How to ace the GMAT in 28 days: Day 26 (Quantitative review)

Time to write up my Quantitative cribsheet. Here it is:

Numbers & properties of numbers
Integers: all the whole numbers, -2, -1, 0,
0 is an even integer, but not positive or negative
-8 is LESS than -7
What is true for one odd number is generally true for all odd numbers
Natural numbers: all positive integers, not 0 and not negative

Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37
Except for 2, all prime numbers are odd. 1 is not a prime
A prime has only TWO factors: itself and 1

Rational numbers: fractions of two integers, eg. 3/8
Irrational numbers: like ∏ or √2 that can’t be written as rational numbers
Imaginary numbers: like √-2 that can’t exist naturally
Real numbers: all integers, fractions, decimals, rational, and irrational numbers
Absolute value: a number without extras like a negative sign (i.e. its magnitude)
Any product of integers will have factors that are combinations of these integers

The last digit of the square of an integer is the same as the last digit of the last digit of the integer squared, i.e 94^2 ends in a 6 because the last digit of 4^2 ends in a 6

All perfect squares have the last digit 0, 1, 4, 5, 6, or 9. No perfect squares ends in 2, 3, 7, or 8.

Adding, subtracting, multiplying, dividing
When you add or subtract two even integers, you get an even integer
When you add or subtract two odd integers, you get an even integer
When you add or subtract an even and an odd integer, you get an odd integer
When you multiply two even integers, you get an even integer
When you multiply an odd and an even integer, you get an even integer
When you multiply two odd integers, you get an odd integer

When you multiply or divide two positive numbers, the result is positive
When you multiply or divide two negative numbers, the result is positive
When you multiply or divide a positive number by a negative number, the result is negative

When you add two positive numbers, the result is positive
When you add two negative numbers, the result is negative
When you subtract a negative number from a negative number, you end up adding that number

Fractions
Multiply 2 fractions by multiplying their denominators AND numerators, i.e. 2/8 * 3/9 = 6/72
Divide a fraction by another by switching numerator and denominator of the divisor and multiplying. i.e. 2/8 divided by 3/9 equals 2/8 x 9/3 = 18/24
When positive fractions between 0 and 1 are squared they get smaller, eg 1/2 * 1/2 = 1/4
0.125 is 1/8.

The reciprocal of x is 1/x, and the reciprocal of a/b is b/a

You CAN cancel out roots and exponents in a fraction:
(3√2 * 4√5 * 2√3) / (3√3) lets you cancel one 3 and one √3 from the numerator

Roots and exponents
4^3: 4 is the base, 3 is the exponent
4b^2: 4 is the coefficient, b is the base, 2 is the exponent
256^1/4 equals 4th root of 256, ie 4 (4 * 4 * 4 * 4)
8^-2 equals 1/8^2, or 1/64 (a power of -anything equals the reciprocal of anything)
A positive number taken to a even or odd power remains positive (2 * 2 * 2 = 8)
A negative number taken to an odd power remains negative (-2 * -2 * -2 = -8)
A negative number taken to an even power becomes positive, (-2 * -2 = 4)

1/3^8 is the same as 3/3^9 (useful for rearranging fractions to have the same denominator)

64 is the square of both 8 and -8

√3 * √3 = 3, NOT 9 or √9

3^5 * 3^3 = 3^8, NOT 9^8 (add the exponents)
3^5 / 3^3 = 3^2 (subtract the exponents)

3b^5 * 5b^3 = 15b^8, NOT 8b^anything (watch for coefficients and bases)

Anything to the power 0 is 1

6 * 10^2 is 600. 6 * 10^2 * 3 * 10^2 is 18 * 10^4 but should be expressed as 1.8 * 10^5

To simplify an exponent term (i.e. √20) find a square in the radical (20 = 5 x 4, and 4 is a perfect square) and bring the root of the square outside (making 2√5.) I.e. √50 is the same as √(25*2) and therefore simplifies to 5√2

Ratios & proportions
1:4 means ‘in the ration 1 to 4’

Look for similarities in two proportionate figures: e.g 1.919 and 0.1919 may look different, but one is just the other multiplied by 10

Algebra
Monomial: single term expression, such as 4x or ax^2
Polynomial: expressions with more than one term, such as a^2 – b^2 (binomial) or ab^2 +2ac +b (trinomial) or ax^2 +bx +c (quadratic polynomial)

Beware of ‘equations’ that don’t equal anything; they’re just terms and MEANINGLESS without a result or other term equal to them

x^2 – y^2 = (x+y)(x-y)
x^2y^2 = (xy)^2
7x + -10x + 22 is the same as -3x = -22

3x + 4y – 7z plus 2x – 2y +8z can be added to result in 5x +2y + z (like terms can be added or subtracted). This includes exponents, so (4x^2 -6xy -12y^2) – (8x^2 – 12xy +4y^2) is the same as
-4x^2 +6xy -16y^2 (REMEMBER MINUS SIGNS!)

To multiply out expressions, multiply each term: 4x(x-3) = 4x^2 – 12x
To divide expressions, divide each term: (16x^2 +4x) / 4x = 4x +1 (THE ONE IS IMPORTANT!)

When solving two algebraic fractions that equal each other, try to make the numerators addable or subtractable – that means the denominators must be equal. Eg. 2x / (4+2x) = 6x / (8x+6)…. multiply the first fraction by 3/3 to give 6x / (12+6x), which means (12+6x) must be equal to (8x+6).

(x^2 + 2xy + y^2) * (x-y) : just multiply each term in the first part by x then -y
To give first x^3 + 2x^2y +xy^2 and -x^2y -2xy^2 -y^3, then finally add them to get the combined result: x^3 +1x^2y -1xy^2 – y^3

Use FOIL to multiply out a pair of terms in brackets and get a quadratic polynomial: (4x-5)(3x+8) = 12x^2 +32x -15x -40

Factoring polynomials: find a ‘common factor’ like -7 in -14x^3 – 35x^6, remember you’ve taken the minus sign outside too, and make it -7(2x^3 + 5x^6) then do the same with x^3 to get -7x^3(2 +5x^3) remembering x^3 divided by x^3 is 1, not x

Factoring quadratics: x^2 + 5x + 6 and do a reverse FOIL: the first term in each bracket must be x or -x, the outside terms and inside terms have to together equal 5x, and the last terms must equal 6. So the last terms in each bracket will sum to 5 and multiply to 6, they must be 2 and 3. Giving the result in binomial factors, (x +2) (x +3)

There’s a shortcut if both terms in a polynomial are perfect squares and the second one a minus, such as x^2 – 4 or x^2 -16. They’ll always give the factors (x-a)(x+a) where a is the root of the second term.

Linear equations (easy) such as 4x + 10 = -38 , just distribute the numbers to get 4x = -48

Simultaneous equations (hard) come in pairs such as 4x + 5y = 30 and x + y/2 = 10. First solve for y (such as 5y = 30-4x) and substitute it into the second to get x + (30-4x)/2 = 10 which is solvable for x

Quadratic equations (harder): when a quadratic polynomial (i.e. one containing a square) is set equal to zero, such as ax^2 + bx + c = 0. If a quadratic equation can’t be easily factored, use the rearrangement x = -b ± √(b^2-4ac) / 2a to solve it.

(x/2)/2 is the same as x/x – 2/x. and 1 – 2/x. DON’T ‘cancel out’ the 2 and -2; it won’t work

Total distance = speed * time
Total production = work rate * time

Remember (x+y / x) is the same as (x/y + y/y)

Functions: f(x) means ‘f of x’ such as f(x) = 2x^2 +3. For x = 2, just substitute in 2 for x. f(2) = 11. In functions, f(x) does NOT mean f * x.
Eg. for f(x) = (x-2)^2, find f(2x-2)… first substitute 2x-2 where you see x, to get f(2x-2) = (2x-2-2)^2. That’s (2x-4)(2x-4), which is 4x^2 -8x -8x +16, or 4x^2 – 16x +16.

Domain: the set of all numbers that could be an input, or value, of f(x). It’s all possible values of x that are real numbers.
For example, in f(x) = x+4 / x-2, x can’t be 2 since that’d make the denominator 0 (and the number unreal.) So the domain of f(x) ≠ 2.
Conversely, in f(n) = 3 √n+2, the n+2 can’t be negative (because the root of a negative number isn’t real) so n must equal -2 or more. f(n) ≥ -2.

Ranges: |vertical bars mean range| the set of all numbers that could be an output, or result, of f(x). It’s all possible values of f(x).

Range can be expressed as 6 < x < 12 (meaning x is 7, 8, 9, 10, or 11) or 6 ≤ x ≤ 12 (meaning x is 6, 7, 8, 9, 10, 11, or 12)
Read the question carefully – is it asking you to find the range of x or of f(x)? They’re different

Geometry

Angles in a triangle always sum to 180, and in a quadrilateral to 360
Useful right angled triangles:
3:4:5, 5:12:13, 8:15:17, 7:24:25
A 30,60,90 degrees triangle is a right angled triangle that’s half an equilateral
A 30,60,90 triangle has length of sides x, x√3, 2x
A 45,45,90 degrees triangle has sides x, x, x√2 and is half a square with sides x

Area of any triangle is half the base * perpendicular height.

Parallelograms have 2 pairs of parallel sides, and the opposite angles are equal.
Area of any parallelogram is base * perpendicular height

If all you know about a square is the length of its diagonal, the area will be d^2/2.

Trapezoids have four sides, two of which are parallel. Its area is average of the 2 unequal bases * perpendicular height.

When two lines intersect, they form 2 pairs of complementary (equal and opposite) angles

A circle’s circumference is 2∏r, its area is ∏r^2.
A chord cuts across a circle, and the longest possible chord is the diameter.
A cylinder’s volume is ∏r^2h (height) and its surface area is 2∏rh+2∏r^2
A sphere’s surface area is 4∏r^2, and its volume 4/3∏r^3

A triangle CAN intersect a circle at just one point; touching counts as an intersect

Co-ordinates
x is the horizontal axis, y the vertical axis. (0,0) is the origin.
Quadrant I is the top right, II the top left, III the bottom left, and IV the bottom right

Slope is the steepness of a line (postive if rising left to right, negative if falling.) To find slope, divide change in horizontal distance by change in vertical distance of the line. Ie. a line between co-ordinates (0,2) and (4,0) falls 2 down for every 4 across, making the slope -2/4 or -1/2.

A formula like y = mx + b describes the line by showing y as a function of x, where m is the slope and b is the y-intercept, where the line crosses the y-axis.

To recognise if a graph has a function, see if one vertical line can cross the line in more than one place. If it can, the line is not the graph of a function.

Domains and ranges of graphs are the same as domains and ranges of functions. A graph’s domain is all the values of x that can go into the equation as inputs; its range is all the values of x produced as outputs when you work it out,

Sets
set A U set B is the union: the set of all elements in A and B
set A ∩ set B is the intersection: the set of all elements common to both A and B
set A C set B is the subset: all elements in A are also in set B

A = {1,2,3} means that set A is a set of these 3 numbers
|3| means the set has 3 elements (NOT totals 3)

Combinations & permutations
Permutations = number of ways a set of items can be arranged in specific orders
Combinations = number of ways a set of items can be arranged if order doesn’t matter

Factorial = product of all natural numbers in a set, e.g. {5,4,3,2,1} would be 5*4*3*2*1, known as n! (Remember 0! always equals 1.)

So a set of n objects such as {a, b, c) has 6 permutations (3 objects, 3*2*1 or 6). 3! is 6.

With a set of n objects but a different number of slots to fit them into (r) the permutations formula is n! / (n-r)!
Eg. To find how many ways 6 objects can be arranged in 4 slots, 6*5*4*3*2*1 / 4*3*2*1 = 30.

With a set of n objects, a different number of slots to fit them into, but no need for any specific orders, the combinations formula is n!/r!(n-r)!
Eg. To find how many ways 6 objects can fill 4 slots without repeating the same group of 4, it’s 6*5*4*3*2*1 / 4*3*2*1(2*1) or 15.

You can’t simplify factorials: 10!/5! is not 2!

Averages
Mean average is total of the all the values divided by number of values

Median is the middle value if all the values are arranged in order – EVEN IF the middle value is a long way from one end and looks ‘skewed’; this is why nobody uses median much!

Mode is the value that appears most often among the values

Watch out for weighted means, ie. 3 people score 12 and 1 scores 10; the mean is 36+10 / 4, not 22/4.

Range in averages is the largest value minus the smallest value, i.e. values from -4 to 8 have a range of 12.

Standard deviation is how far the values spread out from the mean. A regular Bell Curve will always have 68% of the values within 1 SD, 95% of them within 2 SDs, and 99.7% within 3 SDs.High SD means the values are spread out; small SD means they’re clustered closely around the mean.

SD = √ ((sum of (each number minus the average) ^2 / number of values)

Probability
probability of E for a single event:
P(E) = number of outcomes involving occurence of E / number of possible outcomes

probability for multiple events if both can’t happen at once, like rolling a 5 and 6 with one die:
P(A or B) = P A + P B (add the probabilities of A and B to get the probability of either happening)

probability for multiple events if two or more can happen together or separately, like drawing a playing card that’s either a royal or a Club or both:
P(A or B) = P A + P B – P(A and B)

probability of multiple events happening together, like drawing a card that’s both a royal and a Club:
P (A and B) = P A * P B

probability of multiple events if the second event depends on the first, like pulling another 5p from a bag of coins from which some 5p’s have already been taken:
P (A and B) = P A * P(B given A)

Answering data sufficiency questions
About half the 37 Quant questions are data sufficiency, with this set of answers to apply to two statements:

A Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D EACH Statement ALONE is sufficient.
E Statements (1) and (2) TOGETHER are NOT sufficient.

The method for solving:

– First, READ THE QUESTION and note down what information you HAVE
– Then note down what information you NEED to answer the question
– Then see if STATEMENT 1 is sufficient to answer it alone; don’t look at statement 2
– If yes, eliminate B, C and E; the answer will be A or D
– Then see if STATEMENT 2 gives enough missing info to answer it alone; don’t look back
– If yes and so was statement 1, eliminate A; the answer is D.
– If yes but statement 1 did not, the answer is B
– If no, but statement 1 adds enough info to provide the answer, eliminate B; the answer is C
– If no, the answer is A.

WRITE DOWN WHAT YOU NEED – many ds questions hide info that’s easy to skim over, i.e. asking about Tom, Dick, and Harry when the Statements only apply to Tom and Dick, or asking how many things are left over when 2/3 and 1/6 are taken away without telling you how many things you started with.

ALWAYS think about: could the numbers be NEGATIVE (and hence provide two solutions not one), go for SUBSTITUTION to test things out, and substitute MORE THAN ONE set of numbers to check the rule applies in more than one case.

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