How to Test for Primes Using Square Roots and Python

4/13/2018

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Finding Prime Numbers

Prime numbers are very important, yet many students do not see the value of learning them. Primes have several applications, most importantly in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. One key challenge is to find prime numbers. Interestingly, Prime numbers and their properties were first studied extensively by the ancient Greek mathematicians. Euclid, for example, proved that there are infinitely many prime numbers.

Just to refresh our memory, a number greater than 1 is called a prime number, if it has only two factors, namely 1 and the number itself.

Proof by Contradiction

One of the first known proofs is the method of contradiction. It is used to calculate prime factors of large numbers. Calculating prime factors of small numbers is easy. For example, the factors of 17 is 1 and 17, so it is a prime number. What about large numbers? Let's look at the proof by contradiction method.

If a number n is not a prime, it can be factored into two factors a and b, such that n = a*b. For example, let's say a * b = 100, for various pairs of a and b.

If a = b, then they are equal, we have a*a = 100, or a^2 = 100, or a = 10, the square root of 100. If one of the numbers is less than 10, then the other has to be greater to make it to 100. For example, take 4 x 25 = 100. 4 is less than 10, the other number has to be greater than 10. In other words, if a * b, if one of them goes down, the other number has to get bigger to compensate so the product stays at 100. Put mathematically, the numbers revolve around the square root of their product.

Let's test if 101 is prime number. You could start dividing 101 by 2, 3, 5, 7, etc, but that is very tedious. A better way is to take the square root of 101, which is roughly equal to 10.049875621. So you only need to try the integers up through 10, including 10. 8, 9, and 10 are not themselves prime, so you only have to test up through 7, which is prime.

Because if there's a pair of factors with one of the numbers bigger than 10, the other of the pair has to be less than 10. If the smaller one doesn't exist, there is no matching larger factor of 101.

Let's now build an algorithm using this method to test any number for primality.

Algorithm in Python

import math

def isPrime(num):
if (num < 2):
return False
else:
for i in range(2, int(math.sqrt(num)) + 1):
if num % i == 0:
return False
return True

print(isPrime(33))
print(isPrime(0))
print(isPrime(47))
print(isPrime(1047))
print(isPrime(11))
print(isPrime(59392847))

N.B: Do not just copy the code because you have to be careful with indentation in python.

Try the above algorithm and let us know if you found it useful or have alternative solutions.

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Challenge Yourself with this Tricky but Simple Problem on the SAT

4/6/2018

$Math Recusion SAT$

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Problem

N.B: This problem is for SAT Subject Math Level 2

$If \quad x_0 \quad and \quad x_{x+1} = \sqrt {4 + x_n}, \quad then \quad x_3 = $

(A) 2.65
(B) 2.58
(C) 2.56
(D) 2.55
(E) 2.54

Solution

Answer is C.

This is a simple but tricky problem. It is simple to apply but you have to think recursively.

For n = 0, we have:

$x_{0+1} = \sqrt {4 + x_0} $

$x_1 \quad = \quad \sqrt {4 + 3} \quad = \quad \sqrt{7} \quad = \quad 2.65 $

$x_2 \quad = \quad \sqrt {4 + x_1} \quad = \quad \sqrt {4 + 2.65} \quad = \quad 2.58 $

$x_3 \quad = \quad \sqrt {4 + x_2} \quad = \quad \sqrt{4 + 2.58} \quad = \quad 2.56$

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How to Bound Equations that Get Larger on the ACT

4/4/2018

Photo by Antoine Dautry on Unsplash

Problem:

In the equation r = 4/(2 + k), k represents a positive integer. As k gets larger without bound, the value of r:

F. gets closer and closer to 4.
G. gets closer and closer to 2.
H. gets closer and closer to 0.
J. remains constant.
K. gets larger and larger

Answer:

Answer is H.

As k gets larger and larger without bound, the expression 4/(2+k) becomes 4 divided by an increasingly large number. For example, think about the trend between the following fractions:

4/100,
4/10,000,
4/1,000,000, ...

Looking at it this way, you can see that the expression for r gets closer and closer to zero.

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How to Enjoy Jane Eyre on the ACT

4/2/2018

Seeing me, she roused herself: she made a sort of effort to smile, and framed a few words of congratulations; but the smile expired, and the sentence was abandoned unfinished. She put up her spectacles and pushed her chair back from the table.

“I feel so astonished,” she began, “I hardly know what to say to you, Miss Eyre. I have surely not been dreaming, have I? Sometimes I half fall asleep when I am sitting alone and fancy things that have never happened. It has seemed to me more than once when I have been in a doze, that my dear husband, who died fifteen years since, has come in and sat down beside me; and that I have even heard him call me by my name, Alice, as he used to do. Now, can you tell me whether it is actually true that Mr. Rochester has asked you to marry him? Don’t laugh at me. But I really thought he came in here five minutes ago, and said that in a month you would be his wife.” [10]

“He has said the same thing to me,” I replied.

“He has! Do you believe him? Have you accepted him?”

“Yes.”

She looked at me bewildered.

“I could never have thought it. He is a proud man; all the Rochesters were proud: and his father at least, liked money. He, too, has always been called careful.

He means to marry you?” “He tells me so.”

She surveyed my whole person: in her eyes I read 30 that they had there found no charm powerful enough to solve the enigma.

“It passes me!” she continued; “but no doubt it is true since you say so. How it will answer I cannot tell: I really don’t know. Equality of position and fortune is often advisable in such cases; and there are twenty years of difference in your ages. He might almost be your father.” [22]

“No, indeed, Mrs. Fairfax!” I exclaimed, nettled; “he is nothing like my father! No one, who saw us 40 together, would suppose it for an instant. Mr. Rochester looks as young, and is as young, as some men at five and twenty.”

“Is it really for love he is going to marry you?” she asked.

I was so hurt by her coldness and skepticism, that the tears rose to my eyes.

“I am sorry to grieve you,” pursued the widow; “but you are so young, and so little acquainted with men, I wished to put you on your guard. It is an old saying that ‘all is not gold that glitters’; and in this case I do fear there will be something found to be different to what either you or I expect.” [30]

“Why?—am I a monster?” I said: “Is it impossible that Mr. Rochester should have a sincere affection for me?”

“No: you are very well; and much improved of late; and Mr. Rochester, I dare say, is fond of you. I have always noticed that you were a sort of pet of his. There are times when, for your sake, I have been a little uneasy at his marked preference, and have wished to put you on your guard; but I did not like to suggest even the possibility of wrong. I knew such an idea would shock, perhaps offend you; and you were so discreet, and so thoroughly modest and sensible, I hoped you might be trusted to protect yourself. Last night I cannot tell you what I suffered when I sought all over the house, and could find you nowhere, nor the master either; and then, at twelve o’clock, saw you come in with him.

“Well never mind that now,” I interrupted impatiently; “it is enough that all was right.” [40]

“I hope all will be right in the end,” she said: “but, believe me, you cannot be too careful. Try and keep Mr. Rochester at a distance: distrust yourself as well as him. Gentlemen in his station are not accustomed to marry their governesses.”

Questions:

1. When Mrs. Fairfax says, “Gentlemen in his station are not accustomed to marry their governesses,” she is expressing her belief that:

A. Mr. Rochester is incapable of loving Miss Eyre.
B. Mr. Rochester will treat Miss Eyre like a governess when they are married.
C. Mr. Rochester may not be sincere about his feeling towards Miss Eyre
D. Mr. Rochester may not really have asked Miss Eyre to marry him.

2. It can be reasonably inferred from the conversation that Mrs. Fairfax believes Miss Eyre will:

F. recognize that Mr. Rochester actually wants to marry Mrs. Fairfax.
G. marry Mr. Rochester much sooner than originally planned.
H. no longer desire to marry Mr. Rochester.
J. potentially regret her decision to agree to marry Mr. Rochester.

3. Mrs. Fairfax’s opinion about Miss Eyre and Mr. Rochester’s relationship can best be exemplified by which of the following quotations from the passage?

A. “Mr. Rochester looks as young, and is as young, as some men at five and twenty.”
B. “How it will answer I cannot tell: I really don’t know.”
C. “He is a proud man; all the Rochesters were proud.”
D. “But I really thought he came in here five minutes ago, and said that in a month you would be his wife.”

4. The phrase “you were so discreet, and so thor- oughly modest and sensible” (lines 36–37) is used by Mrs. Fairfax to:

F. explain why Miss Eyre should not marry Mr. Rochester.
G. explain why it is likely that Mr. Rochester really does not plan on marrying Miss Eyre.
H. explain why Mrs. Fairfax had not discussed Mr. Rochester’s feelings toward Miss Eyre before.
J. insult Miss Eyre and let her know that Mrs. Fairfax was disappointed in her.

5. The passage makes it clear that Miss Eyre and Mr. Rochester:

A. get married.
B. do not really know each other well enough to become engaged.
C. will not live happily because they will be shunned by society.
D. have a relationship that is not typical in their society.

Answers:

C. The sentence before the quote states, “but, believe me, you cannot be too careful. Try and keep Mr. Rochester at a distance: distrust yourself as well as him.” Mrs. Fairfax is suggesting that Mr. Rochester’s feelings should not be trusted because they may not be genuine. This best supports answer choice C.
J. Mrs. Fairfax states, “It is an old saying that ‘all is not gold that glitters’; and in this case I do fear there will be something found to be different to what either you or I expect.” This shows that Mrs. Fairfax believes Miss Eyre will discover that things may not turn out as she hoped or expected and may regret her decision. The other answer choices are not supported by the passage.
B. Mrs. Fairfax says, “Gentle-men in his station are not accustomed to marry their governesses.” She is pointing out a difference in Miss Eyre and Mr. Rochester’s position and fortune and hinting that this difference is not a good thing. Mrs. Fairfax also clearly indicates that she is uncertain about Miss Eyre’s future with Mr. Rochester. This information best supports answer choice B.
H. Mrs. Fairfax is explaining that she would have cautioned Miss Eyre against forming a relationship with Mr. Rochester, but Miss Eyre had seemed mature and wise enough to conclude on her own that forming an intimate relationship with Mr. Rochester would be unwise. This best supports answer choice H.
D. Mrs. Fairfax states, “Gentlemen in his station are not accustomed to marry their governesses.” The words, “not accustomed to” imply that this is not a common occurrence, and that their relationship is not typical. The other answer choices are not supported by the passage.

Ref: McGraw Hill

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How to Deal with Factored Form on the ACT

3/29/2018

Photo by Antoine Dautry on Unsplash

Question

Which of the following is a factored form of 3x^3y^3 + 3xy?

A. 3xy(x^2y^2 + 1)
B. 3(3x^2y^2)
C. (3x + 3y)(3x + 3y)
D. 3x^2y^2(xy)
E. 3x(x^2y^2 + 3)

Answer

The correct answer is A.

This problem requires you to find the Greatest Common Factor. The Greatest Common Factor is 3xy, because each term has at least 1 factor of 3, 1 factor of x, and 1 factor of y. When you factor 3xy out of 3x^3y^3 you are left with x^2 y^2 , and when you factor 3xy out of 3xy, you are left with 1. Therefore, when factored, 3x^3 y^3 + 3xy = 3xy(x^2 y^2 + 1).

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How to Classify the 400 Known Planets on the SAT

3/19/2018

Astronomers have found over 400 planets orbiting stars. The discovered planets have a variety of compositions, masses, and orbits. Despite the variety, the universal rules of physics and chemistry allow scientists to broadly categorize these planets into just a few types: Gas Giant, Carbon Orb, Water World, and Rocky Earth. Table 1 shows the composition of the various planet types and typical mass ranges relative to Earth.

Table 2 shows a sampling of planets orbiting various stars described in Table 1. These planets are merely numbered 1-7. Table 2 details the masses and orbital radii of the planets.

Questions:

1. The data in Table 1 and Table 2 support which of the following statements?

A. Gas Giant planets have the largest orbital radii.
B. Orbital radius is directly related to mass.
C. Orbital radius is inversely related to mass.
D. The data does not support a correlation between mass and orbital radius.

2. According to Table 1 and Table 2, which of the following stars has the most massive Gas Giant planet orbiting it?

F. Gliese 777
G. OGLE TR 132
H. PSR 1257
J. Gleise 581

3. If a new planet were discovered, with a mass of 325, an orbital radius of 1.5, and a composition of mostly hydrogen, what would be its most likely classification?

A. Carbon Orb
B. Water
C. Rocky Earth
D. Gas Giant

Answers

1. D
2. F
3. D

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How to Calculate the Diameter of your Waffle on the SAT

3/15/2018

Question

A waffle ice cream cone is pictured above. If the volume of the cone is 14.4pi cubic inches, what is the diameter of the cone (in inches)?

Answer

Answer is 6.

Solution:
The volume of a cone is found by using the formula V= 1/3 x pi x r^2 x h, where r is the radius of the circular base and h is the height. Input the known data and solve for r.

14.4pi = 1/3 pi r^2 (4.*)
14.4 = 1.6 r^2
9 = r^2
3 = r

or, d, the diameter is 6.

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How to Calculate a Gymnast's Sway on the SAT?

3/12/2018

Question:

A gymnast has a routine in which he sways back and forth on a high bar, making an arc that measures 135 deg. As he swings, the bottom of his shoes create an arc that measures 9 feet. At the conclusion of his routine, he swings completely around for one full circle around the bar. What is the circumference of that circle (answer in feet)?

Answer:

Answer is 24.

Solution: A circle measures 360 deg. Find what portion a 135 deg arc is of a circle and then use that information to create a proportion. We get:

135/260 which can be reduced to 3/8.

So, 3/8 = 9 feet/x feet,
or 3/8 = 9/x,
or, 3x = 72
or, x = 24

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Challenge Your Chemistry Knowledge on the SAT

3/7/2018

Image from Pixabay

Questions:

1. An element consists of three isotopes in the relative abundance given below. What is the atomic mass of this element?

30.00% = 40.00 amu
50.00% = 41.00 amu
20.00% = 42.00 amu

(A)40.90
(B)41.00
(C)41.90
(D)42.20
(E)42.90

2. The total number of electrons that can be accommodated in the fourth principal energy level is _____

(A)2
(B)8
(C)18
(D)32
(E)50

3. If the set of quantum numbers n = 3, l = 1, ml = 0, ms = Â±1/2 represents the last electron to be added to complete the ground state electron configuration of an element, which one of the following could be the symbol for the element?
(A)Na
(B)Si
(C)Th
(D)V
(E)Zn

4. Which element has the following electron configuration?
1s22s22p63s23p64s23d4

(A)Cr
(B)Mn
(C)Mo
(D)S
(E)Se

5. Oxygen-15 has a half-life of 9.98 minutes. How much of a 20.0 g sample of oxygen-15 remains after 60.0 minutes?

(A)0.156 g
(B)0.312 g
(C)0.625 g
(D)1.25 g
(E)2.50 g

6. Which of the following atoms would have the largest second ionization energy?

(A)Mg
(B)Cl
(C)S
(D)Ca
(E)Na

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How I Improved My 240 Common Spelling Mistakes in English In One Day

2/20/2018

Image from Pixabay

A

·       absence
·       acceptable
·       accessible
·       accommodation
·       accomplish
·       achievement
·       acquire
·       address
·       advertisement
·       advice – (noun)
·       advise – (verb)
·       amateur
·       apartment
·       appearance
·       argument
·       athletic
·       attendance

B

·       basically
·       beginning
·       belief – indicating the noun
·       believe – indicating the verb
·       beneficial
·       business

C

·       calendar
·       campaign
·       category
·       cemetery
·       challenge
·       characteristic
·       cigarette
·       clothes
·       column
·       committee
·       commitment
·       completely
·       condemn
·       conscience
·       conscientious
·       conscious
·       controversy
·       convenient
·       correspondence
·       criticism

D

·       deceive
·       definitely
·       definition
·       department
·       describe
·       despair
·       desperate
·       development
·       difference
·       difficult
·       disappointed
·       discipline
·       disease

E

·       easily
·       effect
·       eighth
·       either
·       embarrass
·       encouragement
·       enemy
·       entirely
·       environment
·       especially
·       exaggerate
·       excellent
·       existence
·       experience
·       experiment

F

·       familiar
·       February
·       finally
·       financial
·       foreign
·       foreigner
·       formerly
·       forty
·       fourth

G

·       general
·       generally
·       genius
·       government
·       grammar
·       grateful
·       guarantee
·       guidance

H

·       happily
·       height
·       heroes
·       humorous
·       hypocrite

I

·       ideally
·       imaginary
·       immediate
·       incredible
·       independent
·       influential
·       insurance
·       intelligent
·       interference
·       interrupt
·       introduce
·       island
·       its – for possession
·       it’s – for “it is” or “it has”

J

·       jealous
·       jealousy

K

·       kneel
·       knowledge

L

·       later
·       legitimate
·       length
·       library
·       lightning
·       likely
·       loneliness
·       lose (verb)
·       loose (adjective)
·       lovely
·       luxurious

M

·       maintain
·       maintenance
·       manageable
·       management
·       manufacture
·       marriage
·       married
·       millionaire
·       misspell
·       mischievous
·       money
·       mortgage
·       muscle
·       mysterious

N

·       naturally
·       necessary
·       neighbor / neighbour
·       ninety
·       noticeable
·       nowadays

O

·       obedient
·       obstacle
·       occasional
·       occurred
·       official
·       opinion
·       opportunity
·       opposition
·       ordinary
·       originally

P

·       particular
·       peculiar
·       perceive
·       performance
·       permanent
·       personal
·       personnel
·       physical
·       physician
·       piece
·       pleasant
·       possession
·       possible
·       possibility
·       potatoes
·       practically
·       prefer
·       privilege
·       professor
·       professional
·       pronounce / pronunciation
·       psychology
·       psychological

Q

·       quantity
·       quality
·       questionnaire
·       queue
·       quizzes

R

·       realistic
·       realize
·       really
·       receipt
·       receive
·       recognize
·       recommend
·       religion
·       religious
·       remember
·       representative
·       restaurant
·       rhythm
·       ridiculous
·       roommate

S

·       sacrifice
·       safety
·       scared
·       scenery
·       schedule
·       secretary
·       sentence
·       separate
·       similar
·       sincerely
·       strength
·       surprise
·       suspicious
·       success
·       successful

T

·       technical
·       technique
·       temperature
·       temporary
·       their (possessed by them)
·       there (not here)
·       they’re (contraction of “they are”)
·       themselves – not themself

U

·       undoubtedly
·       unforgettable
·       unique
·       until

V

·       valuable
·       village
·       violence
·       violent
·       vision
·       volume

W

·       weather – indicating climate – The weather is nice today.
·       Wednesday
·       weird
·       whether – (indicating if)
·       which
·       woman – (singular)
·       women – (plural)
·       worthwhile
·       width
·       writing

X Y Z

·       yacht
·       young

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How to Build a Great Vocabulary 'Very' Quickly

1/26/2018

Photo by Amador Loureiro on Unsplash

Building a vocabulary is hard. You have to read a lot. We mean a lot. Read. Read. Read. However, when you are faced with standardized tests, like the ACT, SAT, GRE, GMAT, HSPT, ISEE, SSAT, LSAT, etc. you don't have much time to build a great vocabulary, if you already don't have one.

Here is a handy tip to quickly build a great one if you are pressed for time.

Replace the word 'very' with a new word!

parched instead of "very dry"
squalid instead of "very dirty"
terrified instead of "very afraid"
furious instead of "very angry"
exquisite instead of "very beautiful"
hideous instead of "very ugly"
vivacious instead of "very lively"
immense instead of "very big"
tiny instead of "very small"
spacious instead of "very roomy"
precious instead of "very valuable"
mindful instead of "very aware"
precise instead of "very accurate"
rudimentary instead of "very basic"
meticulous instead of "very careful"
rotten instead of "very bad"
courteous instead of "very civil"
vivid instead of "very colorful"
brilliant instead of "very clever"
meticulous instead of "very clean"
fascinating instead of "very interesting"
envious instead of "very jealous"
scorching instead of "very hot"
imminent instead of "very likely"
perilous instead of "very dangerous"
filthy instead of "very dirty"
arduous instead of "very difficult"
tenacious instead of "very determined"
fervent instead of "very eager"
effortless instead of "very easy"
skeptical instead of "very dubious"
swift instead of "very fast"
renowned instead of "very famous"
ferocious instead of "very fierce"
doting instead of "very fond"
lithe instead of "very graceful"
exasperating instead of "very frustrating"
ponderous instead of "very heavy"
hirsute instead of "very hairy"
famished instead of "very hungry"
infirm instead of "very ill"
peerless instead of "very rare"
conscientious instead of "very responsible"
pristine instead of "very pure"
depraved instead of "very immoral"
humongous instead of "very large"
opulent instead of "very lavish"
forlorn instead of "very lonely"
novel instead of "very new"
precise instead of "very specific"
acerbic instead of "very sour"
remorseful instead of "very sorry"
stern instead of "very strict"
vacuous instead of "very stupid"
hideous instead of "very ugly"
unjust instead of "very unfair"
improbable instead of "very unlikey"
extraordinary instead of "very unusual"
precious instead of "very valuable"
abusive instead of "very violent"
sage instead of "very wise"
fledgling instead of "very young"
untamed instead of "very wild"
expansive instead of "very wide"
hale instead of "very healthy"
vital instead of "very important"
succulent instead of "very juicy"
adored instead of "very loved"
blessed instead of "very lucky"
furious instead of "very angry"
fretful instead of "very anxious"
tedious instead of "very boring"
stunning instead of "very beautiful"
poised instead of "very confident"
frigid instead of "very cold"
baffled instead of "very confused"
inquisitive instead of "very curious"
contorted instead of "very deformed"
fragile instead of "very delicate"
temperamental instead of "very emotional"
melodramatic instead of "very dramatic"
obese instead of "very fat"
gregarious instead of "very friendly"
ecstatic instead of "very happy"
frank instead of "very honest"
gifted instead of "very intelligent"
infantile instead of "very childish"
casual instead of "very informal"
severe instead of "very intense"
pessimistic instead of "very negative"
apparent instead of "very obvious"
destitute instead of "very poor"
germane instead of "very relevant"
boisterous instead of "very rowdy"
optimistic instead of "very positive"
crooked instead of "very corrupt"
precarious instead of "very risky"
unyielding instead of "very firm"
exorbitant instead of "very expensive"
ebullient instead of "very enthusiastic"

As a bonus, below is a list of the top 100 words on the SAT.

Top 100 words on the SAT

abate
become less in amount or intensity
abdicate
give up, such as power, as of monarchs and emperors
aberration
a state or condition markedly different from the norm
abstain
choose not to consume
adversity
a state of misfortune or affliction
aesthetic
characterized by an appreciation of beauty or good taste
amicable
characterized by friendship and good will
anachronistic
chronologically misplaced
arid
lacking sufficient water or rainfall
asylum
a shelter from danger or hardship
benevolent
showing or motivated by sympathy and understanding
bias
a partiality preventing objective consideration of an issue
boisterous
full of rough and exuberant animal spirits
brazen
unrestrained by convention or propriety
brusque
marked by rude or peremptory shortness
camaraderie
the quality of affording easy familiarity and sociability
canny
showing self-interest and shrewdness in dealing with others
capacious
large in the amount that can be contained
capitulate
surrender under agreed conditions
clairvoyant
someone who can perceive things not present to the senses
collaborate
work together on a common enterprise or project
compassion
a deep awareness of and sympathy for another's suffering
compromise
an accommodation in which both sides make concessions
condescending
characteristic of those who treat others with arrogance
conditional
imposing or depending on or containing an assumption
conformist
someone who follows established standards of conduct
conundrum
a difficult problem
convergence
the act of coming closer
deleterious
harmful to living things
demagogue
a leader who seeks support by appealing to popular passions
digression
a message that departs from the main subject
diligent
quietly and steadily persevering in detail or exactness
discredit
the state of being held in low esteem
disdain
lack of respect accompanied by a feeling of intense dislike
divergent
tending to move apart in different directions
empathy
understanding and entering into another's feelings
emulate
strive to equal or match, especially by imitating
enervating
causing weakness or debilitation
ephemeral
anything short-lived, as an insect that lives only for a day
evanescent
tending to vanish like vapor
exemplary
worthy of imitation
extenuating
partially excusing or justifying
florid
elaborately or excessively ornamented
forbearance
a delay in enforcing rights or claims or privileges
fortitude
strength of mind that enables one to endure adversity
fortuitous
occurring by happy chance
foster
providing nurture though not related by blood or legal ties
fraught
filled with or attended with
frugal
avoiding waste
hackneyed
repeated too often; overfamiliar through overuse
haughty
having or showing arrogant superiority
hedonist
someone motivated by desires for sensual pleasures
hypothesis
a tentative insight that is not yet verified or tested
impetuous
characterized by undue haste and lack of thought
impute
attribute or credit to
inconsequential
lacking worth or importance
inevitable
incapable of being avoided or prevented
intrepid
invulnerable to fear or intimidation
intuitive
spontaneously derived from or prompted by a natural tendency
jubilation
a feeling of extreme joy
lobbyist
someone who is employed to persuade how legislators vote
longevity
the property of having lived for a considerable time
mundane
found in the ordinary course of events
nonchalant
marked by blithe unconcern
opulent
rich and superior in quality
orator
a person who delivers a speech
ostentatious
intended to attract notice and impress others
parched
dried out by heat or excessive exposure to sunlight
perfidious
tending to betray
pragmatic
concerned with practical matters
precocious
characterized by exceptionally early development
pretentious
creating an appearance of importance or distinction
procrastinate
postpone doing what one should be doing
prosaic
lacking wit or imagination
prosperity
the condition of having good fortune
provocative
serving or tending to excite or stimulate
prudent
marked by sound judgment
querulous
habitually complaining
rancorous
showing deep-seated resentment
reclusive
withdrawn from society; seeking solitude
reconciliation
the reestablishment of cordial relations
renovation
the act of improving by renewing and restoring
restrained
under control
reverence
a feeling of profound respect for someone or something
sagacity
the ability to understand and discriminate between relations
scrutinize
examine carefully for accuracy
spontaneous
said or done without having been planned in advance
spurious
plausible but false
submissive
inclined or willing to give in to orders or wishes of others
substantiate
establish or strengthen as with new evidence or facts
subtle
difficult to detect or grasp by the mind or analyze
superficial
of, affecting, or being on or near the surface
superfluous
more than is needed, desired, or required
surreptitious
marked by quiet and caution and secrecy
tactful
having a sense of what is considerate in dealing with others
tenacious
stubbornly unyielding
transient
lasting a very short time
venerable
profoundly honored
vindicate
show to be right by providing justification or proof
wary
marked by keen caution and watchful prudence

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How to Save Words on the Endangered List from Verbal Abuse

1/16/2018

Here is how you stop the abuse of words.

Often words are misused. A good example is the case of mixed doubles: pair of words or phrases, like affect and effect. Some are not even words, like irregardless. Following are words who are endangered by bad usage. These words have been bloodied and mauled. You can however rescue them with proper usage.

1. Fortuitous.
Well, you may think this word means fortunate or lucky. Actually, fortuitous means accidental or by chance. The root of the word comes from Latin fortuitus, from forte ‘by chance.'

It was entirely fortuitous that I washed my car before we decided to drive it on the beach.

2. Fulsome.
You may think this means abundant. Actually, it means overdone or disgustingly excessive.

Andy's fulsome speech got on my nerves.

3. Affect/effect.
Affect is usually a verb, and it means to impact or change. Effect is usually a noun, an effect is the result of a change.

The problem affected Matt's performance.
The termites had a devastating effect on the house.

4. Ago/since.
Use one or the other, but not both.

Buster died five days ago. It's been five days since Buster died.

You cannot say:

It has been five days ago since Buster died.

5. Any more/anymore.
Use any more if you mean any additional, while use anymore if you mean nowadays or no longer.

He won't be chasing any more jobs.
He doesn't do this anymore.

6. Alternate/alternative.
The first means one after the other, while the second means one instead of the other.

Running requires the alternate use of your left and right legs.
The alternative to a taxi is Uber or Lyft.

7. Every one/everyone.
If you can substitute everybody, then the single word everyone is correct; if not, use two words, every one.

Everyone fears Hilton's children. Every one of them is a terror.

8. Every day/everyday.
The single word, everyday, is an adjective. It is usually found before a noun.

Sarah loves her everyday diamonds.

Expressing time every day is two words.

Matt wears a black turtle neck every day.

9. Any one/anyone.
If you can substitute anybody, then the single word anyone is correct. If not, use two words, any one.

Anyone can complain.
Any one of his colleagues would agree with his position.

10. Any place/ anyplace.
The single word anyplace is the right choice. It is used informally and the word anywhere is better English.

11. Any time/anytime.
The single word is correct.

He will take a free ride anytime.

12. A while/awhile.
Students often confuse the two words. Awhile means "for a time", while a while means "a period of time."

Doug rested awhile.
Greta dozed for a while.

13. One of the ... if not like.
Do you think something doesn't sound right in the sentence below?

Michael was one of the best, if not the best, player on the team.

Anything wrong? Well, you may say that it sounds right, but actually it is not. If you remove the phrase "if not the best" the sentence reads: Michael was one of the best player on the team. But is that you meant to say? No. So it is better to put the qualifying phrase at the end like:

Michael was one of the best player on the team, if not the best.

14. Only.
This word can be tricky. It can mean "alone," "solely," or "and no other" and can appear anywhere in a sentence. Make sure to put it in the right place, which is right before the word or phrase you want to single out as the lone wolf. Let's take the following example.

Peter says he saw the crime.

Now see how placing "only" in different places results in different meanings.

Only Peter says he saw the crime. (Peter, and no one else, says he saw the crime.)
Peter only says he saw the crime. (Peter says, but can't prove, he saw the crime.)
Peter says only he saw the crime. (Peter says he, and no one else, saw the crime.)
Peter says he only saw the crime. (Peter says he saw, but didn't hear, the crime.)
Peter says he saw only the crime. (Peter saw just the crime and nothing else.)

Remember, it is easy to make a mistake with "only". So watch out!

15. Reason ... is because.
Let's look at the following sentence.

The reason Samantha stayed home is because Tabatha was crying.

Can you hear something wrong, like perhaps an echo? Because means 'for the reason that,' so the example says: The reason Samantha stayed home is for the reason that Tabatha was crying. Use one or the other, not both.

The reason Samantha stayed home is that Tabatha was crying.
Samantha stayed home is because Tabatha was crying.

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How to Avoid Danglers on the GMAT

12/12/2017

Photo by denis harsch on Unsplash

A key to Scoring High on the GMAT

One of the most common errors in Sentence Correction on the GMAT is an incorrect, or dangling, modifier. These modifiers are often hard to detect. However, it is crucial to detect them if you want to score high on the GMAT. This blog attempts to eliminate confusion and frustration of students who are familiar with dangling modifiers.

What are Dangling Modifiers, or "Danglers"?

A dangler is a phrase that is used at the start of a sentence to describe something, but that something is not the subject doing the main action of the sentence. Since dangling modifiers don't attach to what comes right after them, they "dangle." The result is that they can be read as describing the subject of the sentence when they actually don't, which can be funny or just confusing. Let's see some examples.

Born at the age of forty-five, the baby was a great comfort to Mrs. Wheaton. [As the sentence is structured, the baby--not his mother--was forty-five. The opening phrase, born at the age of forty-five, is attached to the baby, so that is what it describes.]

The Usual Suspect

Watch out for an ing word if it's near the front of a sentence. Most likely, it is a dangling modifier. To find it, ask a whodunit question. Who is doing the talking, reading, singing, walking, etc? You may be surprised by what you find. Often, the danglers are participles.

With Present Participles

Walking down the street, the Statue of Liberty appeared. [It was not the statue that was walking.]

With Past Participles

Trapped underwater, John recounted his miraculous rescue. [Well, John was not trapped at the time he recounted it.]

With Prepositional Phrases

Often prepositions, a words that show position or direction, can lead your astray.

With his silver trim and deep blue color, Tom found his car. [Does Tom have the silver trim and deep blue color or the car?]
[Incorrect] At the age of ten, my father bought me a cat. [Did the father buy the cat when he was ten? Or is the son ten years old?]
[Correct] At the age of ten, I got a cat from my father.

Pin the Tail on the Donkey

Often adjectives get pinned to the wrong part of a sentence and become danglers.

[Incorrect] Sluggish and overweight, the vet said our dog needed more exercise. [The description sluggish and overweight should be pinned on the dog and not the vet.]
[Correct] Sluggish and overweight, our dog needed more exercise, the vet said.

Hitch your Wagon

A dangling adverb at the front of a sentence is similar to a horse that's hitched to the wrong wagon. Such adverbs are easy to spot because they often end in 'ly'. When you see one, make sure it is hitched to the right verb.

[Incorrect] Miraculously we watched as the surgeon operated with a simple tweezer.
[Correct] Miraculously, the surgeon operated with a simple tweezer as he watched.

The Infinitive Trouble

Some of the hardest danglers to see begin with to.A sentence that starts with an infinitive (a verb usually preceded by to, like to say, to laugh) cannot be left to dangle. The opening phrase has to be attached to whoever or whatever is performing the action.

[Incorrect] To crack an egg properly, the yolk is be left intact. [The way it is structured, the yolk is the one doing the cracking.]
[Correct] To crack an egg properly, you must leave the yolk intact.
[Correct} To crack an egg properly, leave the yolk intact.

Likely Problems

Can't find a dangler? It might hiding as a 'like' or as an 'unlike'. Consider this likely example.

[Incorrect] Like Kim, Eva's house was expensive. [The phrase like Kim is a dangler because it's attached the wrong thing: Eva's house. Is Eva or her house like Kim?]
[Correct] Like Kim, Eva paid a lot of money for her house.
[Correct] Like Kim's Eva's house was expensive. [Note we are comparing the right things here.]

Source: Woe is I, by Patricia T. O'Conner

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You Need to Know How to Fix Comma Splices on the SAT

12/6/2017

Image from Pixabay

The SAT Writing and Language section will test you on your knowledge of punctuation. Often students are unprepared especially if grammar has not been emphasized in school. Unfortunately, more and more schools are not teaching grammar. However, it is essential you understand grammar rules if you want to score high on the SAT.

What is a Comma Splice?

When a comma alone is used to separate two independent clauses, the result is known as a comma splice. Comma splices are always incorrect.

Can you detect a Comma Splice?

Potatoes are used in many different cuisines, farmers around the world grow many varieties of them.
My family bakes cakes and cookies on weekends, we then sit around and enjoy everything we make together.
I didn't like the movie, it was way too long.

How to fix a Comma Splice?

Comma splices are often signaled by the construction comma + pronoun. Let's try to fix the above sentences. There are many ways to do it, and the SAT does not prefer any specific method. Some questions will require you to correct them with a period, while others will require you to fix them using a semicolon, a comma with FANBOYs, or even another formulation.

Incorrect: Potatoes are used in many different cuisines, farmers around the world grow many varieties of them.

Correct:

Potatoes are used in many different cuisines. Farmers around the world grow many varieties of them.
Potatoes are used in many different cuisines, and farmers around the world grow many varieties of them.
Potatoes are used in many different cuisines; farmers around the world grow many varieties of them.

Incorrect: My family bakes cakes and cookies on weekends, we then sit around and enjoy everything we make together.

Correct:

My family bakes cakes and cookies on weekends, and we then sit around and enjoy everything we make together.
My family bakes cakes and cookies on weekends. We then sit around and enjoy everything we make together.

Incorrect: I didn't like the movie, it was way too long.

Correct:

I didn't like the movie. It was way too long.
I didn't like the movie because was way too long.

Another option is to turn one of the independent clauses into a dependent clause, often by using a subordinating conjunction like, because, while, or although. We just saw an example above when we used 'because'.

You can also combine sentences with participles (-ing) to create dependent clauses.

Incorrect: Tomatoes were originally small, they became large only recently.
Correct: Tomatoes were originally small, becoming large only recently.

Summary

We can fix comma splices by:

Splitting the sentences into two independent clauses separated by a period.
Using a semicolon to separate the clauses.
Using a comma with FANBOYs.
Turning one clause into a subordinate clause by using either a subordinating conjunction like, because, while or although -- or using a participle (-ing).

Practice Makes Perfect

Here are some sentences you can test your understanding of how to fix comma splices.

John was a great explorer who found the fountain of youth on the top of a mountain, he also found that it couldn't be brought down to the world.
The recaptured sense of her own childhood came back to her when she met the two young boys, they seemed to face life as she had faced it.
There are more than 10,000 festivals in Germany, they are the world's biggest and strangest.

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Want to Soar on the ACT? You Need to Know How to Tackle Pronoun Usage

11/28/2017

Pronouns are small (I, me, he, she, it), but they are among the biggest troublemakers in the language.

For example, in Shakespeare's Hamlet, Ophelia cried "Woe is me," which in today's formalized English should be "Woe is I." See the trouble?

The English section of the ACT contains 75 questions, which you must answer in 45 minutes. More than 50 percent of the English section tests standard English conventions, such as sentence structure and formation, punctuation and usage. Pronouns are a part of these conventions.

You all know what pronouns are, so we will not discuss that here. You also know how to use them: a substitute for a noun. Things get complicated when pronouns take on different guises, depending on the roles in plays in the sentence. Some pronouns are so well disguised that you may not be able to tell one from another. The usual suspects are: that and which; it's and its; who's and whose; who and whom; everybody and nobody; and their, they're, and theirs.

The 'Which' Trials

Image from Pixabay

Nobody likes a dog that bites.
Nobody likes a dog which bites.

Which sentence sounds right to you? If both did, you got 'which-ed'! The first sentence with 'that' is the correct usage. A lot of students (and adults) have problems with that-versus-which. Here are two rules that can help you figure out whether a clause should start with that or which.

If you can drop the clause and not lose the point of the sentence, use which. If you can't, use that.
A which clause goes inside commas. A that clause doesn't.

Let's look at these sentences:

Buster's bulldog, which had one white ear, won best in show.
The dog that won best in show was Buster's bulldog.

The point of each sentence is that Buster's bulldog won. What happens when we remove the that or which clause? In the first example, the which clause is disposable. We can just say: Buster's bulldog won best in show. In the second example, however, the that clause (that won best in show) is essential. Without it the sentence would read: The dog was Buster's bulldog, which is a different meaning that the dog that won best in show was Buster's bulldog.

Who vs. Whom

When do you 'who' and when do you use 'whom'? Keep it simple! The most important thing to know is that who does something (it's a subject, like she), and whom has something done to it (it's an object, like her). Sometimes it can get tricky. Let's see the following examples:

Amanda invited only girls [who or whom] she thought used math for fashion design.

Is it 'who' or 'whom'? If you strip off all the words between the subject and verb, you end up with who ... used math for fashion. 'Who' did something (used math for fashion), so it's the subject. 'Whom' does not work.

Tom wouldn't tell Miss Marple [who or whom] he invited to his bowling game.

First strip the sentence down to the basic clause, [who or whom] he invited. You can see that whom is the object--he did something to (invited) whom--even though whom comes ahed of both the subject and the verb.

More than Meets the I

Many smart people hesitate about I vs me, he vs him, she vs her, and they vs them. How do we use them correctly? It all depends on the context. Let's look at the following examples:

Mary loves pasta more than I. Here it means that she likes pasta more than I do.
Mary loves pasta more than me. Here it means that she loves pasta more than she loves me!

So the usage of I vs me depends on the context, i.e., what you are trying to communicate.

The Many Selves

Students (and adults) often confuse the usage of 'I' and 'myself' and the rest of the 'self' crew (yourself, himself, herself, itself, ourselves, yourselves, themselves). The key point about the 'self' crew is that they should not take the place of ordinary pronouns I and me, she or her, and so on. They are used only for two purposes:

To emphasize. I made the cake myself. The detective himself was the murderer.
To refer to the subject. She hates herself. And you call yourself a mechanic! They consider themselves lucky to be alive. The problem surprisingly solved itself.

There are some other pesky pronouns, but that topic is for another blog.

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You Need to Know This to Tackle the Subjunctive Mood

11/26/2017

[Infographic provided by Grammar.net]

Were or Was?

People often confuse the usage between 'I was' and 'I were'. When people say things like, "I wish it were impossible," some people would ask, were or was? Why not I wish it was impossible? Well, in English there is a special way of speaking wishfully. We say, I wish I were in love again, and not I wish I was in love again.

Grammarians call it the 'subjunctive mood'. It is when we are talking about things that are desirable, as opposed to things as they really are. It is to separate the 'what if' from the 'what is'. When we're in a wishful mood, was becomes were:

I wish I were in Disneyland. (I'm not in Disneyland.)
I wish people were not so obnoxious. (People are so obnoxious.)
I wish I were able to do more exercise. (I am not able to do more exercise.)

Iffy Situation

The word 'if' can make all the difference to the meaning of a statement like I was faster becomes quite different when we insert our little word: if I were faster.

Why is this? It is because "what if" means something that's untrue. When that happens, the subjunctive mood kicks in, and was becomes were. This happens when a sentence or a clause starts with if, and what's being talking about is contrary to fact:

If I were the President, I would cut spending. (I'am not the President.)
We could go to shopping to the mall, if it were a weekday. (It is not a weekday.)

The above is true only for those if statements that are contrary to fact. In cases where the statement may be true, was remains was:

If I was rude, I apologize. (I may have been rude.)
If it was Sunday, I must have gone to bed early. (It may have been Sunday.)

The same rules apply to if statements that start with as if or as though:

She acts as if she were infallible. (She's not infallible.)
She behaves as though food were scarce. (Food is not scarce.)

Simple? Feel free to comment if you have any questions.

Reference: Woe is I, by Patricia T. O'Connor

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How to Use the Unloved Semicolon and the Formidable Colon

11/22/2017

[Infographic provided by Grammar.net]

The semicolon is one of the most useful but least used punctuation marks.

The Semicolon

Many people avoid the semicolon. Some even seem to dislike it, but it does not have to be that way. The source of avoidance or dislike is the lack of understanding of the proper role of the semicolon. If a comma is a yellow light and a period is a red light, the semicolon is a flashing red--one of the lights you drive through a brief pause.

Here's when to use it.

1. Use a semicolon to separate clauses when there's no and in between.

Sam's hat flew off his head; it sailed into the distance.

2. Use semicolons to separate items in a series when there's already a comma in one or more of the items.

Amanda's favorite things were her robe, a pink sweater from Grandma; her slippers; her overstuffed chair, which had once been her grandfather's; mystery stories, especially by Agatha Christie and Sherlock Holmes.

The Colon

Think of the colon as punctuation's master of ceremonies. Use it to present something: a statement, a series, a quotation, or instructions. But remember that a colon is an abrupt stop, almost like a period. Use one only if you want your sentence to brake completely. Here is how to do it.

1. Use a colon instead of a comma, if you wish, to introduce a quotation.

I said to her: "Jane, please pick up a bottle of wine on your way over. But don't spend too much time looking. Just pick a Merlot."

Many people prefer to introduce a longer quotation with a colon instead of a comma.

2. Use a colon to introduce a list, if what comes before the colon could be a small sentence in itself (it has both a subject and a verb).

John brought three wines: a Bordeaux, a Merlot, and a Chardonnay.

Just don't use the colon to separate a verb from the rest of the sentence. In John's shopping cart were: a Bordeaux, a Merlot, and a Chardonnay. If you don't need a colon, why use one? In John's shopping cart were a Bordeaux, a Merlot, and a Chardonnay.

And that's it folks. Wasn't that easy?

Reference: Woe is I, by Patricia T. O'Connor

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13 Comma Rules You Need to Know to Soar on Tests

11/2/2017

The comma is a small mark, but it is perhaps the most important punctuation in grammar. Despite that, comma confusion is one of the most common grammatical problems that students face. This blog attempts to help students with proper comma usage.

Short Summary (TL;DR)

Comma Usage:

The pause
Separate parts of speech
Never separate subject from its verb
After Introductory phrase, clause, word
Separate clauses when using FANBOYS
Relative clauses
Non-relative clauses
"that" clauses
'Keep' the essential and 'cut out' the non-essentials
Geography, Dates, Addresses and Titles
Series or list
Shift between the main discourse and a quotation
Contrasts

1. The Pause

Commas, commas, commas. They often are a source of confusion. How do you use without getting lost in grammatical jargon? Thanks to Patricia T. O’Conner's book on grammar, Woe is I: The Grammarphobe’s Guide to Better English in Plain English, commas can easily be understood. We let her tell you about them in her own words:

"When you talk, your voice, with its pauses, stresses, rises and falls, shows how you intend your words to fit together. When you write, punctuation marks are the road signs (stop, go, yield, slow, detour) that guide the reader, and you wouldn’t be understood without them.

"If you don’t believe me, try making sense out of this pile of words:
Who do you think I saw the other day the Dalai Lama said my Aunt Minnie.

"There are at least two possibilities:

"Who do you think I saw the other day?" the Dalai Lama said. "My Aunt Minnie."
"Who do you think I saw the other day? The Dalai Lama!" said my Aunt Minnie."

"Punctuation isn’t some subtle, old-fashioned concept that’s hard to manage and probably won’t make much of a difference one way or another. It’s not subtle, it’s not difficult and it can make all the difference in the world.

2. Separate the Parts of Speech

If you get commas right, you will get most of your punctuation right. How do we use them?

Long and short division
Use a comma to separate big chunks (clauses) of a sentence with and between them.

Samantha hadn’t left the city in months, and by Sunday she was climbing the walls.

If there’s no 'and' in between, use a semi-colon instead:

Samantha hadn’t left the city in months; by Sunday she was climbing the walls.

Use commas to separate a series of things or actions.

She packed a toothbrush, a blow-dryer, her swimsuit and her teddy bear.
She finished packing, paid some bills, ate a few Oreos and watered the plants.

In a series, you can leave out the comma before "and". It’s just a matter of taste. 'And' can also be thought of as a separator, a break, so a comma often is unnecessary.

3. Comma with Subjects and their Verbs

With few exceptions, a comma should not separate a subject from its verb.

Incorrect: My friend Amanda, is a wonderful dancer.

Writers are often tempted to insert a comma between a subject and verb this way because speakers sometimes pause at that point in a sentence. But in writing, the comma only makes the sentence seem stilted.

Correct: My friend Amanda is a wonderful singer.

Be especially careful with long or complex subjects:

Incorrect: The things that cause me joy, may also cause me pain.
Correct: The things that cause me joy may also cause me pain.
Incorrect: Navigating through snow, sleet, wind, and darkness, is a miserable way to travel.
Correct: Navigating through snow, sleet, wind, and darkness is a miserable way to travel.

4. Comma After Introductions

Introductory Clauses
Introductory clauses are dependent clauses that provide background information or "set the stage" for the main part of the sentence, the independent clause. For example:

If they want to win, athletes must exercise and practice every day.

Introductory phrases
Introductory phrases also set the stage for the main action of the sentence, but they are not complete clauses. Phrases don't have both a subject and a verb that are separate from the subject and verb in the main clause of the sentence. Common introductory phrases include prepositional phrases, appositive phrases, participial phrases, infinitive phrases, and absolute phrases.

To stay in shape for competition, athletes must exercise every day. (Introductory infinitive phrase, main clause)
Barking insistently, Smokey got us to throw his ball for him. (Introductory participial phrase, main clause)
A popular and well respected mayor, Bailey was the clear favorite in the campaign for governor. (introductory appositive phrase, main clause)
The wind blowing violently, the townspeople began to seek shelter. (introductory absolute phrase, main clause)
After the adjustment for inflation, real wages have decreased while corporate profits have grown. (introductory prepositional phrases, main clause)

Introductory words (SHFM)
Introductory words like however, still, furthermore, and meanwhile create continuity from one sentence to the next.

The coaches reviewed the game strategy. Meanwhile, the athletes trained on the Nautilus equipment.
Most of the evidence seemed convincing. Still, the credibility of some witnesses was in question.

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Top 15 Things You Need to Know About Physics on the SAT

10/30/2017

1. Newton's First Law of Motion
Three fundamental principles, called Newton's First, Second and Third Laws, form the basis of classical, or Newtonian, mechanics and have proved valid for all mechanical problems not involving speeds comparable with the speed of light and not involving atomic or subatomic particles. Newton's First Law states that a particle not subjected to external forces remains at rest or moves with constant speed in a straight line. This is also known as the Law of Inertia.

Newton's first law of motion is often stated as: An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.

2. Newton's Second Law of Motion
The second law is the one that tells you how to calculate the value of a force. Force (measured in Newtons) is one of the fundamental physical properties of a system and comes in many forms. You might feel it as a push or pull (a mechanical force), while it is the value of your weight (the gravitational force of the Earth pulling on you) and can be seen in the repulsion or attraction of magnets or electric charges (electromagnetic force).

The Law is often stated as: The acceleration of a particle is directly proportional to the resultant external force acting on the particle and is inversely proportional to the mass of the particle, or a = F/m.

3. Newton's Third Law of Motion
If two particles (or bodies) interact, the force exerted by the first particle on the second particle (called the action force) is equal in magnitude and opposite in direction to the force exerted by the second particle (called the reaction force).

Suppose you are watching the lift off of a rocket, like the ones by SpaceX. You hear a deafening roar and see burning gases shooting from the exhaust vents of the rockets. At that moment, the rocket moves slowly upward. You can infer that the force for the lift off comes from the burning gases pushing against the shuttle rockets. Why does the shuttle system move in the opposite direction of the gases?

The forces on the rocket are similar to the forces in a collision between two tennis balls. When the balls collide, they are propelled in opposite directions. The rockets force burning gases downward through the exhaust vents. In response to these downward forces, the shuttle system moves upward. The motion of the rocket demonstrates Newton's third law of motion. When one object exerts a force upon a second object, the second object exerts an equal and opposite force upon the first object.

The third law of motion states that every action has an equal and opposite reaction. You can see equal and opposite forces interact when you blow up a balloon and release it, it moves in the opposite direction. The force propelling the balloon is equal and opposite to the force of the air leaving the balloon.

4. Newton's Law of Universal Gravitation
Newton's Law of Universal Gravitation states that a particle attracts every other particle in the universe using a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

5. Einstein's Special Theory of Relativity
Einstein's theory about the relationship between space and time is based on two postulates: (1) that the laws of physics are invariant in all inertial systems and (2) that the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source.

6. Zeroth Law of Thermodynamics
The principle states that if two bodies are each in thermal equilibrium with a third body, then the first two bodies are in thermal equilibrium with each other.

7. First Law of Thermodynamics
The First Law of Thermodynamics states that heat is a form of energy, and thermodynamic processes are therefore subject to the principle of conservation of energy. This means that heat energy cannot be created or destroyed. It can, however, be transferred from one location to another and converted to and from other forms of energy.

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How to Find the Volume of a Cylinder like an Amazon Echo

10/23/2017

The volume of a 3 dimensional solid is the amount of space it occupies.

Since the end (base) of a cylinder is a circle, the area of that circle is given by the formula:

\[area = π r^2\]

The volume of a cylinder is found my multiplying the area of one end of the cylinder by its height. So the formula of the volume of a cylinder is:

\[volume = π r^2 h\]

where:
π  is Pi, approximately 3.142
r  is the radius of the circular end of the cylinder
h  height of the cylinder

Our problem involves calculating the volume of Amazon Echo. The height of the Echo is 235mm while the diameter of the base is 84 mm or 8.4cm . Let's first find the area of the base.

The area of the base of Amazon Echo is:

\[area = π r^2\]

\[= 3.142 x 8.4^2\]

= 221.56 cm^2

To calculate the volume, we multiply the area with the height of the Echo, which is 235mm or 23.5cm

So, volume of Alexa Echo = 221.56 * 23.5 = 5,206.6224 cm^3

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How Quickly Can You Solve These Easy Quant Problems On The GMAT?

10/10/2017

Photo by Chris Liverani on Unsplash

The GMAT is a tricky test, so make sure you can solve the easy problems. Test yourself with the following problems.

Questions

1. The value of an investment triples every 10 years. By what factor does the value increase over a 30-year period?

(A) 3
(B) 6
(C) 9
(D) 27
(E) 30

2. A chemist is making a 50% alcohol solution. How many milliliters of distilled water must the chemist add to 600 milliliters of an 80% alcohol solution to obtain a 50% solution?

(A) 180
(B) 300
(C) 360
(D) 480
(E) 600

3. An investor receives interest on two simple interest investments, one at 3%, annually, and the other at 2%, annually. The two investments together earn $900 annually. The amount invested at 3% is $20,000. How much money is invested at 2%?

(A) $10,000
(B) $12,000
(C) $15,000
(D) $20,000
(E) $35,000

4. If the diameter of a circle is 14, then the area of the circle is

(A) 7pi
(B) 14pi
(C) 28pi
(D) 49pi
(E) 196pi

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Please Don't Get These Easy Problems Wrong On The SAT Physics

10/3/2017

Question 1:

An astronomer discovers a planet with two moons. One moon is 2 times as far from the center of the planet as the other and 3 times the mass of the other moon. What is the ratio of the gravitational force on the first moon to the gravitational force on the second moon?

(A) 0.33
(B) 0.40
(C) 0.66
(D) 0.75
(E) 0.90

Question 2:

A 1-newton force and a 4-newton force act in opposite directions. What is the magnitude of the resultant force, in newtons?

(A) 0
(B) 1
(C) 3
(D) 4
(E) 5

Question 3:

An object starts from rest and accelerates at 6.0 m/s^2 for 4.0 seconds. How far does it travel?

(A) 8.0 m
(B) 12 m
(C) 48 m
(D) 96 m
(E) 288 m

Question 4:

Force A and Force B act on an object. Let theta equal the angle between the directions of the two forces. At what value of theta is the resultant force the greatest?

(A) 0 deg
(B) 45 deg
(C) 60 deg
(D) 90 deg
(E) 180 deg

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Challenge Yourself with these Sentence Corrections on the GMAT

9/29/2017

Instructions: The following questions present a sentence, part of which or all of which is underlines. Beneath the sentence you will find five ways of phrasing the underlined part. Choose the best answer.

Questions

1. In 1980, lack of preparation reduced Costa Rica's coffee production to about 30 million tons, nearly 20 percent less than those of the 1979 harvest.

(A) less than those of the 1979 harvest.
(B) less than the 1979 harvest.
(C) less than 1979.
(D) fewer than 1979.
(E) fewer than that of India's 1979 harvest.

2. Veronica needs to buy a new dress, resole her dancing shoes, and the visit to the doctor needs to be rescheduled.

(A) and the visit to the doctor needs to be rescheduled.
(B) and visiting the doctor needs to be rescheduled for later in the day.
(C) and reschedule her visit to the doctor.
(D) and doctor's visiting needs to be rescheduled.
(E) and visit the doctor.

3. After I called all of your friends, you was surprised when they showed up for the party.
(A) you was surprised
(B) you were surprise
(C) you was surprising
(D) you were surprised
(E) surprised was your feeling

4. Nancy, like many nurses, work long hours.

(A) work long hours.
(B) work long hour.
(C) works long hours.
(D) work for long hours.
(E) work for longer hours.

5. The audience, already amazed by the special effects, gasp when the curtain rose on the second act's new set.

(A) gasp when the curtain rose
(B) gasped when the curtain rose
(C) gasp when the curtain roses
(D) gasp upon seeing the curtain rise
(E) gasps when the curtain rose

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This is How to Calculate if a Hurricane will Wreck your House

9/12/2017

Question

In a hurricane, the air (density 1.2 kg/m^3) is blowing over the roof of a house at a speed of 110 km/h.

(a) What is the pressure difference between inside and outside that tends to lift the roof?
(b) What would be the lifting force on a roof of area 93 m^2?

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Can You Solve this Tricky Roots Problem on the SAT?

9/7/2017

Photo by chiara conti on Unsplash

N.B: SAT Subject Test Math Level 2 Question

Question

What is the sum of the roots of the equation?

\[(x - \sqrt{2})(x^2 - \sqrt{3x} + \pi) = 0\]

(A) -0.315
(B) -0.318
(C) 1.414
(D) 3.15
(E) 4.56

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