Essential GRE Quant Skills

Question 1

How do we convert an Improper Fraction to a Mixed Number?

e.g., 9/2

Answer 1

Divide the numerator by the denominator.
The quotient becomes the whole number, the remainder becomes the new numerator, and the denominator remains.

e.g., 7/2? 7 divides by 2 3 times, with a remainder of 1.

Thus 7/2 = 3 and 1/2

Question 2

How do we convert a Mixed Number to an Improper Fraction?

Answer 2

Multiply the denominator by the whole number. Add the original numerator to that product, then put it over the original denominator.

e.g., 4 and 1/3. 4 times 3 = 12. Add the 1 in the numerator.

Thus , 4 and 1/3 = 13/3

Question 3

What is the Least Common Denominator (LCD)?

Answer 3

The smallest figure that each of your numbers evenly divides into.

Tip: begin with the largest denominator to save time.

For example, the LCD of 1/2, 1/3, and 1/12 can’t be less than 12 because 12 doesn’t evenly divide into anything less than 12.

Question 4

What is the elementary way to find the LCD of 2 or more numbers?

The way TTP suggests using to start out.

Answer 4

The simplest way is to list out the multiples of each denominator, then select the smallest one that is found for all of them.

Question 5

If we have two fractions, what is a trick to determine whether they are equivalent fractions?

Answer 5

The quickest way to test if two fractions are equivalent is to cross-multiply the numerators and denominators of each. If equivalent, you’ll get the same number.

e.g., 1/2 and 2/4. 1x4 = 4. Also, 2x2 = 4, so they’re equivalent.

Question 6

What are equivalent fractions?

If we have two fractions, what is a trick to determine whether they are equivalent fractions?

Answer 6

These are fractions that represent the same portion of the whole, but with different values.

The quickest way to test if two fractions are equivalent is to cross-multiply the numerators and denominators of each. If equivalent, you’ll get the same number.

e.g., 1/2 and 4/8. 3/4 and 6/8.

Again, two fractions a/b and c/d are equivalent if a x d = b x c.

Question 7

How do we add & subtract fractions?

Answer 7

When there is a shared denominator, we add/sub the numerators and place above the existing denominator.

If there is not a shared denominator, we must find one before doing so.

e.g., 1/4 + 2/4 = 3/4. Similarly, 3/4 = 1/4 + 1/4 + 1/4

a/b + c/b = (a+c)/b. This applies to subtraction, too.

Question 8

Can we break up fractions with the same denominator?

e.g., is 3/5 = 1/5 + 2/5?

Answer 8

Yes.
a/b +- c/b = (a+-c)/b
So, it must be true that:
(a+-c)/b = a/b +- c/b

e.g., 5/6 - 4/6 = 1/6

Referred to as the Distributive Property of Division.

Question 9

Can we break up fractions with the same numerator, but different denominators?

e.g., is 3/5 = 3/4 + 3/1?

Answer 9

No.
b/a +- b/c =/= b/(a+-c)

No, 3/5 =/= 3/4 + 3/1

This makes sense visually, but can get confusing when just using variables.

Question 10

How does the Bowtie Method for getting a shared denominator look as a formula?

Answer 10

a/b + c/d
= ad/bd + bc/bd
= (ad+bc)/bd

e.g., 1/3 + 3/5 = (1x5 + 3x3)/15 = (5 + 9)/15 = 14/15

Question 11

How do we add a fraction to a whole number?

Answer 11

We can simply “attach” the whole number to the fraction (proper or improper).

Or, we can convert the whole number to an improper fraction and add them.

A b/c = [(c x A) + b ] / c

This works with proper and improper fractions.

e.g., 4 + 1/6 = [(6x4) + 1]/6. Note that 2 + 3/5 = 2 and 3/5 = 13/5

Question 12

How do we subtract a fraction from a whole number?

Answer 12

The main method is to convert the whole number to a fraction, then subtract.

A - b/c = [(cxA)-b]/c

There is another method, but I find it confusing, so we’ll stick to this for now.

e.g., 5 - 2/3 = [(3x5) -2]/3 = (15-2)/3 = 13/3 or 4 and 1/3

Question 13

How do we multiply fractions?

Answer 13

We multiply the numerators and denominators across, then simplify.
a/b x c/d = ac/bd

e.g., 2/5 x 3/4 = 6/20 = 3/10

Question 14

How do we divide fractions?

Answer 14

We take the reciprocal of the divisor (i.e., the number after the division sign or below the division bar), then multiply the fractions.

a/b / c/d = a/b x d/c = ad/bc

e.g., 1/3 / 3/4 = 1/3 x 4/3 = 4/9

Question 15

Multiplying a whole number and a fraction?

Answer 15

Put the whole number over 1 and multiply.

Question 16

What are the two ways we can simplify fractions we are multiplying?

Answer 16

1) Top-and-bottom simplification
2) Cross simplification

Remember two things:
* you can cross simplify two non-adjacent fractions
* you can do top-and-bottom and cross simplification in the same problem.

It doesn’t matter the order you do them in.

e.g., 35/64 x 1/2 x 24/45 = 7/64x

Question 16

What is “Top-and-Bottom” simplification?

Relates to multiplying fractions

Answer 16

Top-and-bottom simplification is when we divide the numerator and denominator by the same value.
(i.e., we remove a factor that occurs in both)

Ex: 20/25 = (4x5)/(5x5) = 4/5

Question 16

What is “cross simplification”?

When can it be used: when adding or multiplying?

Answer 16

Cross simplification is when we divide the numerator of one fraction and the denominator of another fraction by the same value.
(i.e., we remove a factor found in each)

Ex: 6/7 x 7/10 = 3/7 x 7/5

Question 16

How do we get the reciprocal of a whole number?

e.g., what is the reciprocal of 5?

Answer 16

Put 1 over the number.

e.g., 5 = 1/5

Question 16

How do we get the reciprocal of a fraction?

e.g., what is the reciprocal of 1/6?

Answer 16

In simple terms, flip the numerator and the denominator.

We put 1/1 over the fraction.

e.g., 1/6 = (1/1)/(1/6) = 1 x (6/1) = 6/1 or 6

Question 16

What is the reciprocal of -1?

What is the reciprocal of 0?

Answer 16

The reciprocal of -1 is -1.

-1 = -1/1, which when flipped, is 1/-1, which is -1.

0 is the only number without a reciprocal.

0 = 0/1, which when flipped, is 1/0, which is undefined.

Question 16

What does the product of a number and its reciprocal equal?

Answer 16

It always equals 1.

Question 16

What are our methods for approaching complex fractions?

e.g., try both for [(1/3) + (1/5)] / [(1/3) + (1/4)]

Answer 16

Method 1: make the numerator and denomiator as single fractions, then divide.

Method 2: multiply both the numerator by the (LCD/1) and the denominator by the (LCD/1) of the complex fraction, then simplify.

e.g., if we have [(1/3) + (1/5)] / [(1/3) + (1/4)], our LCD is 60, as the LCD of 3, 5, and 4 is 60.

So we multiply both the numerator by (60/1) and the denominator by (60/1), then simplify.

Question 16

How can we use the Bowtie Method to compare the size of fractions?

Answer 16

Say we have two fractions, A (which is a/b) and B (which is c/d).

If the product of ad > the product of bc, (a/b) > (c/d).

You have to start with the denominators and multiply up.

e.g., 4/5 and 3/7. (5x3) < (7x4) so 4/5 is bigger.

Question 17

How can converting to a common denominator help us compare the size of fractions?

Answer 17

If our set of positive fractions are all under the same denominator, the larger the numerator, the larger the value of the fraction.

e.g., 2/5, 3/5, 4/5. They are increasing in size.

Question 18

What are the ways in which we can compare the size of fractions?

Answer 18

* Bowtie Method: when comparing (a/b) and (c/d), if bc>ad, (a/b) is larger

* Common Denominator Method: convert all numbers to common denominator. The larger the numerator, the larger the fraction.

* Common Numerator Method: convert all numbers to a common numerator. The larger the denominator, the smaller the fraction.

Question 19

If we multiply the numerator and denominator of a fraction by the same, constant value, does the value of the fraction change?

e.g., does 3/4 x 2/2 change the value of the fraction?

Answer 19

No, the value does not change if we multiply both the top and bottom. They would be equivalent fractions.

e.g., 3/4 x 2/2 = 6/8. These are equivalent fractions.

Question 20

If we add the same, constant value to the numerators and denominators of a fraction, does the value of the fraction change?

e.g., if we +2 to num & denom of 3/4, is it the same fraction?

Answer 20

We get a completely new value if we add to the numerator and denominator.

e.g., +2 to the numerator and denominator of 3/4 gives 5/6. This is NOT the same.

Question 21

How do we handle rounding when the digit we’re rounding to is a 9?

e.g., Round 5.298 to the nearest hundredths

Answer 21

We can think of it as rounding .29 to .30. Do not drop the 0, however.

Question 22

What does “percent” mean in literal terms?

Answer 22

Percent means “per 100” or “divide by 100.”

Question 23

How do we convert a percent to a fraction?

e.g., What is 42% in fraction form?

Answer 23

We put the fraction over 100, then simplify.

e.g., 42% = 42/100 = 21/50

Question 24

How do we convert a fraction to a percent?

e.g., 3/5 is what percent?

Answer 24

Multiply by 100 and attach the percent sign.

e.g., 3/5 x (100/1) = 300/5 = 60%

Question 25

How do we convert a decimal to a percent?

e.g., 0.004 to a percent = ?

Answer 25

We multiply by 100 (or move the decimal to the right two spaces) and add the percent sign.

e.g., 0.004 = 0.4%

Question 26

What is the Principal Square Root?

e.g., principal square root of 121 = ?

Answer 26

The principal square root is the non-negative square root of a number.

Thus, the principal square root can be greater than or equal to 0, but not less than 0.

e.g., 121 has two square roots, 11 and -11, but its principal square root is 11.

sqrt(0) = 0, which is the principal square root.

Question 27

What is an important consideration when using the square root (radical) symbol?

e.g., What is sqrt(4)?

Answer 27

When using the radical symbol, we consider only the positive square root.

e.g., sqrt(4) is 2, it is never -2!

Question 28

How do we treat squared values when our base is a fraction?

e.g., (a/b)^2 = ?

or (2/3)^2 = ?

Answer 28

We square both the numerator and the denominator.

However, be mindful that the denominator cannot be equal to 0.

e.g., (a/b)^2 = (a^2)/(b^2)

or (2/3)^2 = (2^2) / (3^2) = 4 / 9

Question 29

How do we treat radicals when we’re taking the square root a fraction?

e.g., sqrt( (x/y) ) = ?

or e.g., sqrt( (9/4) ) = ?

Answer 29

We take the square root of both the numerator and the denominator, assuming the numerator is greater than or equal to 0 and the denominator is greater than 0.

e.g., sqrt( (x/y) ) = sqrt(x) / sqrt(y)

or e.g., sqrt(9) / sqrt(4) = 3 / 2

Question 30

What is unique about squares and square roots of numbers between 0 and 1?

e.g., what’s the relationship between x, x^2, and sqrt(x) when 0 < x < 1

Answer 30

When 0 < x < 1:

x^2 < x < sqrt (x)

e.g., if x = 1/4, x^2 = 1/16, and sqrt(x) = 1/2

Since 1/16 < 1/4 < 1/2 … x^2 < x < sqrt(x)

Question 31

When squaring a large number, how can the units digit help us find the answer?

Answer 31

The squared units digit will be reflected in the units digit of the correct answer.

E.g., 9,007^2 = 81,126,049 – 7^2 = 49, so units digit is a 9.

Same goes for decimals:

E.g., 504.13^2 = 254,147.0569

Especially helpful when all answers have unique units digit.

Question 32

A perfect square cannot have what numbers in the units digit?

Answer 32

2, 3, 7, or 8.

Question 33

What is a helpful trick when comparing the size of negative numbers?

Answer 33

If the number is greater/larger when positive, it is lesser/smaller when negative.

Question 34

Note: you can re-express numbers to make adding/subtracting easier.

Answer 34

e.g., 996 + 578 = (1000 - 4) + 578

Question 35

How can we make 1,000,000,000,000 - 888,888,888,888 easier to approach?

Answer 35

We can rewrite 1T as 999,999,999,999(+1) and the operation is much easier.

Question 36

What is 0!

What is 1!

Answer 36

0! = 1
1! = 1

Question 37

How can shortening factorials make our math quicker?

e.g., ( 10! ) / ( 8! x 3! )

Answer 37

Since larger factorials contain smaller factorials, we can cancel them out and save time.

For example, 4! = 4 x 3 x 2 x 1, and 3! = 3 x 2 x 1.
Thus, 4! = 4 x 3!

e.g., (10!) / (8! x 3!) = (10 x 9 x 8!) / (8! x 3!)
= (10 x 9) / (3!) = (10 x 9)/ (3 x 2)
= 5 x 3 = 15

Question 38

How does factoring factorials work?

e.g.,

9! - 8! - 7! = ?

Answer 38

By shortening factorials, we can find common factors and pull them out as we would any factor.

e.g., 9! - 8! - 7!
= [(9x8x7!) - (8x7!) - (7!)]
= 7! [(9x8) - (8) - (1)]
= 7! (72 - 8 - 1) = 7! (63)

Question 39

What are all of the non-negative one-digit integers?

What is their sum?

Answer 39

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

0+1+2+3+4+5+6+7+8+9 = 45

Question 40

True or false: if two numbers are positive, the larger the number, the larger the term when squared.

e.g., which is larger:
0.22^2
or
0.202^2

Answer 40

True – when we have two positive numbers, the larger the value, the larger the value is squared.

e.g., 0.22^2 = 0.0484
0.202^2 = 0.040804

Question 41

How do we use the distributive property to simplify this problem:

16(456) + 10(456) + 24(457) = ?

Answer 41

We can use the distributive property on 24(457), then factor:

16(456) + 10(456) + 24(457)
= 16(456) + 10(456) + 24(456+1)
= 16(456) + 10(456) + 24(456) + 24(1)
= 456 (16+10+24) + 24

Question 42

**How do we take the reciprocal of a complex fraction?

e.g., what is the reciprocal of [(a/b) / (c/d)]?

Or [(2/3) / (4/5)]?

Answer 42

When taking the reciprocal of a complex fraction, you swap the entire numerator and denominator.

e.g., the reciprocal would be [(c/d) / (a/b)].

The other would be [(4/5) / (2/3)].

Question 43

What is the only positive integer whose reciprocal is greater than or equal to itself?

Answer 43

The only positive integer whose reciprocal is greater than or equal to itself is 1.

For all others, as the integer grows, the reciprocal is smaller:
2 –> 1/2
10 –> 1/10
100 –> 1/100

Question 44

Is 1/(a/b) = b/a?

Answer 44

Yes. Don’t forget.