Quant Notes

Terminating decimal

A fraction is a terminating decimal only if the denominator is 2 or 5

GoldBach believe

Any even number greater than or equals 4 can be presented as the sum of 2 primes Examples :

• 4 = 2 + 2
• 8 = 3 + 5

What fraction of X

In a certain class, 5/12 of all the students are girls and 1/4 of all the students are girls who take Spanish. What fraction of the girls take Spanish? I answered it as 1/4 , but fraction of girls means the take the number of girls as the denominator. I don’t have it but the fraction is there so 1/4 * 12/5 = 3/5

Comparison

Take care of assumptions when comparing 2 unknowns. Unless a constraint explicitly mentioned; you can’t assume any relation about the them m/p > n/p so that doesn’t mean m > n , Without knowing the signs of the variables, do not assume that m is greater than n. While it certainly could be (e.g., m = 4, n = 2, and p = 1), if p is negative, the reverse will be true (e.g., m = 2, n = 4, and p = –1).

Root vs Exp If x < 1 : $\sqrt[Y]{X}>X>X^Y$ If X > 1: $X^Y>X>\sqrt[Y]{X}$

Number of factors

Take care when asking for the factors “Does the questions require the +ve factors only or all factors”

Example There’re 4 positive factors for 6 , 8 factors in total {1, 2, 3, 6, -1, -2, -3, -6}

Range

$range=largest-smallest$

Ratios

When there’s an equation like 1 Dollar = 100 Cents, and then asked, what 2 Dollars actually worth in cents ?

so the NUMBER of cents is 200, that’s a value not an equation to continue with. Generally we have the equation 1D = 100C so when asked what 2D worth, we should multiply both sides by 2 to get a correct relation 2D = 200C.

we can’t actually present this equation in a meaningful ratio, they are of different units, and even the value of each paper is different so we can’t sum them up.

TIP

Don’t fucken use ratios when you see such an equation, just continue as is.

solve manhattan 5lb 3rb edition, chapter 20, problems 15, 34

|X| absolute value

If the problem is |X| = 2, that means what are the X points at which the distance from zero is 2, which are -2 and 2.

what is |0| means? means what is the distance from zero to zero, which is zero

Ratio with different units

Parts of a ratio that have different units can’t sum up

iron (ounces) to carbon (ounces) to steel (sheets) is 98 : 2 : 1. Because there are different units (ounces and sheets), the Part numbers do not sum to the Whole number as they typically do

Never underestimate the GRE

Does adding 2 to the numerator and denominator of a fraction produces equivalent fraction ?

MAYBE , not always. example adding 2: 6 / 6 → (6 + 2) / (6 + 2) → 8 / 8 SAME RESULT (1) so adding same value to both sides of a fractions results the same value if both parts are the same.

Divisible by factors too

If a number is a multiple of some number, then it’s also a multiple of the factors of that number.

Example: if Y’s factors are {a, b, c}, and Y divides X then {a, b, c} also divides X

MAX value X

If 50! / 4^x is an integer, what is the maximum value of x ?

You can’t check how many 4 in range from 1 to 50 as 4 is not a prime number, so you need to check how many 2 there which equals 2x then x = number of 2x / 2

Number Line

Any real number (rational or irrational) can be presented on the number line

System of linear solution

A system of linear solutions can have one, infinite, or zero solutions

1. $x+y=1,2x+2y=2$ x, and y have infinite solutions, can’t isolate and solve for one variable

2. $x+y=1,5x+5y=7$ There’s no solution for these equations

3. $x+y=1,3x+2y=6$ These equations have a single solution for x, and y

Quadratic equation solution

May have zero, one, or two solutions depending on the value under the DISCRIMINANT $\sqrt{b^2-4ac}$ if b2 - 4ac is: 5. +ve the equation has 2 solutions 6. -ve the equation has 0 solutions 7. 0 the equation has 1 solution

Fucken remember If the problem says you consumed 30% then you have the fuck 70% remaining. don’t multiply by 0.3 but 0.7

X^3 Equations

These equations should have 1, 2, 3 roots

1. 1 root, then the equation can be represented as (X + a)
2. 2 roots, then the equation can be represented as (X + a)(X + b)
3. 3 roots, the normal case the equation is presented as (x + a)(x + b)(x + c)

Example If 3 is a root of the equation y = x^3 − 7x^2+ 2(8x−6) and there is exactly one more root in the equation, what is the other root?

Answer As the equation has only 2 roots, one of then is 3. then the equation can be written as $x^3-7x^2+16x-12=(x-3)(x+b{)}^2$ if we applied the multiplication and compared the number then $(x-3)(x+b{)}^2=(x-3)(x^2+2bx+b^2)=x^3+(2b-3)x^2+(b^2-6b)x-3b^2$ so now we can conclude that $2b-3=-7$ and hence $b=-2$ so the other root is 2

Work and Rates

If both rates have the same +ve or -ve effect then the rates must be added, if they are against each other then subtract the rates.

Compound Vs Simple interests

Both are equal for the first year only, then compound becomes better. As the second interest calculations for the compound will take the first interest into account.

Example Invest 1000$ for 2 years, with interest rate 10% annually If the interest is simple: then after 2 years I will have 1000 + (2)(0.1)(1000) = 1200 while if the interest was compounded then the total will be (1000)(1.1)^2 = 1210

Notice for a single year both will make the same gain.