A) 12x + 9.5y ≤ 220

x + y ≥ 20

B) 12x + 9.5y ≤ 220

x + y ≤ 20

C) 12x + 9.5y ≥ 220

x + y ≤ 20

D) 12x + 9.5y ≥ 220

x + y ≥ 20

]]>

Free Image from Pixabay

Unmanned spacecraft taking images of Jupiter’s moon Europa have found its surface to be very smooth with few meteorite craters. Europa’s surface ice shows evidence of being continually resmoothed and reshaped. Cracks, dark bands, and pressure ridges (created when water or slush is squeezed up between 2 slabs of ice) are commonly seen in images of the surface. Two scientists express their views as to whether the presence of a deep ocean beneath the surface is responsible for Europa’s surface features.

*Scientist 1*

A deep ocean of liquid water exists on Europa. Jupiter's gravitational field produces tides within Europa that can cause heating of the subsurface to a point where liquid water can exist. The numerous cracks and dark bands in the surface ice closely resemble the appearance of thawing ice covering the polar oceans on Earth. Only a substantial amount of circulating liquid water can crack and rotate such large slabs of ice. The few meteorite craters that exist are shallow and have been

smoothed by liquid water that oozed up into the crater from the subsurface and then quickly froze.

Jupiter’s magnetic field, sweeping past Europa, would interact with the salty, deep ocean and produce a second magnetic field around Europa. The spacecraft has found evidence of this second magnetic field.

*Scientist 2*

No deep, liquid water ocean exists on Europa. The heat generated by gravitational tides is quickly lost to space because of Europa’s small size, as shown by its very low surface temperature (–160°C). Many of the features on Europa’s surface resemble features created by flowing glaciers on Earth. Large amounts of liquid water are not required for the creation of these features. If a thin layer of ice below the surface is much warmer than the surface ice, it may be able to flow and cause cracking and movement of the surface ice. Few meteorite craters are observed because of Europa’s very thin atmosphere; surface ice continually sublimes (changes from solid to gas) into this atmosphere, quickly eroding and removing any craters that may have formed.

A deep ocean of liquid water exists on Europa. Jupiter's gravitational field produces tides within Europa that can cause heating of the subsurface to a point where liquid water can exist. The numerous cracks and dark bands in the surface ice closely resemble the appearance of thawing ice covering the polar oceans on Earth. Only a substantial amount of circulating liquid water can crack and rotate such large slabs of ice. The few meteorite craters that exist are shallow and have been

smoothed by liquid water that oozed up into the crater from the subsurface and then quickly froze.

Jupiter’s magnetic field, sweeping past Europa, would interact with the salty, deep ocean and produce a second magnetic field around Europa. The spacecraft has found evidence of this second magnetic field.

No deep, liquid water ocean exists on Europa. The heat generated by gravitational tides is quickly lost to space because of Europa’s small size, as shown by its very low surface temperature (–160°C). Many of the features on Europa’s surface resemble features created by flowing glaciers on Earth. Large amounts of liquid water are not required for the creation of these features. If a thin layer of ice below the surface is much warmer than the surface ice, it may be able to flow and cause cracking and movement of the surface ice. Few meteorite craters are observed because of Europa’s very thin atmosphere; surface ice continually sublimes (changes from solid to gas) into this atmosphere, quickly eroding and removing any craters that may have formed.

- Scientist 1:Sublimation

Scientist 2: Filled in by water - Scientist 1: Filled in by water

Scientist 2: Sublimation - Scientist 1: Worn smooth by wind

Scientist 2: Sublimation - Scientist 1: Worn smooth by wind

Scientist 2: Filled in by water

- Europa has a larger diameter than does Jupiter.
- Europa has a surface made of rocky material.
- Europa has a surface temperature of 20°C.
- Europa is completely covered by a layer of ice.

- is being shaped by the movement of ice.
- is covered with millions of meteorite craters.
- is the same temperature as the surface of the Arctic Ocean on Earth.
- has remained unchanged for millions of years.

- No meteorites have struck Europa for millions of years.
- Meteorite craters, once formed, are then smoothed or removed by Europa's surface processes.
- Meteorite craters, once formed on Europa, remain unchanged for billions of years.
- Meteorites frequently strike Europa's surface but do not leave any craters.

- Nitrogen
- Methane
- Chlorine
- Oxygen

- Frozen nitrogen
- Water ice
- Dissolved salts
- Molten magma

- Pressure ridges
- Cracks
- Meteorite craters
- Dark bands

Free Image from Pixabay

Idioms are often challenging because idiom questions cannot be figured out by applying a rule. An **idiom** is a word or phrase which **means** something different from its literal **meaning**. For **example**, "it's raining cats and dogs" is a common **idiom** in English, but it's not meant to be taken literally: cats and dogs are not falling from the sky! The SAT and the ACT tests you on your idiomatic usage Can you figure out the errors in the following sentences?

- Because she laughed when Jerry fell down, Mary was accused of being devoid at sympathy.
- Sherri insists at being annoying.
- Jason tends being worrisome.
- Greek Yogurt, an excellent source of calcium and protein, serves to be a digestive aid.
- Sam is confused at his best friend's behavior.
- Amanda has a mastery in the violin.
- Glass walls and dividers can also be to replace solid walls as a means through distributing natural light more freely.

The SAT and the ACT may test you on two types of idioms: **prepositional idioms and idioms with gerunds/infinitives**.

In a prepositional idiom, the preposition determines the meaning. For example, consider the pair **at ease** and **with ease**. The prepositions 'at' and 'with' change the meaning of the phrase. “At ease” refers to a state of relaxation: “He stood at ease during the parade.” “With ease” is used to refer to a sense of effortlessness as “She completed the gymnastics routine with ease.”

Some common prepositional idioms on the SAT and the ACT are:

- anxious about
- advise against
- celebrate as
- accompanied by
- advocate for
- impressed by
- tolerance for
- excuse from
- inquire into
- succeed in
- impose on
- argue over
- characteristic of
- adapt to
- correlate with
- struck by
- blame for
- obvious from
- a wealth of
- draw on
- think over
- deprive of
- focus on
- sympathize with
- unique to
- prey on
- rebel against

Gerunds and infinitives are sometimes referred to as verb complements. They may function as subjects or objects in a sentence.

A**gerund **is a verb in its ing (present participle) form that functions as a noun that names an activity rather than a person or thing. Any action verb can be made into a gerund.

An**infinitive** is a verb form that acts as other parts of speech in a sentence. It is formed with to + base form of the verb. Ex: to buy, to work.

Let's look at some examples. Here's an example with the gerund in bold:*I neglected ***doing** my homework.

The sentence is also correct if you use an infinitive:*I neglected ***to do** my homework.

Some examples of gerund and infinitive idioms on the SAT and the ACT are:

A

An

Let's look at some examples. Here's an example with the gerund in bold:

The sentence is also correct if you use an infinitive:

Some examples of gerund and infinitive idioms on the SAT and the ACT are:

- regarded as (being)
- viewed as (being)
- in the hope(s) of being
- effective in/at being
- accustomed/used to being
- succeed in/at being
- insist on being
- accused of being
- deny being
- postpone being
- resent being
- mind being
- risk being
- in danger of being
- deserve to be
- promise to be
- refuse to be
- threaten to be
- inclined to be
- decline to be
- seek/strive to be
- choose/decide to be
- neglect to be
- offer to be
- attempt to be
- struggle to be
- want/wish to be
- enable somebody to be
- encourage somebody to be
- ask to be

Now that you know some of the idioms asked on the SAT and the ACT, turn over to see the answer for the quiz.

- Because she laughed when Jerry fell down, Mary was accused of being devoid at sympathy.
- Sherri insists at being annoying.
- Jason tends being worrisome.
- Greek Yogurt, an excellent source of calcium and protein, serves to be a digestive aid.
- Sam is confused at his best friend's behavior.
- Amanda has a mastery in the violin.
- Glass walls and dividers can also be to replace solid walls as a means through distributing natural light more freely.

- Because she laughed when Jerry fell down, Mary was accused of being devoid
**of**sympathy. - Sherri insists
**on**being annoying. - Jason tends
**to be**worrisome. - Greek Yogurt, an excellent source of calcium and protein, serves
**as**a digestive aid. - Sam is confused
**by**his best friend's behavior. - Amanda has a mastery
**of**the violin. - Glass walls and dividers can also be to replace solid walls as a means
**of**distributing natural light more freely.

Resources:

1. https://www.dailywritingtips.com/5-pairs-of-prepositional-idioms/

2. https://www.gingersoftware.com/content/grammar-rules/verbs/gerunds-and-infinitives/

3. https://canvas.instructure.com/courses/987665/pages/idioms

]]>1. https://www.dailywritingtips.com/5-pairs-of-prepositional-idioms/

2. https://www.gingersoftware.com/content/grammar-rules/verbs/gerunds-and-infinitives/

3. https://canvas.instructure.com/courses/987665/pages/idioms

To refresh your memory, a sequence is a set of things (usually numbers) that are in order. In an Arithmetic Sequence

For example, the sequence 5, 7, 9, 11, 13, 15 ... is an arithmetic progression with common difference of 2.

We can write an Arithmetic Sequence as a rule:

xn = a + d(n−1)

How would you write it using Python? Try it yourself. If you cannot figure it out, see code on the next page.

def is_arithmetic(seq):

d = seq[1] - seq[0]

for index in range(len(seq) - 1):

if not (seq[index + 1] - seq[index] == d):

return False

return True

print(is_arithmetic([3, 7, 11, 15]))

print(is_arithmetic([3, 8, 11, 15]))

]]>d = seq[1] - seq[0]

for index in range(len(seq) - 1):

if not (seq[index + 1] - seq[index] == d):

return False

return True

print(is_arithmetic([3, 7, 11, 15]))

print(is_arithmetic([3, 8, 11, 15]))

Free Image from Pixabay

N.B. The questions for the SAT subject physics test.

Heat is added to a 2.0-kg block of ice. The specific heat of water is 4.2 x 10^3 J/Kg'C and the heat of fusion of ice is 3.3 x 10^5 KJ/Kg.

1. How much heat is required to melt the block of ice?

(A) 1.2 x 10^4 J

(B) 6.5 x 10^4 J

(C) 1.7 x 10^5 J

(D) 6.6 x 10^5 J

(E) 8.4 x 10^5 J

(A) 1.2 x 10^4 J

(B) 6.5 x 10^4 J

(C) 1.7 x 10^5 J

(D) 6.6 x 10^5 J

(E) 8.4 x 10^5 J

2. Once the ice melts, heat is added until the temperature of the water reaches 30'C. How much heat is required to raise the the temperature from 0'C to 30'C?

(A) 8.4 x 10^4 J

(B) 2.5 x 10^5 J

(C) 3.0 x 10^5 J

(D) 6.0 x 10^5 J

(E) 1.9 x 10^6 J

(A) 8.4 x 10^4 J

(B) 2.5 x 10^5 J

(C) 3.0 x 10^5 J

(D) 6.0 x 10^5 J

(E) 1.9 x 10^6 J

The amount of energy required to melt the ice is found by Q = mH = (2 kg)(3.3 x 10^5 KJ/Kg) = 6.6 x 10^5 J.

The specific heat required can be found using Q = mc.dt = (2 kg)(4.2 x 10^3 J/Kg'C) (30'C) = 2.5 x 10^5 J.

\[\sqrt[5]{\sqrt[4]{\sqrt[3]{\sqrt{n}}}} \] | is equal to |

(A) (B) (C) (D) (E) | \[n^{\frac{1}{17}}\] \[n^{\frac{1}{19}}\] \[n^{\frac{1}{60}}\] \[n^{\frac{1}{120}}\] \[n^{\frac{77}{60}}\] |

\[\sqrt[5]{\sqrt[4]{\sqrt[3]{\sqrt{n}}}} \] | can be written as | \[(((n^{\frac{1}{2}})^{\frac{1}{3}})^{\frac{1}{4}})^{\frac{1}{5}}\] |

When raising a power to a power, multiply the exponents.

\[\frac{1}{2} x \frac{1}{3} x \frac{1}{4} x \frac{1}{5} = \frac{1}{120}\]

\[\sqrt[5]{\sqrt[4]{\sqrt[3]{\sqrt{n}}}} \] | \[= n^{\frac{1}{120}}\] |

Free Image from Pixabay

Hybrid quant questions on the GMAT contain multiple steps with twists and turns. You must complete each step quickly to finish the problem in under two minutes. Don't underestimate how much time it takes to mull over how the different parts fit together and to transition from one line of thinking to another.

Try the following problem.

Try the following problem.

A student cuts 80 rectangles from construction paper, all of which are at least 10 inches in length and in width, and 20 percent of the rectangles that are greater than 10 inches long are exactly 10 inches wide. If 40 of the rectangles have a length of exactly 10 inches and 50 of the rectangles are greater than 10 inches wide, how many rectangles have a perimeter of greater than 40 inches? (Note: Assume that width and length are interchangeable; in other words, width does not have to be shorter than length.)

(A) 18

(B) 22

(C) 32

(D) 58

(E) 66

(A) 18

(B) 22

(C) 32

(D) 58

(E) 66

If you are able to figure out that any rectangles with one dimension greater than 10 inches would have to have a perimeter greater than 40, you would be able to eliminate three answer choices! The problem states that 40 rectangles have a length of exactly 10 inches, so the other 30 rectangles mist have a length greater than 10 inches. At least 40 rectangles, then, have a perimeter greater than 40 inches, so answers (A), (B) and (C) cannot be correct.

Which leaves (D) and (E). It is okay to guess here, if you cannot quickly figure the answer. You don't want to waste any time. A single wrong answer is not going to hurt your score! Don't get frustrated and just move on.

]]>Which leaves (D) and (E). It is okay to guess here, if you cannot quickly figure the answer. You don't want to waste any time. A single wrong answer is not going to hurt your score! Don't get frustrated and just move on.

Free Image from Pixabay

Each question or incomplete statement is followed by five possible answers or completions. Sleect the one choice that is the best answer.

1. All of the following contribute to variation in a population EXCETP

(A) mutation

(B) isolation

(C) sexual reproduction

(D) conjugation

(E) genetic drift

(A) mutation

(B) isolation

(C) sexual reproduction

(D) conjugation

(E) genetic drift

2. Tendons connect _________________ to ____________________; ligaments connect __________________ to ________________.

(A) bone to bone; bone to muscle

(B) bone to muscle; bone to bone

(C) bone to bone; muscle to muscle

(D) muscle to muscle; bone to bone

(E) ligaments to bone; tendons to bones

(A) bone to bone; bone to muscle

(B) bone to muscle; bone to bone

(C) bone to bone; muscle to muscle

(D) muscle to muscle; bone to bone

(E) ligaments to bone; tendons to bones

3. A solution of pH of 5 is _______ times more acidic than a solution with a pH of 7.

(A) 1/10

(B) 1/100

(C) 10

(D) 100

(E) 1,000

(A) 1/10

(B) 1/100

(C) 10

(D) 100

(E) 1,000

4. Farmers have successfully bred Brussels sprouts, broccoli, kale, and cauliflower from the mustard plant. This demonstrates

(A) convergent evolution

(B) coevolution

(C) adaptive radiation

(D) natural selection

(E) artificial selection

(A) convergent evolution

(B) coevolution

(C) adaptive radiation

(D) natural selection

(E) artificial selection

1. (B)

Variation in a population results from an influx or development of new genetic material. Conjugation is a primitive form of sexual reproduction carried out by bacterial and algae. If a population is isolated, there can be no flow of genetic material.

Variation in a population results from an influx or development of new genetic material. Conjugation is a primitive form of sexual reproduction carried out by bacterial and algae. If a population is isolated, there can be no flow of genetic material.

2. (B)

Tendons connect bone to muscle; ligaments connect bone to bone.

Tendons connect bone to muscle; ligaments connect bone to bone.

3. (D)

A solution of pH 5 has H^+ concentration of 1 x 10^-5 M or -0.00001 M. A solution of ph7 has H^+ concentration of 1 x 10^-7 M or -0.0000001 M. You can see that 0.00001 M is 100 times more concentrated than 0.0000001 M.

A solution of pH 5 has H^+ concentration of 1 x 10^-5 M or -0.00001 M. A solution of ph7 has H^+ concentration of 1 x 10^-7 M or -0.0000001 M. You can see that 0.00001 M is 100 times more concentrated than 0.0000001 M.

4. (E)

Artificial selection is the selective breeding of domesticated plants and animals to develop desired traits.

]]>Artificial selection is the selective breeding of domesticated plants and animals to develop desired traits.

Free Image from Pixabay

A certain platoon is made up of 3 squads, each of which has 4 soldiers. When the platoon lines up to enter the mess hall, the squads are allowed to be in any order but the soldiers must line up within their squads according to certain rules. The soldiers in the first squad can line up any way they want as long as they stay with their squad. The squad leader of the second squad insists that the soldiers in that squad be in one particular order. the third squad leader wants the soldiers in that squad to line up in order from either tallest to shortest or shortest to tallest. How many different ways can the platoon line up?

Answer is 288.

Solution:

This question involves the "groups of groups" pattern. First consider how many ways the groups (squads) can be arranged. There are 3 distinct squads, so there are 3! = 3 x 2 x 1 = 6 different ways. For the squad that is permitted to choose any order they wish, there are 4! = 4 x 3 x 2 x 1 = 24 different ways they can line up. The squad that lines up by height can only have 2 variations and the remaining squad only has one way to line up within the squad. Therefore, the total number of ways that the platoon can line up is 6 x 24 x 2 x 1 = 288.

]]>This question involves the "groups of groups" pattern. First consider how many ways the groups (squads) can be arranged. There are 3 distinct squads, so there are 3! = 3 x 2 x 1 = 6 different ways. For the squad that is permitted to choose any order they wish, there are 4! = 4 x 3 x 2 x 1 = 24 different ways they can line up. The squad that lines up by height can only have 2 variations and the remaining squad only has one way to line up within the squad. Therefore, the total number of ways that the platoon can line up is 6 x 24 x 2 x 1 = 288.

Free image from Pixabay

The Verbal Reasoning section features Sentence Equivalence questions on the GRE. In each sentence, one word will be missing, and you must identify two correct words to complete the sentence. The correct answer choices, when used in the sentence, will result in the same meaning for both sentences. This question type tests your ability to figure out how a sentence should be completed by using the meaning of the entire sentence.

In the questions below, select the **two** answer choices that, when inserted into the sentence, fit the meaning of the sentence as a whole **and **yield complete sentences that are similar in meaning.

1. Her last-minute vacation was _______________________ compared to her usual trips, which are planned down to the last detail.

A. expensive

B. spontaneous

C. predictable

D. satisfying

E. impulsive

F. atrocious

A. expensive

B. spontaneous

C. predictable

D. satisfying

E. impulsive

F. atrocious

2. After staying up all night, she felt extremely _____________________; however, she still an three miles with her friends.

A. apprehensive

B. lethargic

C. controversial

D. sluggish

E. vigorous

F. energetic

A. apprehensive

B. lethargic

C. controversial

D. sluggish

E. vigorous

F. energetic

3. Although the lab assistant openly apologized for allowing the samples to spoil, her _________________ did not appease the research head, and she was let go.

A. insincerity

B. frankness

C. falsehoods

D. candor

E. inexperience

F. hesitation

A. insincerity

B. frankness

C. falsehoods

D. candor

E. inexperience

F. hesitation

4. He was unable to move his arm after the stroke; in addition, the stroke ____________________ his ability to speak.

A. appeased

B. satisfied

C. impeded

D. helped

E. hindered

F. assisted

A. appeased

B. satisfied

C. impeded

D. helped

E. hindered

F. assisted

5. The firefighter, desperate to save the children on the second floor of the fiery house, rushed into their bedroom; his colleagues, more wary of the ____________________ structure, remained outside.

A. stalwart

B. precarious

C. stout

D. irrefragable

E. tottering

F. fecund

A. stalwart

B. precarious

C. stout

D. irrefragable

E. tottering

F. fecund

The phase "compared to her usual trips" is a clue that the first half of the sentence will have an opposite meaning to the second half; that is, her "last-minute" vacation was apparently not planned in detail.

You can use this phrase to make a prediction such as: "her last minute vacation was unplanned compared to her usual trips, which are planned down to the last detail."

Something that is unplanned is done without much preparation or careful thought. Both choice

For answers below, try to reason them on your own. If you are not sure, do not hesitate to contact us!

2. B, D

3. B, D

4. C, E

5. B, E

]]>Free image from Pixabay

\[\frac{1}{(a + b)^\frac{-1}{2}} = (a + b)^{\frac{-1}{2}}\]

If the above is true, which of the following must be true?

\[(A) \quad a = 0\]

\[(B) \quad \sqrt{a + b} = - 1\]

\[(C) \quad \sqrt{a + b} = 0\]

\[(D) \quad a + b = 1\]

\[(E) \quad (a + b)^2 = 0\]

Here is an expression that is equal to the reciprocal of itself. Can you think of numbers that have those properties?

Answer: (D)

Don't let the fractions in the exponents throw you off. The first step of such problems is to cross-multiply.

Don't let the fractions in the exponents throw you off. The first step of such problems is to cross-multiply.

\[\frac{1}{(a + b)^\frac{-1}{2}} = (a + b)^{\frac{-1}{2}}\]

\[1 = \frac{1}{(a + b)^\frac{-1}{2}} \quad * \quad (a + b)^{\frac{-1}{2}}\]

\[(a+b)^{-1} = {\frac{1}{a + b}}\]

Since this fraction is equal to 1, we know the reciprocal is also equal to 1.

1 = a + b, or choice (D).

]]>One of the most amazing properties of the natural numbers -- 1, 2, 3, 4, 5, 6, 7 ... -- is that they go on forever. However, you can't write down the largest number. If you think a number is the largest, you can always add 1 to it and make it larger. Hence, the sequence of natural numbers is infinite.

So how to we deal with large numbers? For example, the earth has a mass of about 5.98 sextillion metric tons. Wait a minute! What is sextillion??

It is 1,000,000,000,000,000,000,000.

And the Great Lakes contain roughly 52.92 duodecillion water molecules !! What is duodecillion?

It is 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

How do you represent such large numbers easily? And what if we have to add or multiply them?

Archimedes, the great scientist and mathematician of antiquity, came up with a solution. Can you guess it?

Exponents!!

Yes, you can use exponents to reduce the number of zeroes to deal with. Archimedes wrote a letter to a local monarch, in which he proposed some tools and methods to handle arbitrarily large numbers. He informed the monarch that he could use this method to count the number of grains of sand in the universe.

Archimedes was fed up with people saying you couldn’t calculate the number of grains of sand on a beach. He believed that one could, and he calculated not just how many grains of sand there were on the beach, but how many there were in the universe. The trouble Archimedes faced was the Greek number system. It was a primitive system in which letters became numbers: A = 1, B = 2, C = 3, etc. So Archimedes invented a new classification of numbers: the exponents.

Archimedes thought that to count the number of grains of sand in the universe he needed numbers up to the eighth order, i.e. (10^8)^8 = 10^64, which is equal to:

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

We call this great innovation that Archimedes developed the**Exponential Notation**. He uses exponents of powers of 10. And the idea is to express large numbers using powers of 10, by keeping track of the number of digits in the number. So using exponential notation, we would write 100 as 10 squared. And 1,000 as 10 cubed. 10,000 -- 10 to the 4th. 100,000 -- 10 to the 5th power. So a number like 4,270,000 would be written as 4.27 times 10 to the 6th, or 4.27 x 10^6.

The advantage of this exponential notation that Archimedes discovered is, that it makes it very easy to multiply arbitrarily large numbers. And that's for the following reason. If we wanted to multiply a million (10^6) times 100 million (10^8), all we have to do is add the exponents! So 6 + 8 = 14 and you get 10^14 as the result. In other words, 10^6 x 10^8 = 10^(6+8) = 10^14.

How would we apply the exponential notation to estimating the time since the Big Bang?

Astronomers have estimated that the Big Bang is about is about 13.8 billion years old. That was a number that was obtained from the recent Planck mission. We can write it in exponential notation, as 13.8 x 10^10, as 1 Billion is 10^9. We can rewrite it in the standard form as 1.38 x 10^9 years. That's the number of years in the history of the universe.

What about the number of seconds in the history of the universe? Well, we have to take that large number-- 1.38 times 10 to the 10th power, or 1.38 x 10^10 -- and multiply it by the number of seconds in every year.

Well, how many seconds are there in a year? There are 60 seconds in a minute, 60 minutes in an hour, 24 hours every day, and 365 days in a normal year. So the total number of seconds in a year is 60 times 60 times 24 times

365, which works out exactly to 31,536,000 seconds in a year. In exponential notation, we have to convert that number. And we get 3.15 times 10 to the 7th power, or 3.15 x 10^7.

So to calculate the number of seconds in the history of the universe, we take the number of years in the history of the universe --1.38 x 10^10 -- and multiply it by the number of seconds in a year -- 3.15 x 10^7.

We multiply the constants, 1.38 times 3.15, which is about 4.4. And then we add the exponents, 10 and 7 to get 17. So, approximately, the number of seconds in the history of the Big Bang as we know it is 4.4 x 10^17 seconds.

Isn't that impressive?

So how to we deal with large numbers? For example, the earth has a mass of about 5.98 sextillion metric tons. Wait a minute! What is sextillion??

It is 1,000,000,000,000,000,000,000.

And the Great Lakes contain roughly 52.92 duodecillion water molecules !! What is duodecillion?

It is 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

How do you represent such large numbers easily? And what if we have to add or multiply them?

Archimedes, the great scientist and mathematician of antiquity, came up with a solution. Can you guess it?

Exponents!!

Yes, you can use exponents to reduce the number of zeroes to deal with. Archimedes wrote a letter to a local monarch, in which he proposed some tools and methods to handle arbitrarily large numbers. He informed the monarch that he could use this method to count the number of grains of sand in the universe.

Archimedes was fed up with people saying you couldn’t calculate the number of grains of sand on a beach. He believed that one could, and he calculated not just how many grains of sand there were on the beach, but how many there were in the universe. The trouble Archimedes faced was the Greek number system. It was a primitive system in which letters became numbers: A = 1, B = 2, C = 3, etc. So Archimedes invented a new classification of numbers: the exponents.

Archimedes thought that to count the number of grains of sand in the universe he needed numbers up to the eighth order, i.e. (10^8)^8 = 10^64, which is equal to:

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

We call this great innovation that Archimedes developed the

The advantage of this exponential notation that Archimedes discovered is, that it makes it very easy to multiply arbitrarily large numbers. And that's for the following reason. If we wanted to multiply a million (10^6) times 100 million (10^8), all we have to do is add the exponents! So 6 + 8 = 14 and you get 10^14 as the result. In other words, 10^6 x 10^8 = 10^(6+8) = 10^14.

How would we apply the exponential notation to estimating the time since the Big Bang?

Astronomers have estimated that the Big Bang is about is about 13.8 billion years old. That was a number that was obtained from the recent Planck mission. We can write it in exponential notation, as 13.8 x 10^10, as 1 Billion is 10^9. We can rewrite it in the standard form as 1.38 x 10^9 years. That's the number of years in the history of the universe.

What about the number of seconds in the history of the universe? Well, we have to take that large number-- 1.38 times 10 to the 10th power, or 1.38 x 10^10 -- and multiply it by the number of seconds in every year.

Well, how many seconds are there in a year? There are 60 seconds in a minute, 60 minutes in an hour, 24 hours every day, and 365 days in a normal year. So the total number of seconds in a year is 60 times 60 times 24 times

365, which works out exactly to 31,536,000 seconds in a year. In exponential notation, we have to convert that number. And we get 3.15 times 10 to the 7th power, or 3.15 x 10^7.

So to calculate the number of seconds in the history of the universe, we take the number of years in the history of the universe --1.38 x 10^10 -- and multiply it by the number of seconds in a year -- 3.15 x 10^7.

We multiply the constants, 1.38 times 3.15, which is about 4.4. And then we add the exponents, 10 and 7 to get 17. So, approximately, the number of seconds in the history of the Big Bang as we know it is 4.4 x 10^17 seconds.

Isn't that impressive?

References:

1. https://sites.google.com/site/largenumbers/home/2-1/Larger_Numbers_in_Science

2. https://www.famousscientists.org/how-archimedes-invented-the-beast-number/

]]>1. https://sites.google.com/site/largenumbers/home/2-1/Larger_Numbers_in_Science

2. https://www.famousscientists.org/how-archimedes-invented-the-beast-number/

Image from Pixabay

The Fibonacci sequence was first observed by the Italian mathematician Leonardo Fibonacci in 1202. He was investigating how fast rabbits could breed under ideal circumstances. He made the following assumptions:

Fibonacci asked how many*pairs* of rabbits would be produced in one year.

Can you create the numbers yourself? Remember to count the 'pairs' of rabbits and not the individual ones. Try it.

Were you able to come up with the Fibonacci numbers? If not, here is how you would do it.

The pattern comes out to be 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233.

Fibonacci numbers are of interest to biologists and physicists because they are frequently observed in various natural objects and phenomena. For example, the branching patterns in trees and leaves are based on Fibonacci numbers.

On many plants, the number of petals is a Fibonacci number: buttercups have 5 petals; lilies and iris have 3 petals; some delphiniums have 8; corn marigolds have 13 petals; some asters have 21 whereas daisies can be found with 34, 55 or even 89 petals.

- Start with one male and one female rabbit. Rabbits can mate at the age of one month, so by the end of the second month, each female can produce another pair of rabbits.
- The rabbits never die.
- The female produces one male and one female every month.

Fibonacci asked how many

Can you create the numbers yourself? Remember to count the 'pairs' of rabbits and not the individual ones. Try it.

Were you able to come up with the Fibonacci numbers? If not, here is how you would do it.

- You start with one pair of rabbits - (1 pair).
- At the end of the first month, there is still only one pair - (1 pair).
- At the end of the second month, the female has produced a second pair, so there are 2 pairs - (2 pairs).

- At the end of the third month, the original female has produced another pair, so now there are 3 pairs - (3 pairs).
- At the end of the fourth month, the original female has produced yet another pair, and the female born two months earlier has produced her first pair, making a total of 5 pairs - (5 pairs).
- And so on ...

The pattern comes out to be 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233.

Fibonacci numbers are of interest to biologists and physicists because they are frequently observed in various natural objects and phenomena. For example, the branching patterns in trees and leaves are based on Fibonacci numbers.

On many plants, the number of petals is a Fibonacci number: buttercups have 5 petals; lilies and iris have 3 petals; some delphiniums have 8; corn marigolds have 13 petals; some asters have 21 whereas daisies can be found with 34, 55 or even 89 petals.

How can we create a rule (algorithm) for the fibonacci series (sequence)?

First, the terms are numbered from 0 onwards like this:

*n = 0 1 2 3 4 5 6 7 8 9 10 ...*

xn =0 1 1 2 3 5 8 13 21 34 55 ...

What rule can we create here? Well, if you look, x3 = x2 + x 1 (2 = 1 + 1) and x4 = x2 + x3 (3 = 1 + 2), etc.

So we can write the rule (algorithm) as: xn = x(n-1) + x(n-2).

where:

Example: term 7 is calculated as:

x7= x(7-1) + x(7-2)

= x6 + x5

= 13 + 8

= 21

First, the terms are numbered from 0 onwards like this:

xn =0 1 1 2 3 5 8 13 21 34 55 ...

What rule can we create here? Well, if you look, x3 = x2 + x 1 (2 = 1 + 1) and x4 = x2 + x3 (3 = 1 + 2), etc.

So we can write the rule (algorithm) as: xn = x(n-1) + x(n-2).

where:

**xn**is term number "n"**x(n-1)**is the previous term (n-1)**x(n-2)**is the term before that (n-2)

Example: term 7 is calculated as:

x7= x(7-1) + x(7-2)

= x6 + x5

= 13 + 8

= 21

Let's write programs in Python to calculate the Fibonacci numbers.

1. With looping:

def fib(n):

a,b = 1,1

for i in range(n-1):

a,b = b,a+b

return a

print(fib(1))

print(fib(2))

print(fib(3))

print(fib(4))

print(fib(5))

print(fib(6))

print(fib(7))

print(fib(8))

a,b = 1,1

for i in range(n-1):

a,b = b,a+b

return a

print(fib(1))

print(fib(2))

print(fib(3))

print(fib(4))

print(fib(5))

print(fib(6))

print(fib(7))

print(fib(8))

1. With recursion:

def fibR(n):

if n==1 or n==2:

return 1

return fibR(n-1)+fibR(n-2)

print(fibR(1))

print(fibR(2))

print(fibR(3))

print(fibR(4))

print(fibR(5))

print(fibR(6))

print(fibR(7))

print(fibR(8))

if n==1 or n==2:

return 1

return fibR(n-1)+fibR(n-2)

print(fibR(1))

print(fibR(2))

print(fibR(3))

print(fibR(4))

print(fibR(5))

print(fibR(6))

print(fibR(7))

print(fibR(8))

N.B: No not copy and paste the python code as identation is important.

]]>The greatest common divisor (GCD) or the highest common factor (HCF) of two numbers is the largest positive integer that perfectly divides the two given numbers. Solving this problem for a specific set of numbers is easy. For example, find the GCD of 12 and 18. The The divisors of 12 are 1, 2, 3, 4, 6, 12 and for 18 are 1, 2, 3, 6, 9, 18. The common factors are 1, 2, 3, and 6. So the greatest common factor is 6.

How would you find the GCD for any number? Here the problem is more challenging. Here is one solution. Let's take two integers a and b passed to a function which returns the GCD. In the function, we first determine the smaller of the two number since the GCD (HCF) can only be less than or equal to the smallest number. For example, the GCD of 12 and 14 can only be less than 12 and not greater. We then use a for loop to go from 1 to that number.

In each iteration, we check if our number perfectly divides both the input numbers. If so, we store the number as the GCD. At the completion of the loop we end up with the largest number that perfectly divides both the numbers.

Below is the algorithm in python.

In each iteration, we check if our number perfectly divides both the input numbers. If so, we store the number as the GCD. At the completion of the loop we end up with the largest number that perfectly divides both the numbers.

Below is the algorithm in python.

def computeGCD(a, b):

if a < b:

smaller = a

else:

smaller = b

for i in range(1, smaller+1):

if (a % i == 0) & (b % i == 0):

gcd = i

return gcd

print(computeGCD(24, 16))

print(computeGCD(48, 256))

N.B: Do not cut and paste the above code. Make sure the indentation is correct.

if a < b:

smaller = a

else:

smaller = b

for i in range(1, smaller+1):

if (a % i == 0) & (b % i == 0):

gcd = i

return gcd

print(computeGCD(24, 16))

print(computeGCD(48, 256))

N.B: Do not cut and paste the above code. Make sure the indentation is correct.

The above method is easy to understand and implement but not efficient. A much more efficient method to find the GCD (HCF) is the Euclidean algorithm.

The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. That is a mouthful! Let's make it simple by taking an example. 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 147 (252 − 105). Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that occurs, they are the GCD of the original two numbers.

A more efficient version of the algorithm shortcuts these steps, instead we divide the greater by smaller and take the remainder. Now, divide the smaller by this remainder. Repeat until the remainder is 0.

For example, if we want to find the H.C.F. of 54 and 24, we divide 54 by 24. The remainder is 6. Now, we divide 24 by 6 and the remainder is 0. Hence, 6 is the required GCD.

A more efficient version of the algorithm shortcuts these steps, instead we divide the greater by smaller and take the remainder. Now, divide the smaller by this remainder. Repeat until the remainder is 0.

For example, if we want to find the H.C.F. of 54 and 24, we divide 54 by 24. The remainder is 6. Now, we divide 24 by 6 and the remainder is 0. Hence, 6 is the required GCD.

Python code for Euclidean Algorithm

def euclidAlgo(a, b):

while (b):

a, b = b, a % b

return a

print(euclidAlgo(24, 16))

print(euclidAlgo(48, 256))

while (b):

a, b = b, a % b

return a

print(euclidAlgo(24, 16))

print(euclidAlgo(48, 256))

Python code for Euclidean Algorithm using recursion:

def euclidAlgo(a, b):

if (b == 0):

return a

else:

return euclidAlgo(b, a % b)

print(euclidAlgo(24, 16))

print(euclidAlgo(48, 256))

if (b == 0):

return a

else:

return euclidAlgo(b, a % b)

print(euclidAlgo(24, 16))

print(euclidAlgo(48, 256))

Sources: Wikipedia; https://www.programiz.com/python-programming/examples/hcf

]]>Free Image from Pixabay

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.

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.

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))

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.

]]>Free Image from Pixabay

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

(B) 2.58

(C) 2.56

(D) 2.55

(E) 2.54

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:

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

For n = 0, we have:

\(x_{0+1} = \sqrt {4 + x_0} \)

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

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

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

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

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 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.

]]>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.

Free Image from Pixabay

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.”

“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.”

A. Mr. Rochester is incapable of loving Miss Eyre.

F. recognize that Mr. Rochester actually wants to marry Mrs. Fairfax.

A. “Mr. Rochester looks as young, and is as young, as some men at five and twenty.”

F. explain why Miss Eyre should not marry Mr. Rochester.

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.

- 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

]]>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)

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).

Free Image from Pixabay

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.

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.

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

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

A. Carbon Orb

B. Water

C. Rocky Earth

D. Gas Giant

1. D

2. F

3. D

]]>2. F

3. D

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 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.

]]>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.

Free Image from Pixabay

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 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

]]>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

Image from Pixabay

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

To solve this problem, first multiply the percent abundance by the atomic mass of a given isotope and then add the products together:

30.00% = 40.00 amu: 30% is 310%. So if 10% of 40 is 4, then 34 = 30% of 40, which is equal to 12 amu.

50.00% = 41.00 amu, and 50% is 1 /2 of 41, which is 20.5 amu.

20.00% = 42.00 amu, and 20% = 210%, so 10% of 42 = 4.2, and 20% = 24.2, which is 8.4 amu.

Now add those three numbers together to get your answer: 12 + 20.5 + 8.4 = 40.9 amu.

An another way to solve is to estimate: 50% of the element exists as the 41.00 amu isotope. Now, 30% of the remaining element is the 40.00 amu isotope, and only 20% of the element exists as the 42.00 isotope. Therefore you can estimate that the average of these three amounts should be less than 41 since there is more of the lighter isotope. The only answer choice that’s less than 41 is A.

The fourth principal energy level has four sublevels: s, p, d, and f. If the sublevel is completely filled, then s = 2 electrons, p = 6 electrons, d = 10 electrons, and f = 14 electrons; thus 2 + 6 + 10 + 14 = 32 total electrons for a full fourth principal energy level.

The set of quantum numbers given was n = 3, l = 1, ml = 0, ms = ±1/2. If n = 3, this means that it’s a third energy level electron; if l = 1, then it’s a p-sublevel electron; if ml= 0, then it’s in the middle position of the set of three p orbitals. The only tricky thing is that ms = + or - 1/2. This means it’s either a p2 or a p5 electron. However, if it were p5, then one of the answer choices would be argon (a noble gas), but it isn’t listed, so it must be the p2, which makes silicon the correct answer.

The configuration given is 1s22s22p63s23p63d44s2. The 3d4 is the important part—it means the element we desire is in the first row of the d-block elements and is the fourth element in that block, so it is Cr, or chromium.

The half-life is given as 9.98 minutes, which is close to 10 minutes. The total time is given as 60.0 minutes, so the sample undergoes six half-lives. Start with this mass and keep cutting it in half; each 10-minute half-life should be represented with an arrow, and you can even put numbers under each arrow if you want, in order to keep track.

20.010.05.02.51.250.6250.3125, which is B.

This question asks about the second ionization energy. Remember that the second ionization energy of any element is always larger than its first ionization energy. The second ionization energy is significantly larger if the second electron comes from a completed sublevel or principal energy level. Na’s first electron removed is 3s1, while the second to be removed comes from 2p6. There is a huge increase in the amount of energy needed to remove that second electron because of the change in principal energy levels.

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

]]>· absence

· acceptable

· accessible

· accommodation

· accomplish

· achievement

· acquire

· address

· advertisement

· advice – (noun)

· advise – (verb)

· amateur

· apartment

· appearance

· argument

· athletic

· attendance

· basically

· beginning

· belief – indicating the noun

· believe – indicating the verb

· beneficial

· business

· calendar

· campaign

· category

· cemetery

· challenge

· characteristic

· cigarette

· clothes

· column

· committee

· commitment

· completely

· condemn

· conscience

· conscientious

· conscious

· controversy

· convenient

· correspondence

· criticism

· deceive

· definitely

· definition

· department

· describe

· despair

· desperate

· development

· difference

· difficult

· disappointed

· discipline

· disease

· easily

· effect

· eighth

· either

· embarrass

· encouragement

· enemy

· entirely

· environment

· especially

· exaggerate

· excellent

· existence

· experience

· experiment

· familiar

· February

· finally

· financial

· foreign

· foreigner

· formerly

· forty

· fourth

· general

· generally

· genius

· government

· grammar

· grateful

· guarantee

· guidance

· happily

· height

· heroes

· humorous

· hypocrite

· ideally

· imaginary

· immediate

· incredible

· independent

· influential

· insurance

· intelligent

· interference

· interrupt

· introduce

· island

· its – for possession

· it’s – for “it is” or “it has”

· jealous

· jealousy

· kneel

· knowledge

· later

· legitimate

· length

· library

· lightning

· likely

· loneliness

· lose (verb)

· loose (adjective)

· lovely

· luxurious

· maintain

· maintenance

· manageable

· management

· manufacture

· marriage

· married

· millionaire

· misspell

· mischievous

· money

· mortgage

· muscle

· mysterious

· naturally

· necessary

· neighbor / neighbour

· ninety

· noticeable

· nowadays

· obedient

· obstacle

· occasional

· occurred

· official

· opinion

· opportunity

· opposition

· ordinary

· originally

· particular

· peculiar

· perceive

· performance

· permanent

· personal

· personnel

· physical

· physician

· piece

· pleasant

· possession

· possible

· possibility

· potatoes

· practically

· prefer

· privilege

· professor

· professional

· pronounce / pronunciation

· psychology

· psychological

· quantity

· quality

· questionnaire

· queue

· quizzes

· realistic

· realize

· really

· receipt

· receive

· recognize

· recommend

· religion

· religious

· remember

· representative

· restaurant

· rhythm

· ridiculous

· roommate

· sacrifice

· safety

· scared

· scenery

· schedule

· secretary

· sentence

· separate

· similar

· sincerely

· strength

· surprise

· suspicious

· success

· successful

· technical

· technique

· temperature

· temporary

· their (possessed by them)

· there (not here)

· they’re (contraction of “they are”)

· themselves – not themself

· undoubtedly

· unforgettable

· unique

· until

· valuable

· village

· violence

· violent

· vision

· volume

· weather – indicating climate – The weather is nice today.

· Wednesday

· weird

· whether – (indicating if)

· which

· woman – (singular)

· women – (plural)

· worthwhile

· width

· writing

· yacht

· young

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.

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

- 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.

- 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