Q1) Basic math (fractions, decimals, converting, squares and square roots, algebra) Flashcards
What is the difference between PROPER and IMPROPER fractions?
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
When would converting to mixed fractions be useful?
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
Two unknowns, linear equation with multiplication, is it solvable?
e.g., 3c + 3a = ac
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
How can you tell two numbers are reciprocals?
Product of the two numbers is 1
Two ways to simplify complex fractions
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)
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
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
What are strategies for comparing fractions?
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
Multiplying and Dividing decimals
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
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?
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
How do you convert terminating decimals in fractions?
e.g., 3.4
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
** Memorize the following base fractions and their decimal approximations:
1/6 **
1/7 **
1/8
1/9
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
What is the square root of 4?
2
> not -2
Concept: Principal square root of a number is its NON NEGATIVE SQUARE ROOT
Sqrt ( positive known number ) = positive number
What is the value of x?
Sqrt(x) = 5
25
Concept: Sqrt ( positive known number ) = positive number
If 0 < x < 1, what must be true of x, x^2 and sqrt(x)?
x^2 < x < sqrt(x)
Most helpful when dealing with UNKNOWNS or DECIMALS that cannot be converted to fractional form easily
What is the largest if the equation is (Decimal)^2?
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
Strategies when dealing with DECIMALS
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
What are the only unit digits that a perfect square can have?
0
1
4
5
6
9
Perfect squares NEVER end in 2,3,7, or 8
Distributive property and FACTORING
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
Ways to simplify expressions
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
0! and 1!
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!
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
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
Repeating decimals:
What is x equal if 0.368368368 = 368x
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
Re-expressing numbers
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)
Comparing decimals AND fractions
e.g., 9/20 vs 0.47
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)
