GMAT MATH REFERENCE
1. How to use the PERCENT/DECREASE formulas
X 100%
% Increase = Amount of increase
Original whole
% Decrease = X 100%
Amount of decrease
Original whole
2. How to recognize MULTIPLES OF 2, 3, 4, 5, 6, 9, 10 and 12
• 2: Last digit is even.
• 3: Sum of digits is multiple of 3.
• 4: Last two digits are multiple of 4
• 5: Las digit 5 or 0
• 6: Sum of digits is a multiple of 3 and last digit is even
• 9: Sum of digits is multiple of 9
• 10: Last digit 0
• 12: Sum of digits is multiple of 3 and last two multiple of 4
3. How to find an AVERAGE and how to use it to find the SUM
Sum of terms
Number of terms
Average = Sum = (Average) x (Number of terms)
4. CONSECUTIVE NUMBERS
Average: Is simply the average of the smallest number and the largest number.
Count: The number of integers from A to B inclusive is B-A+1.
Sum:
Sum = (Average) x (Number of terms)
5. MEDIAN, MODE & RANGE
Median: Put the numbers in numerical order and take the middle number.
Mode: The Mode is the number that appears most often.
Range: Is the difference between the highest and the lowest values.
6. RATIO
How to use actual numbers to find a Ratio
How to use ratio to find actual numbers: Example:
Ratio = of
to
The ratio of boys to girls is 3 to 4.
If there are 135 boys, how many girls are there?
3 = 135
4 x
7. How to use actual numbers to determine a RATE
Identify the quantities and the units to be compared. Keep the units straight.
Example: Anders typed 9450 words in 3,5 hours. What was his rate in words per minute?
First convert 3,5 hours to 210 minutes. Then set up a rate with words on top and minutes on
bottom.
HINT: Unit before per goes on top, and unit after per goes on the bottom
8. How to count the NUMBER OF POSSIBILITIES
In most cases, you won’t need to apply the combination the permutation
formulas on the GMAT. The number of possibilities is generally so small that
the best approach s just to write them out systematically and count them.
9. How to calculate a simple PROBABILITY
Example: What is the probability of throwing a 5 on a fair six-sided die?
There is one favourable outcome (throwing 5). There are six possible outcomes (one for each
side of the die) PROBABILITY= 1/6
Probability = Number of favorable outcomes
Total number of possible outcomes
10. How to FACTOR certain POLYNOMIALS
(a+b)² = a² + 2ab + b² (a-b)² = a² - 2ab + b²
a²- b² = (a-b)(a+b)
11. Geometry – HYPOTENUSE – Pythagorean Theorem
h² = a² + b²
12. SPECIAL RIGHT TRIANGLES
5-12-13 30º-60º-90º
3-4-5 45º-45º-90º
13
5
3 12
5
4 √3 2
1
60º 45º
√2
1
1
45º
13. CIRCUMFERENCE of a CIRCLE
Circumference = 2πr
14. AREA of a CIRCLE
Area = πr²
15. SLOPE of a LINE
Slope= change in y
change in x
16. COMBINED PERCENT INCREASE/DECREASE
Always start with 100 as original value and see what happens
17. SIMPLE INTEREST problem
Interest = (principal) x (interest rate) x (time)
decimals years
18. COMPOUND INTEREST
Final balance = (principal) x (1 + interest rate)
(time)(c)
C
Where C = the number of times compounded annually
19. How to use the ORIGINAL AVERAGE and NEW AVERAGE
to figure out WHAT WAS ADDED or DELETED
Number added = (new sum) – (original sum)
Number deleted = (original sum) – (new sum)
Example: the average of five numbers is 2. after one number is deleted, the new average is -3.
what number was deleted?
Find the original sum from the original average = 5x2=10
Find the new sum from the new average = 4x(-3) = (-12)
The difference between the original sum and the new sum is
Number deleted = 10 – (-12)= 22
20. How to find an AVERAGE RATE
Average A per B = Total A Average Speed = Total distance
Total B Total time
21. How to solve a WORK problem
In a work problem, you are given a rate at which people or machines perform
work individually, and asked to compute the rate at which they work together (or
vice-versa). The work formula states:
1/r + 1/t = 1/t
22. How to determine a COMBINED RATIO
Multiply one or both ratios by whatever you need to in order to get the terms
they have in common to match.
Example: The ratio of a to b is 7:3. The ratio of b to c is 2:5. What is the ratio of a to c?
Multiply each member of a:b by 2 and multiply each member of b:c by 3 and you get
a:b= 14:16 and b:c = 6:15. Now that the b´s match, you can just take a and c and say a:c = 14:15
23. How to solve a DILUTION or MIXTURE problem
You have to determine the characteristics of the resulting mixture when
substances with different characteristics are combined. Or, alternatively, you
have to determine how to combine substances with different characteristics to
produce a desired mixture. There are two approaches to such problems:
Straight forward
Example: If 5 pounds of raisins that cost $1 per pound are mixed with 2 pounds of almonds that
cost $2,40 per pound, what is the cost per pound of the resulting mixture?
Solution: 5(1) + 2(2,4) = 9,8 ….. The cost per pound is 9,8/7= $1,40
Balancing method
Example: How many liters of a solution that is 10% alcohol by volume must be added to 2 liters
of a solution that is 50% alcohol by volume to create a solution that is 15% alcohol by volume?
Solution: Make the weaker and stronger (or cheaper and more expensive, etc.) substances
balance. That is: (%/price difference between the weaker solution and the desired solution) x
(amount of weaker solution) = (%/price difference between the stronger solution and the desired
solution) x (amount of stronger solution)
n(15-10) = 2(50-15)
n = 70/5 = 14. So 14 liters of the 10% solution must be added.
24. How to solve a GROUP problem involving BOTH/NEITHER
Some GMAT word problems involve two groups with overlapping members,
and possibly elements that belong to neither group. It’s easy to identify this type
of question because the words “both” and/or “neither” appear in the question.
Group1 + Group 2 + Neither – Both = Total
Example: Of the 120 students at a certain language school, 65 are studying French, 51 are
studying Spanish, and 53 are studying neither language. How many are studying both French and
Spanish?
Solution: 120 = 65 + 51 + 53 – B --- B = 169 – 120 = 49
25. How to solve a PERMUTATION problem
If you are asked to find the number of ways to arrange a smaller group that’s
being drawn from a larger group, you can use the permutation formula
P = n! Where n = # in the larger group
(n-k)! k = # you are arranging
26. How to solve a COMBINATION PROBLEM
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, and a different formula is called for:
C = n! Where n = # in the larger group
k!(n-k)! k = # you are arranging
27. How to solve a MULTIPLE-EVENT PROBABILITY problem
Many hard probability questions involve finding the probability of a certain
outcome after multiple events (a coin being tossed several times, etc.). These
questions come into two forms: those in which each individual event must occur
a certain way, and those in which individual events can have different outcomes.
To determine multiple event probability where each individual event must
occur a certain way:
• Figure out the probability for each individual event.
• Multiply the individual probabilities together.
Example: if 2 students are chosen at random from a class with 5 girls and 5 boys, what’s the
probability that both students chosen will be girls?
Solution: The probability that the first student chosen will be girl is 5/10 = 1/2, and since there
would be 4 girls left the probability that the second student chosen will be a girl is 4/9. Thus, the
probability the both students chosen will be girls is: 1/2 x 4/9 = 2/9
To determine multiple event probability where individual events can have
different outcomes:
• Find the total number of possible outcomes by determining the number
of possible outcomes for each individual event and multiplying these
numbers together.
• Find the number of desired outcomes by listing out the possibilities.
Example: If a fair coin is tossed 4 times, what’s the probability that at least 3 of the 4 tosses will
come up heads?
Solution: There are 2 possible outcomes for each toss, so after 4 tosses there are a total of
2x2x2x2 = 16 possible outcomes. List out all the possibilities where “at least 3 of the 4 tosses”
come up heads:
HHHT
HHTH
HTHH
THHH
HHHH
So there’s a total of 5 possible desired outcomes. Thus, the probability that at least 3 of the 4
tosses will come up heads is:
Number of desired outcomes = 5
Number of possible outcomes 16
28. QUADRATIC EQUATIONS
29. How to solve a SEQUENCE problem
In a sequence problem, the nth term in the sequence is generated by performing
an operation, which will be defined for you, on either n or on the previous term
in the sequence. Familiarize yourself with sequence notation and you should
have no problem.
Example: What is the difference between the fifth and fourth terms in the sequence 0, 4, 18, …
whose nth term is n²(n-1).
Solution: Use the operation given to come up with the values for your terms:
n5 = 5² (5-1) = 25(4) = 100
n4 = 4² (4-1) = 16(3) = 48
So, the difference between the fifth and the fourth terms in the sequence is = 100 – 48 = 52
30. How to find the MAXIMUM and MINIMUM lengths for a
SIDE of a TRIANGLE
If you now two sides of a triangle, then you now that the third side is somewhere
between the difference and the sum.
Example: 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?
Solution: The third side is greater than the difference (7-3=4) and less than the sum (7+3=10)
31. How to find one angle or the sum of all the ANGLES of a
REGULAR POLYGON
Sum= (n-2) x 180
Degree of one angle = (n-2) x 180
n
x1, x2 = -b +/- √ b² - 4ac
2a
32. How to find the LENGTH of an ARC
Length of arc = n x 2πr
360
nº
r
Length
33. How to find the AREA of a SECTOR
Area of arc = n x πr²
360
nº
Area
34. VOLUME of a CYLINDER
35. VOLUME of a SPHERE
Volume = πr² h
h=height
Volume = ¾ πr³
