Q1) Basic math (fractions, decimals, converting, squares and square roots, algebra)

Question 1

What is the difference between PROPER and IMPROPER fractions?

Answer 1

Proper fractions - numerator < denominator
So proper fraction < 1

Improper fraction - numerator > denominator
So improper fraction > 1

In problems, you might be able to turn this info into an inequality and find relationship between numerators
e.g., x/2 < 1 < y/2
x < 2 < y
x < y
x - y < 0

Question 2

When would converting to mixed fractions be useful?

Answer 2

When comparing values of two fractions

e.g., If p = 9/2 and q = 16/3, is p > q? Is p*q > 20? Is p + q > 9?

It is faster if we compared the whole numbers first vs getting precise answer by simplifying to a common denominator

Question 3

Two unknowns, linear equation with multiplication, is it solvable?

e.g., 3c + 3a = ac

Answer 3

No - even if there is a multiplication (ac), it is still a LINEAR equation with TWO UNKNOWNS
> not sufficient
> if you pick a value for a and solve for c, it could be a fractional answer too

Question 4

How can you tell two numbers are reciprocals?

Answer 4

Product of the two numbers is 1

Question 5

Two ways to simplify complex fractions

Answer 5

Recall: Complex fractions are fractions whose numerator, denominator, or both contain fractions

Strategy 1) Simplify numerator and denominator into single fractions, then divide

** Strategy 2) Multiply both numerator and denominator of the complex fraction by the LCD of the smaller fractions, then simplify
> works because LCD/LCD = 1 (doesn’t change the value of the fraction)
> often FASTER

For both strategies, work INSIDE OUT (i.e., multiply inner complex fraction by LCD/LCD, then find the next LCD)

Question 6

What is an easy way for finding the LCD for multiple fractions / LCM for multiple numbers?

e.g., LCM of 2, 3,4,5,6

Answer 6

List the multiples of the LARGEST NUMBER until there is a number that is divisible by ALL the other numbers

e.g., Focus on 6 –> 6, 12, 18, 24, 30, 36, 42, 48, 54, 60

60 is the LCM

Question 7

What are strategies for comparing fractions?

Answer 7

Strategy 1) When comparing TWO fractions only, do criss-cross method
e.g., Is 2/7 > 3/8? —> cross multiply to see if the statement is still true
Is 16 > 21? –> Statement is False, so we know 2/7 < 3/8

Strategy 2) Estimate relative sizes of fractions by converting to decimal form

**Strategy 3) Knowing that adding to or subtracting from numerator and denominator a non-zero constant will CHANGE the value of the fraction
> If fraction is positive, ADDING a POSITIVE constant to both numerator and denominator will bring the fraction CLOSER TO 1 (either getting larger or smaller)
> If fraction is positive, SUBTRACTING a positive constant to both numerator and denominator will move the fraction AWAY FROM 1 (either getting larger or smaller depending on whether fraction is between 0 and 1, or >1)
—> HOWEVER, YOU NEED TO MAKE SURE the new numerator and new denominator are BOTH STILL POSITIVE
—> e.g., Suppose the fraction is 1/2. Subtracting 5 from numerator and denominator, (1-5)/(2-5) = -4/-3 = 4/3 —> bigger than 1/2 and closer to 1

> > > SUBTRACTION CAN GET INTO NEGATIVE TERRITORY SO BE CAREFUL

**Strategy 4) Compare fractions with the SAME DENOMINATOR

**Strategy 5) Compare fractions with the SAME NUMERATOR
> do this strategy if it’s not easy to find the LCD

Question 8

Multiplying and Dividing decimals

Answer 8

Multiplying decimals
> calculate the product ignoring the decimal point and leading zeros
> count the total number of decimal places to the RIGHT of the decimal point in the two numbers that were multiplied (include trailing 0s)
> move the decimal point to the LEFT the same number of spaces

e.g., 0.2 * 0.035 —> first calculate 2 * 35 =70 (include the trailing zero!)

Since there are 4 decimal places to the right of the decimal point, move the decimal point in 70 four places to the left = 0.007

*if dealing with decimals, powers of 10, AND exponents, leverage the powers of 10 to bring it INSIDE the power
e.g., (1.776)^3 * x10^9 = (1.776 * 10^3)^3

Dividing decimals
> set up as a fractional form (e.g., 10.36/2.8)
> multiply numerator and denominator by powers of 10 until denominator is a whole number
e.g., 103.6/28
> perform long division or reverse solve using answer choices

Question 9

What do you call a decimal that:

1) Has a finite number of digits

2) Has an infinite number of digits

3) Has repeating digits

Which types of decimals can be converted into fractions?

Answer 9

1) Has a finite number of digits = TERMINATING decimal
> ** denominators need to have factors of only 2 or 5 (in prime factorized form, AND most REDUCED FORM)

2) Has a infinite number of digits = NON-TERMINATING decimal (can be comprised of repeating decimals OR non-repeating decimals)

3) Has repeating digits = REPEATING, non-terminating decimal

Decimals can be:
1) Terminating (0.125) –> Can be converted to fractions
> and comprise only of powers of 2 and/or 5 in the denominator of the fraction’s REDUCED form (aka be wary of the NUMERATOR that could cancel out non 2 or 5 in denominator)
> also be ware of exponents in the denom that could equal 0 and make a term = 1

** 2) Non-terminating and repeating (e.g., 0.3333 = 1/3) –> CAN be converted to fractions
> Denominator comprises of numbers other than powers of 2 and 5 in the fraction’s REDUCED form

** 3) Non-terminating and non-repeating (e.g., Pi)
> CANNOT be converted to fractions
> Denominator comprises of numbers other than powers of 2 and 5 in the fraction’s REDUCED form

In other words, ANY FRACTION with an INTEGER numerator and non-zero INTEGER denominator will either terminate or repeat

Question 10

How do you convert terminating decimals in fractions?

e.g., 3.4

Answer 10

1) Multiply the decimal by a power of 10 until the decimal is a WHOLE NUMBER

2) Put the power of 10 in the denominator
e.g., 34/10

3) Then simplify
e.g., 34/10 = 17/5

In other words, for a terminating decimal with n decimal places:
> The numerator is the number without the decimal point
> The denominator is 10^n

Question 11

** Memorize the following base fractions and their decimal approximations:

1/6 **
1/7 **
1/8
1/9

Answer 11

1/6 –> ~0.167 (0.16666)
5/6 –> 0.833…

1/7 –> ~0.143
2/7 –> ~0.286
3/7 –> 0.429
4/7 –>0.571
6/7 –>0.857

1/8 –> 0.125
3/8 –> 0.375
5/8 –> 0.625
7/8 –> 0.875

1/9 –> ~0.111
2/9 –> ~0.222
3/9 –> ~0.333
4/9 –> ~0.444
5/9 –> ~0.556
8/9 –> ~0.889

Why is this important to memorize?
> So we can easily convert from fractions to decimals, OR vice versa

Question 12

What is the square root of 4?

Answer 12

2
> not -2

Concept: Principal square root of a number is its NON NEGATIVE SQUARE ROOT

Sqrt ( positive known number ) = positive number

Question 13

What is the value of x?

Sqrt(x) = 5

Answer 13

25

Concept: Sqrt ( positive known number ) = positive number

Question 14

If 0 < x < 1, what must be true of x, x^2 and sqrt(x)?

Answer 14

x^2 < x < sqrt(x)

Most helpful when dealing with UNKNOWNS or DECIMALS that cannot be converted to fractional form easily

Question 15

What is the largest if the equation is (Decimal)^2?

Answer 15

The larger the decimal, the larger the answer

Concept: Think about a parabola y = x^2
> If x > 0, as x increases, y increases

Note that (decimal)^2 is < decimal

Question 16

Strategies when dealing with DECIMALS

Answer 16

1) Convert to fraction
> Question is asking about values
> only do so if you are dealing with TERMINATING DECIMALS and it’s easy to do so (e.g., lots of squares and square roots involved too)
> remember that you can ALWAYS convert a terminating decimal into fraction by having a denominator of 100

2) Estimation
> Question is asking about values or requires approximation (“closest to which of the following”)
> Or question asks you to calculate squares or sqrt of decimals and it’s hard to do
> round decimals to nearest whole number that makes calculations easier (e.g., nearest perfect square) OR find the RANGES using easier numbers
> Hint: For PS, scan the answer choices to see how far apart they are
e.g., 123.4^2 is between 120^2 and 130^2
> or CONVERT TO MIXED fraction in order to compare to PS answers
e.g., 95/91 = 1 + 4/91 —> closer to 1 vs 3/2

3) Strategic numbers
> Question is asking about BEHAVIOR (e.g., x2 < x, sqrt(x) - x) versus precise values
> Use an easier decimal that can be converted to a fraction but retains similar characteristics to test behavior
e.g., sqrt(0.7776) –> use sqrt(1/4) since both are between 0 and 1

4) Leverage powers of 10 and patterns

Question 17

What are the only unit digits that a perfect square can have?

Answer 17

0
1
4
5
6
9

Perfect squares NEVER end in 2,3,7, or 8

Question 18

Distributive property and FACTORING

Answer 18

Concept: Applies to multiplication of a number by a SUM OR DIFFERENCE of two numbers
> distributive property and factoring are two sides of the same coin!

a(b + c) = ab + ac = (b + c)a = ba + ca ——> b/c of distributive property and associative property of multiplication AND factoring

a(b - c) = ab - ac = (b - c)a = ba - ca

> especially watch out for hidden applications of distributive property in FRACTION

e.g., (1-2+3-4…+49-50) / (2-4+6-8…+98-100)
> can factor out 2 from the denominator

Question 19

Ways to simplify expressions

Answer 19

1) FACTORING common factors (especially in ALL terms)

2) Utilize distributive property and other rules/patterns
e.g., (20 - 1) + (30 - 1) + (40 - 1) … –> combine 20+30+40 … and sum of -1 separately

3) Simplify fractions

***4) RE-EXPRESS NUMBERS as addition or subtraction
4A: when there are common factors or close enough numbers (1 or 2 apart)
e.g., 30(799) + 15(799) + 54(799) + 798 —-> You can factor out 799 from ALMOST all the terms
= 30(799) + 15(799) + 54(799) + 799 - 1
= 799(99 + 1) - 1
= 799(100) - 1
= 79899 —-> notice this pattern (will be something easy to multiply usually)

e.g., 8180 - 80 + 1 ——> 80 shows up a lot
= 80(81 - 1) + 1
= 80^2 + 1
–> rewrite numbers as addition or subtraction with a power of 10
–> includes repeating decimals

4B) When adding large numbers and don’t want to deal with places –> convert to 1000 (or multiple of 10) +/- number
e.g., 999 + 578 = (1000 - 1) + 578 = 1577

4C) When subtracting large numbers and don’t want to deal with places from 1000 or multiple of 10 –> convert to 999…. + 1 since it is easier to subtract from 999 vs 1000+ (to avoid dealing with place values)
e.g., 1,000,000,000 - 123,456,789
= (999,999,999 + 1) - 123,456,789

4D) Repeating decimals or decimals with patterns –> factor out constant and leave in the decimal with 0s and 1s
e.g., 5.0005/9.0009 = 5(1.0001)/9(1.0001) = 5/9
e.g., 0.368368368 = 368(0.001) + 368(0.000001) + 368(0.000000001)

*** 5) Look for time saving properties of addition
e.g., rearrange order of expression to find patterns (e.g., multiple groupings of 100, 80, 50 etc.)

In other words, work on IDENTIFYING PATTERNS:
> re-expressing number so that you can factor more easily and do arithmetic more easily
> grouping numbers so you can do arithmetic more easily

Question 20

0! and 1!

Answer 20

Both equal 1

0! = 1

1! = 1

Recall: n! (Factorial) is simply the product of all the integers from 1 to n, inclusive, with the exception of 0!

Question 21

Fraction arithmetic:

A) Adding and subtracting fractions (e.g., a/b + c/d, a/b - c/d)
B) Multiplying fractions (a/b * c/d)
C) Multiplying fractions using distributive property (e.g., (1/5 + 1/4)*20

Answer 21

A) Adding and subtracting fractions: Need to find COMMON denominator

(ad + cb)/bd
(ad - cb)/bd

Concept: DO NOT just add numerators and denominators

B) Multiplying fractions –> multiply numerator and denominator

a/b * c/d = ac/bd

C) Distributive property still applies to fractions (treat as ONE TERM)

e.g., (A + B)*20 = 20A + 20B
therefore, 20/5 + 20/4
= 4 + 5
= 9

Question 22

Repeating decimals:
What is x equal if 0.368368368 = 368x

Answer 22

When dealing with repeating decimals and trying to understand division or multiplication, try FACTORING and RE-EXPRESSING as sums
> don’t get scared by the number of decimal places

0.368368368 = 0.368 + 0.000368 + 0.000000368
= 368(0.001) + 368(0.000001) + 368*(0.000000001)

Therefore:
368x = 368(0.001) + 368(0.000001) + 368*(0.000000001)
x = 0.001 + 0.000001 + 0.000000001
x = 0.001001001

Question 23

Re-expressing numbers

Answer 23

4) RE-EXPRESS NUMBERS as addition or subtraction
4A: when there are common factors or close enough numbers (1 or 2 apart)
e.g., 30(799) + 15(799) + 54(799) + 798 —-> You can factor out 799 from ALMOST all the terms
= 30(799) + 15(799) + 54(799) + 799 - 1
= 799(99 + 1) - 1
= 799(100) - 1
= 79899 —-> notice this pattern (will be something easy to multiply usually)

e.g., 8180 - 80 + 1 ——> 80 shows up a lot
= 80(81 - 1) + 1
= 80^2 + 1
–> rewrite numbers as addition or subtraction with a power of 10
–> includes repeating decimals

4B) When adding large numbers and don’t want to deal with places –> convert to 1000 (or multiple of 10) +/- number
e.g., 999 + 578 = (1000 - 1) + 578 = 1577

4C) When subtracting large numbers and don’t want to deal with places from 1000 or multiple of 10 –> convert to 999…. + 1 since it is easier to subtract from 999 vs 1000+ (to avoid dealing with place values)
e.g., 1,000,000,000 - 123,456,789
= (999,999,999 + 1) - 123,456,789

4D) Repeating decimals or decimals with patterns –> factor out constant and leave in the decimal with 0s and 1s
e.g., 5.0005/9.0009 = 5(1.0001)/9(1.0001) = 5/9
e.g., 0.368368368 = 368(0.001) + 368(0.000001) + 368(0.000000001)

4E) When multiplying large numbers, try to create multiplications that equal 100s
e.g., 452425124125
= (425) * (5124) * (24125)
= (425) * (5262) * (8125*3)

Question 24

Comparing decimals AND fractions

e.g., 9/20 vs 0.47

Answer 24

Either convert all numbers into fractions or decimals

Converting decimals to fractions –> ALWAYS possible as long it is terminating decimal
> denominator = power of 10, then simplify
> *easier to do

Converting fraction to decimal –> divide (but may be more challenging)

Question 25

Exponent rules

(ab)^2 / a

Answer 25

Remember you can REWRITE bases and then use division rules when we have the same bases

(ab)^2 = a^2 * b^2

e.g., (200)^4 / 20
= (20*10)^4 / 20
= (20^4 * 10^4) / 20
= 20^3 * 10^4

Question 26

Q: A rectangle field with width x is marked with paint into 4ths and 7ths. In terms of x, what is the largest difference between any two consecutive 4th and 7th marks on this side of the field? Assume there are not painted marks at the beginning or end of the field

Answer 26

Concept: Any time you are asked to split something into -ths, think about FRACTIONS and a NUMBER LINE
> draw a picture and compare -ths to plot number line
> understand which “ticks” you are trying to compare

End up with:
0, 1/7, 1/4, 2/7, 3/7, 2/4, 4/7, 5/7, 3/4, 6/7, 2

To find the largest difference between consecutive 4th and 7th “ticks”, we need to COMPARE the fractions easily by converting to SAME denom (28)

Numerator of the ticks of 7ths (denom 28): 4, 8, 12, 16, 20, 24 (excluding last tick)

Numerator of the ticks of 4ths (denom 28): 7, 14, 21 (excluding the last tick)

Therefore the largest difference between 7th and 4th ticks is 3x/28

> however, IGNORE the LAST TICK (not considered a consecutive 4th and 7th mark because both share the last tick)

Question 27

Consecutive integers - how do you find the SUM of a set of consecutive integers?

Answer 27

Option #1) Arithmetic formula for sum
Sum from a1 to an = (average * # of terms)
—> equally spaced sequences
= (a1 + an)/2 * n

Since the average of a set of consecutive integers = median, formula simplifies to:
Median * # of terms

Option #2) Use average formula to rearrange and solve for SUM
> since average = median for set of consecutive integers…

Avg = SUM / N
Median = SUM / N

Question 28

Tip for trying to figure out the decimal value of an improper fraction (once you’ve simplified as much as you can)

e.g., 325/7

Answer 28

So… you get to the point in the question where you CANNOT simplify more …

> convert your improper fraction into a MIXED FRACTION (whole number out front is helpful and then what’s left is still expressed as a fraction)

e.g., 325/7 = 46 + 3/7

(3/7 is a little bit < 0.5)

Question 29

Which operations HAVE the commutative property? Which operations DO NOT have the commutative property?

Answer 29

HAVE commutative property: Addition and multiplication
x + y = y + x (order of operations does not matter)
xy = yx

DO NOT have commutative property: Subtraction and division
x - y =/ y - x
x/y =/ y/x

Question 30

Which operations HAVE the associative property? Which operations DO NOT have the associative property?

Answer 30

HAVE associative property: Addition and multiplication
(x+y)+z = x+(y+z) (grouping of operations does not matter)
(xy)z = x(yz)

DO NOT have associative property: Subtraction and division
(x-y)-z =/ x-(y-z)
(x/y)/z =/ x/(y/z)

Question 31

What is “base 10 notation”?

Answer 31

Equal to “decimal notation” –> refers to expressing numbers using 10 digits (1,2,3,4,5,6,7,8,9,0)

IN CONTRAST to binary numeral system (base 2 number system)