Math Shortcuts Flashcards
two digit number divided by 99
becomes a decimal with that two digit number repeating. For instance –> 49/100 = 0.494949….
FDP: Scientific Notation
FDP: Compensating Decimals
FDP: Solution Less Traveled
Quadratic Formula
The discriminant (b^2 - 4ac) which is under the radical in the quadratic formula can tell you how many solutions.
b^2 - 4ac > 0 –> 2 solutions
b^2 - 4ac = 0 –> 1 solutions
b^2 - 4ac = 0 –> 0 solutions
Conjugate of square root expression involving addition or subtraction
multiply by a negative number in an inequality
flip the sign
On test cases, when I see |x|, I will try
absolute value: + and -
On test cases, when I see x^2, I will try
exponents: 0, 1, and fractions
On smart numbers, I will avoid…
0 and 1 and usually #s that appear in the problem
On smart numbers when choosing 2 variables, I will…
pick two differen tnumbers
decimal raised to exponent = how many decimal places?
of decimals * exponent
decimals for a root = how many decimal places
of decimals / root
=# of decimals * (1/root)
What is the pattern if the denominator is a number equal to a power of 10 - 1 (eg 9, 99, 999, etc)
then the numerator is repeating
How to check if a number will terminate?
The denominator only has 2 and 5 as prime factors once fraction is in smallest form
Digit Place Value questions
example two digit number -A -> how to represent this
A = 10x + y
Find units digit or a remainder after division of 10
you can pay attention to only the digits
x^2 - y^2
(x+y)(x-y)
x^2 + 2xy + y^2
(x+y)(x+y) = (x+ y)^2
x^2 - 2xy - y^2
(x-y)(x-y) = (x+ y)^2
inequality statement –> what is the implication?
xy>0
x and y are both positive or negative
inequality statement –> what is the implication?
xy < 0
one positive, one negative
inequality statement –> what is the implication?
x^2 - x < 0
x^2 < x
0 < x < 1
When combining inequalities can yo subtract?
NO
direct proportions > what to use?
ratios
indirect proportions > what to use?
products
Reciprocals of Inequalities
Only flip the inequality if both are the same sign. If you do not know the sign, then you cannot take reciprocals
squaring inequalities
if both sides are known to be negative then
then flip the inequality sign. if this is not known you can’t do this
squaring inequalities
if both sides are known to be positive then
do not flip the sign
squaring inequalities
if one is positive and one is negative then
likely choose another technique besides squaring
squaring inequalities
if sign is unknown then
you cannot square
Compound interest formula
Rate -> how to express
ALWAYS distance over time
use one unit of time
If object moves the same distance twice at different rates, what’s the average weighted towards?
weighted towards slower!!
For evenly spaced sets what is the relation between the mean and median?
they are equal
For evenly spaced sets the median equals to
(first+ last)/2
Products of x consecutive integers is always divisible by what?
x!
Sums of consecutive integers
Odd: the sum is always a multiple of the number of items (sum = average * # of items)
Even: the sum of all the items is NEVER multiple of then number of items. b
Counting integers formula
last - first +1
Is the average of n consecutive integers an integer?
Yes if n is odd
No, if n is even
Composites
three or more factors (not 1 or prime)
factor foundation rule
if a is a factor of b, and b is a factor of c, then a is a factor of c
x and y are both a multiple of r.
Is x + y a multiple of r?
Yes!
And so would ax + by
What prime number is not odd?
2
If the sum of two prime numbers is odd, what must be one of the primes?
2
If the sum of two prime numbers is even, what must not be one of the primes?
2
Prime
only 2 factors > therefore 1 is not prime
even * even
even
even * odd
even
odd * odd
odd
even +/- even
even
odd +/- odd
even
odd +/- even
odd
4!
24
5!
120
6!
720
arranging group of n without restrictions
n!
if you add(or subtract) a multiple of N to a non multiple of N, is the result a multiple of N
NO
if you add (or subtract) two non multiples of N, is the answer a multiple of N
maybe
if a number has a prime factorization of
a^xb^yc^z,
what is the total number of factors?
(x+1)(y+1)(z+1)
perfect square
squares of other integers (25,4,9)
what does it mean if a number has an odd number of total factors
it is a perfect it must be a perfect square, cube, etc
Otherwise it would be a pair!!
What does it mean if a number’s prime factorization contains any odds?
It is not a perfect square!
remainder formula
dividend = quotient * divisor + remainder
