exponents are tricky.
    • Example: 4% semiannually for 1 year: 1st period = 1.02, 2nd = 1.02 × 1.02 = 1.0404.
  5. Use Answer Choices
    • Plug values back into the formula to verify or eliminate options.

Practice Examples

  1. $3,000 at 3% simple interest for 4 years. What’s the total amount?
    • Solution: SI = 3000 × 0.03 × 4 = 360. Total = 3000 + 360 = $3,360.
  2. $2,000 at 8% compounded annually for 3 years. What’s the interest?
    • Solution: A = 2000 × (1.08)^3 = 2000 × 1.259712 ≈ 2519.42. CI = 2519.42 – 2000 = $519.42.
  3. $P$ doubles in 2 years at compound interest annually. What’s the rate?
    • Solution: 2P = P(1 + R)^2 → 2 = (1 + R)^2 → 1 + R = √2 ≈ 1.414 → R ≈ 0.414 = 41.4%.
  1. Geometry (GRE ONLY)

Triangles: 

  1. Triangles & Diagonals 
    1. Sum of any two sides of a triangle will always be greater than the third 
    2. The internal angles of a triangle must add up to 180 
    3. Sides and angles: Sides correspond to the opposite angles
  1. Types of triangles:
  1. Isosceles Triangle:  is a triangle that has two equal angles and two equal sides
  1. Isosceles Right Triangle: 45 – 45 – 90
    1. X : X : X (Square root of 2) 
  1. Half of a square! 
  1. Equilateral Triangle:  All same angles 
  1. 30 – 60 – 90 Triangle
    1. 30 – 60 – 90 
    2. X  : X (square root 3) : 2X
  1. Right Triangles:  Triangle with a 90 degree angle
    1. Pythagorean Theorem: a^2 + b ^2 = c^2 
    2. Pythagorean Triplets:
      1. 3 – 4 – 5 
        1. 6 – 8 -10 
        2. 9 – 12 – 15 
      2. 5 – 12 – 13 
      3. 8 – 15 – 17 
  1. Perimeter : Area : Diagonals
    1. Perimeter of a triangle = the sum of its sides 
    2. Area = ½ BH
      1. BH need to be perpendicular to each other 
    3. Diagonal of a square:
      1. D= s (square root 2) – make sure it’s a right triangle 
    4. Diagonal of a cube: 
      1. D = S(square root 3) 

Polygons ( n five or more) and Quadrilaterals (four sided shapes)

  1. Quadrilaterals → Any figure with 4 sides  
    1. Trapezoid: one pair of opposite sides are parallel
      1. Area of a trapezoid: (B + B) / 2 x Height 
        1. Height → perpendicular to the bases 
    2. Parallelogram: opposite sides and opposite angles are equal
      1. Area of a parallelogram: Base x Height 
      2. Perimeter: Sum of all sides 
    3. Rectangle: all angles are 90 degrees and opposite sides are equal
      1. Area of a rectangle: Length x Width 
      2. Perimeter: Sum of all sides 
    4. Rhombus: opposite sides and opposite angles are equal – all sides are equal 
    5. Square: all angles are 90 degrees
      1. Area of a rectangle:Length x Width → since equal then it’s (S^2) 
        1. Lengths of all sides are equal 
      2. Perimeter: Sum of all sides 
  2. Polygons: 
    1. (N-2) x 180 = SUM of interior angles of a polygon 
    2. Perimeter: sum of the lengths of all sides 
    3. Area: Refers to the space inside of the polygon 
    4. 3 Dimensions: Surface Area 
      1. Surface area is the SUM of all of the areas of the faces! 
    5. 3 Dimensions: Volume 
      1. Volume = Length x Width x Height 
  3. Maximum and Minimum Area of Quadrilaterals 
    1. Of a Quadrilateral: 
      1. Largest possible area = square 
      2. Minimum possible perimeter = square 
      3. A regular polygon of all sides equal and all angles equal will maximize area for a given perimeter and minimize perimeter for a given area
    2. Of a parallelogram: 
      1. Maximize area by placing two sides PERPENDICULAR to each other 

Circles and Cylinders 

  1. Basic Elements:
    1. Radius: Half distance around the circle
      1. Any line segment connecting the middle of the circle to any point is the radius 
    2. Diameter = 2r
      1. Distance across a circle 
    3. Circumference = 2nr
      1. Distance around a circle 
    4. Area = nr^2 (space inside of the circle)
      1. Area inside of the circle
    5. Central angle (questions about sectors) 
      1. Sector area / Circle Area = Arc Length/ Circumference =  Central Angle/360 
  2. Sectors:
    1. Fractional proportion of a circle aka a semicircle → a sector 
    2. Arc length → portion of the circumference that remains 
    3. Central angle (questions about sectors) 
      1. Sector area / (nr^2)→  Arc Length/ Circumference (2nr) →   Central Angle/360 
  3. Inscribed versus Central Angles 
    1. Vertex in a corner rather that at the center 
    2. Central angle defines the arch 
    3. Arc is 60 degrees 
    4. An inscribed angle is equal to half the arc it intercepts 
    5. If one of the sides of an inscribed triangle is the diameter of the circle → THE TRIANGLE MUST BE A RIGHT TRIANGLE 
  4. Cylinders and surface area
    1. Surface Area: (2 circles + rectangle) → 2 (nr^2) + (2nr)(H) 
      1. H = width 
      2. Length = the circumference 
      3. Only information needed is:
        1. The radius 
        2. Height of the cylinder 
    2. Volume: 
      1. V = nr^2 
  5. Coordinate Geo

y=mx+b

slope=rise/run

  1. Exterior angles

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