\[\sqrt[5]{\sqrt[4]{\sqrt[3]{\sqrt{n}}}} \] | 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.

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

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

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.

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.

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.

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).

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.

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

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.

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

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 termites had a devastating effect on the house.

Use one or the other, but not both.

You cannot say:

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.

The alternative to a taxi is Uber or Lyft.

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

The single word, everyday, is an adjective. It is usually found before a noun.

Expressing time every day is two words.

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

Any one of his colleagues would agree with his position.

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

The single word is correct.

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

Greta dozed for a while.

Do you think something doesn't sound right in the sentence below?

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:

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.

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!

Let's look at the following sentence.

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.

Samantha stayed home is because Tabatha was crying.

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.

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.]

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.

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

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

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.

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.

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.

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.

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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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.

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

- 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.

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:

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:

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:

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.

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

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

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).

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.

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?

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.

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

- 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

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:

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.

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.

- 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

- 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

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:

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

- 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.

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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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*:

Grammarians call it the

- 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.)

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:

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:

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

Why is this? It is because "what if" means something that's untrue. When that happens, the subjunctive mood kicks in, and

- 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

- 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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