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Problem Set 1 Solutions by elizabethmcclaire MIT 6.0001 by johnh by Ofekhod Introduction by chenyi by Sherry
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Problem Set 1 Solutions

#1a:
annual_salary = float(input('Enter your annual salary: '))
portion_saved = float(input('Enter the percent of your salary to save, as a decimal:​ '))
total_cost = float(input('Enter the cost of your dream home: '))
current_savings = 0
portion_down_payment = 0.25
down_cost = total_cost * portion_down_payment 
r = 0.04
monthly_salary = annual_salary/12
num_months = 0

while current_savings < down_cost:
    current_savings += monthly_salary * portion_saved + current_savings * r/12
    num_months += 1

num_years = num_months/12  

print('Number of months:', num_months)
print('Number of years:',num_years)

#1b
annual_salary = float(input('Enter your annual salary: '))
portion_saved = float(input('Enter the percent of your salary to save, as a decimal:​ '))
total_cost = float(input('Enter the cost of your dream home: '))
semi_annual_raise = float(input('Enter the semi­annual raise, as a decimal: '))
current_savings = 0
portion_down_payment = 0.25
down_cost = total_cost * portion_down_payment 
r = 0.04
monthly_salary = annual_salary/12
num_months = 0

while current_savings < down_cost:
    current_savings += monthly_salary * portion_saved + current_savings * r/12
    num_months += 1
    if num_months%6 == 0:
        monthly_salary *= (1 + semi_annual_raise)

num_years = num_months/12   
    
print('Number of months:', num_months)
print('Number of years:',num_years)

#1c
def whatrate():
    annual_salary = int(input('Enter the starting salary: '))
    total_cost = 1000000
    semi_annual_raise = 0.07
    current_savings = 0
    portion_down_payment = 0.25
    down_cost = total_cost * portion_down_payment 
    r = 0.04
    monthly_salary = annual_salary/12
    num_months = 36
    high = 10000
    low = 0
    ans = (high + low)/2
    numofsteps = 0
    
    max_savings = 0
    for i in range(1,num_months+1):
        max_savings += monthly_salary + current_savings * r/12
        if i%6 == 0:
            monthly_salary *= (1 + semi_annual_raise)
    if max_savings < down_cost:
        return print('It is not possible to pay the down payment in three years.')

    min_savings = 0
    for i in range(1,num_months+1):
        min_savings += monthly_salary * 0.0001 + current_savings * r/12
        if i%6 == 0:
            monthly_salary *= (1 + semi_annual_raise)
    if min_savings > down_cost:
        return print('You don\'t really need to save, mate.')    
    
    while abs(current_savings - down_cost) > 100:
        current_savings = 0
        monthly_salary = annual_salary/12
        numofsteps += 1
        for i in range(1,num_months+1):
            current_savings += monthly_salary * ans/10000 + current_savings * r/12
            if i%6 == 0:
                monthly_salary *= (1 + semi_annual_raise)
        if current_savings > down_cost:
            high = ans
        else:
            low = ans
        ans = (high + low)/2
    print('Best savings rate:', round(ans/10000,4))
    print('Steps in bisection search:​', numofsteps)

MIT 6.0001

# Lecture 1: What is Computation?
   1.1 Topics of the courses
* iteration and recursion
  * organize and modularize
* ALGORITHMS!
* complexity of algorithms
   1.2 Construction of languages
* primitive constructs - numbers, strings, simple operators
* syntax - 3.2 * 5
* static semantics - syntactically valid strings have meaning; e.g. 3.2 * "hi" #static semantic error
* semantics - meaning associated with a syntactically correct string of symbols with no static semantic errors
        # syntactic + static semantic + semantic = all those errors come from
   1.3 Objects (or type)
      1.3.1 scalar object (can not be subdivided)
* int - represents integers, ex. 5
* float - represent real numbers, ex. 3.27
* bool - represent Boolean values True and False
* Nonetype - special and has one value, None
        # type() - see the type of an object
        # int() or float() - type conversions (cast)
      1.3.2 non-scalar (internal structure)
* list - [1, 2, 3]
   1.4 Expression <object><operator><object>
        # an expression has a value, which has a type
* i + j - the sum
* i - j - the difference
* i * j - the product
* i / j - the division #result is float
** i % j - the remainder when i is divided by j
** i ** j - i to the power of j
        # use parentheses () to tell Python to do these operations first
    1.5 Binding variables and values
* equal sign is an assignment of a value to a variable name
* pi = 3.14159; radius = 2.2; area = pi * (radius ** 2)
        # radius = radius +1 is equal to radius += 1

  
# Lecture 2: Branching and iteration
   2.1 string object type
      2.1.1 strings - letters, special characters, spaces, digits 
* hi = "hello there"; name = "qixiang"; - enclose in quotation marks or single quotes
* greeting = hi + " " + name - concatenate strings
* silly = hi + " " + name * 3 - operations on string
      2.1.2 INPUT/OUTPUT - print("") & input("")
* 2 way of printing sentences - print("My name is", x, "."); print("My name is " + x + " .")
      2.1.3 comparision operators on int, float, string (ascii table) 
* i == j - equality test
* i != j - inequality test
      2.1.4 logic operators on bools
* not a - True if a is False, False if a is True
* a and b - True if both are True
* a or b - True if either or both are True 
   2.2 branching and conditionals (CONTROL FLOW)
      2.2.1 if<condition> + <expression>
      2.2.2 elif<condition> + <expression>
      2.2.3 else: +<expression>
* <condition> has a value True or False
* evaluate expressions in the block if <condition> is True
   2.3 indentation - how you denote (indiate) block of code
* vs == and != 
   2.4 iteration and loops
      2.4.1 while loops
* while <condition>: <expression> 
   # <condition> evaluates to a Boolean; 
      if <condition> is True do <expression>;
      check <condition> again; 
      repeat until <condition> is False
* iteration through nums in a sequence - i = 0; while num < i: <expression>
      2.4.2 for loops
* for <variable> in range(start, stop, step)
   # loop until value is stop -1
      2.4.3 break STATEMENT
* exits only innermost loop!
      2.4.4 comparation between for and while LOOPS
* for LOOPS - 
   # known number of iterations;
      with an inner counter; 
      rewritiable through while LOOPS
* while LOOPS - 
   # unbounded number of iterations; 
      can use a counter but must initialize; 
      MAYNOT rewritiable through for LOOPS

# Lecture 3: String Manipulation, Guess and Check, Approximations, Bisection
   # last 3 are ALGORITHMS for guessing
   3.1 string manipulation
      3.1.1 definition and basic operation
* sequence of case sensitive characters
* compare operation - ==, >, < etc.
* length operation - len()
      3.1.2 indexing
* square brackets to index string
   # s = "abc" index: 0 1 2 or -3 -2 -1
      s[0] == s[-3] = "a"; s[1] == s[-2] = "b"; s[2] == s[-1] = "c"
      3.1.3 slicing
* slice strings using [start: stop: step]
   # s = "abcdefgh"
      s[3: 6: 2] = "df" 
* strings are "IMMUTABLE" - cannot be modified
   # change s = "hello" to s = "yello"
      s[0] = 'y' - ERROR!
      s = 'y' + s[1: len(s)]
      3.1.4 LOOPS RECAP (string loops)
* loop variable 
   - numbers - for var in range(4, 6): <expression>
   - indexing in string - for index in range(len(s)): <expression>
   - CHAR - for char in s: <expression>
      # more pythonic for char in word
* example - Robot cheerleader & Common letter
   3.2 guess and check algorithms (exhuasted enumeration)
* for guess in range(cube + 1) - test from 0 to cube
if guess ** 3 == cube:
print("Cube root of", cube, "is", guess)
   # guess a value for solution
   # check if the solution is correct
   # keep guessing and checking ...unitil find solution or guessed all values
   3.3 approximate solutions (find an approximate solution)
* Initial value
   - epsilon (how close to the target) and increment (how large each step is)
   - target and guess, and num_guesses
* while abs(guess ** 3 - cube) >= epsilon and guess <=cube:
guess += increment
num_guesses += 1
   # good enough approximation
   # start with a guess and increment by small value
   # keep guessing if abs(guess ** 3 - cube) >= epsilon, for some small epsilon
   # decreasing increment size - slower program
      increasing epsilon - less accurate answer
   3.4 bisection search method
* half interval each iteration and new guess is halfway in between
* guess CONVERGES on the order of log2N steps
* Initial low and high value - low = 0, high = guess, guess = (high + low) / 2.0
  While abs(guess ** 3 - cube) >= epsilon:
if guess ** 3 < cube:
low = guess  #re-define the low boundary
else:
high = guess  #re-define the high boundary
guess = (high + low) / 2.0
num_guesses +=1

Introduction

Lecture 1

Concepts

  • computer - calculations, storage, result memorization, - tell them what to do! 
  • 1. statement of facts  2. order/recipe/ the sequence of steps  
  • flow of control: making decisions 
  • 6 primitives --> types of result -- > instruction flow --> output 
  • expression: primitives + programming languages
  • expression - value - type 

[Revisit]

Programming

  • 35:00
  • give names, reuse names 
  • = assignment (left - variable) 


Remaining Question: 

        usa_gold = 46
        uk_gold = 27
        romania_gold = 1
        total_gold = usa_gold + uk_gold + romania_gold
        print(total_gold)
        romania_gold += 1

    total_gold = usa_gold + uk_gold + romania_gold
    print(total_gold)

My question:

Why do we get 75 for the second print? 


My answer: 

From John's notes:    

  # radius = radius +1 is equal to radius += 1

so, romania_gold + = 1 is equal to romania_gold = romania gold + 1 



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Lecture 1. What is Computation?

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Problem Set 1 Solutions

#1a:
annual_salary = float(input('Enter your annual salary: '))
portion_saved = float(input('Enter the percent of your salary to save, as a decimal:​ '))
total_cost = float(input('Enter the cost of your dream home: '))
current_savings = 0
portion_down_payment = 0.25
down_cost = total_cost * portion_down_payment 
r = 0.04
monthly_salary = annual_salary/12
num_months = 0

while current_savings < down_cost:
    current_savings += monthly_salary * portion_saved + current_savings * r/12
    num_months += 1

num_years = num_months/12  

print('Number of months:', num_months)
print('Number of years:',num_years)

#1b
annual_salary = float(input('Enter your annual salary: '))
portion_saved = float(input('Enter the percent of your salary to save, as a decimal:​ '))
total_cost = float(input('Enter the cost of your dream home: '))
semi_annual_raise = float(input('Enter the semi­annual raise, as a decimal: '))
current_savings = 0
portion_down_payment = 0.25
down_cost = total_cost * portion_down_payment 
r = 0.04
monthly_salary = annual_salary/12
num_months = 0

while current_savings < down_cost:
    current_savings += monthly_salary * portion_saved + current_savings * r/12
    num_months += 1
    if num_months%6 == 0:
        monthly_salary *= (1 + semi_annual_raise)

num_years = num_months/12   
    
print('Number of months:', num_months)
print('Number of years:',num_years)

#1c
def whatrate():
    annual_salary = int(input('Enter the starting salary: '))
    total_cost = 1000000
    semi_annual_raise = 0.07
    current_savings = 0
    portion_down_payment = 0.25
    down_cost = total_cost * portion_down_payment 
    r = 0.04
    monthly_salary = annual_salary/12
    num_months = 36
    high = 10000
    low = 0
    ans = (high + low)/2
    numofsteps = 0
    
    max_savings = 0
    for i in range(1,num_months+1):
        max_savings += monthly_salary + current_savings * r/12
        if i%6 == 0:
            monthly_salary *= (1 + semi_annual_raise)
    if max_savings < down_cost:
        return print('It is not possible to pay the down payment in three years.')

    min_savings = 0
    for i in range(1,num_months+1):
        min_savings += monthly_salary * 0.0001 + current_savings * r/12
        if i%6 == 0:
            monthly_salary *= (1 + semi_annual_raise)
    if min_savings > down_cost:
        return print('You don\'t really need to save, mate.')    
    
    while abs(current_savings - down_cost) > 100:
        current_savings = 0
        monthly_salary = annual_salary/12
        numofsteps += 1
        for i in range(1,num_months+1):
            current_savings += monthly_salary * ans/10000 + current_savings * r/12
            if i%6 == 0:
                monthly_salary *= (1 + semi_annual_raise)
        if current_savings > down_cost:
            high = ans
        else:
            low = ans
        ans = (high + low)/2
    print('Best savings rate:', round(ans/10000,4))
    print('Steps in bisection search:​', numofsteps)

MIT 6.0001

# Lecture 1: What is Computation?
   1.1 Topics of the courses
* iteration and recursion
  * organize and modularize
* ALGORITHMS!
* complexity of algorithms
   1.2 Construction of languages
* primitive constructs - numbers, strings, simple operators
* syntax - 3.2 * 5
* static semantics - syntactically valid strings have meaning; e.g. 3.2 * "hi" #static semantic error
* semantics - meaning associated with a syntactically correct string of symbols with no static semantic errors
        # syntactic + static semantic + semantic = all those errors come from
   1.3 Objects (or type)
      1.3.1 scalar object (can not be subdivided)
* int - represents integers, ex. 5
* float - represent real numbers, ex. 3.27
* bool - represent Boolean values True and False
* Nonetype - special and has one value, None
        # type() - see the type of an object
        # int() or float() - type conversions (cast)
      1.3.2 non-scalar (internal structure)
* list - [1, 2, 3]
   1.4 Expression <object><operator><object>
        # an expression has a value, which has a type
* i + j - the sum
* i - j - the difference
* i * j - the product
* i / j - the division #result is float
** i % j - the remainder when i is divided by j
** i ** j - i to the power of j
        # use parentheses () to tell Python to do these operations first
    1.5 Binding variables and values
* equal sign is an assignment of a value to a variable name
* pi = 3.14159; radius = 2.2; area = pi * (radius ** 2)
        # radius = radius +1 is equal to radius += 1

  
# Lecture 2: Branching and iteration
   2.1 string object type
      2.1.1 strings - letters, special characters, spaces, digits 
* hi = "hello there"; name = "qixiang"; - enclose in quotation marks or single quotes
* greeting = hi + " " + name - concatenate strings
* silly = hi + " " + name * 3 - operations on string
      2.1.2 INPUT/OUTPUT - print("") & input("")
* 2 way of printing sentences - print("My name is", x, "."); print("My name is " + x + " .")
      2.1.3 comparision operators on int, float, string (ascii table) 
* i == j - equality test
* i != j - inequality test
      2.1.4 logic operators on bools
* not a - True if a is False, False if a is True
* a and b - True if both are True
* a or b - True if either or both are True 
   2.2 branching and conditionals (CONTROL FLOW)
      2.2.1 if<condition> + <expression>
      2.2.2 elif<condition> + <expression>
      2.2.3 else: +<expression>
* <condition> has a value True or False
* evaluate expressions in the block if <condition> is True
   2.3 indentation - how you denote (indiate) block of code
* vs == and != 
   2.4 iteration and loops
      2.4.1 while loops
* while <condition>: <expression> 
   # <condition> evaluates to a Boolean; 
      if <condition> is True do <expression>;
      check <condition> again; 
      repeat until <condition> is False
* iteration through nums in a sequence - i = 0; while num < i: <expression>
      2.4.2 for loops
* for <variable> in range(start, stop, step)
   # loop until value is stop -1
      2.4.3 break STATEMENT
* exits only innermost loop!
      2.4.4 comparation between for and while LOOPS
* for LOOPS - 
   # known number of iterations;
      with an inner counter; 
      rewritiable through while LOOPS
* while LOOPS - 
   # unbounded number of iterations; 
      can use a counter but must initialize; 
      MAYNOT rewritiable through for LOOPS

# Lecture 3: String Manipulation, Guess and Check, Approximations, Bisection
   # last 3 are ALGORITHMS for guessing
   3.1 string manipulation
      3.1.1 definition and basic operation
* sequence of case sensitive characters
* compare operation - ==, >, < etc.
* length operation - len()
      3.1.2 indexing
* square brackets to index string
   # s = "abc" index: 0 1 2 or -3 -2 -1
      s[0] == s[-3] = "a"; s[1] == s[-2] = "b"; s[2] == s[-1] = "c"
      3.1.3 slicing
* slice strings using [start: stop: step]
   # s = "abcdefgh"
      s[3: 6: 2] = "df" 
* strings are "IMMUTABLE" - cannot be modified
   # change s = "hello" to s = "yello"
      s[0] = 'y' - ERROR!
      s = 'y' + s[1: len(s)]
      3.1.4 LOOPS RECAP (string loops)
* loop variable 
   - numbers - for var in range(4, 6): <expression>
   - indexing in string - for index in range(len(s)): <expression>
   - CHAR - for char in s: <expression>
      # more pythonic for char in word
* example - Robot cheerleader & Common letter
   3.2 guess and check algorithms (exhuasted enumeration)
* for guess in range(cube + 1) - test from 0 to cube
if guess ** 3 == cube:
print("Cube root of", cube, "is", guess)
   # guess a value for solution
   # check if the solution is correct
   # keep guessing and checking ...unitil find solution or guessed all values
   3.3 approximate solutions (find an approximate solution)
* Initial value
   - epsilon (how close to the target) and increment (how large each step is)
   - target and guess, and num_guesses
* while abs(guess ** 3 - cube) >= epsilon and guess <=cube:
guess += increment
num_guesses += 1
   # good enough approximation
   # start with a guess and increment by small value
   # keep guessing if abs(guess ** 3 - cube) >= epsilon, for some small epsilon
   # decreasing increment size - slower program
      increasing epsilon - less accurate answer
   3.4 bisection search method
* half interval each iteration and new guess is halfway in between
* guess CONVERGES on the order of log2N steps
* Initial low and high value - low = 0, high = guess, guess = (high + low) / 2.0
  While abs(guess ** 3 - cube) >= epsilon:
if guess ** 3 < cube:
low = guess  #re-define the low boundary
else:
high = guess  #re-define the high boundary
guess = (high + low) / 2.0
num_guesses +=1

Introduction

Lecture 1

Concepts

  • computer - calculations, storage, result memorization, - tell them what to do! 
  • 1. statement of facts  2. order/recipe/ the sequence of steps  
  • flow of control: making decisions 
  • 6 primitives --> types of result -- > instruction flow --> output 
  • expression: primitives + programming languages
  • expression - value - type 

[Revisit]

Programming

  • 35:00
  • give names, reuse names 
  • = assignment (left - variable) 


Remaining Question: 

        usa_gold = 46
        uk_gold = 27
        romania_gold = 1
        total_gold = usa_gold + uk_gold + romania_gold
        print(total_gold)
        romania_gold += 1

    total_gold = usa_gold + uk_gold + romania_gold
    print(total_gold)

My question:

Why do we get 75 for the second print? 


My answer: 

From John's notes:    

  # radius = radius +1 is equal to radius += 1

so, romania_gold + = 1 is equal to romania_gold = romania gold + 1 



AstraPath Resources+

  • Download Python (https://www.anaconda.com/products/individual#Downloads)
  • Textbook Amazon link (https://http://www.amazon.com/exec/obidos/ASIN/0262529629/ref=nosim/mitopencourse-20)
  • Lecture slides and code (https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-0001-introduction-to-computer-science-and-programming-in-python-fall-2016/lecture-slides-code/)
  • Problem Set 0 (https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-0001-introduction-to-computer-science-and-programming-in-python-fall-2016/assignments/ps0.zip)
  • Problem Set 1 (https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-0001-introduction-to-computer-science-and-programming-in-python-fall-2016/assignments/MIT6_0001F16_ps1.pdf)
  • Problem Set 2 (https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-0001-introduction-to-computer-science-and-programming-in-python-fall-2016/assignments/ps2.zip)
  • Problem Set 3 (https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-0001-introduction-to-computer-science-and-programming-in-python-fall-2016/assignments/PS3.zip)
  • Problem Set 4 (https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-0001-introduction-to-computer-science-and-programming-in-python-fall-2016/assignments/ps4.zip)
  • Problem Set 5 (https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-0001-introduction-to-computer-science-and-programming-in-python-fall-2016/assignments/pset5.zip)
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