MIT60001 Problem Set 1 Solution

This problem set will introduce you to using control flow in Python and formulating a computational solution to a problem. It will also give you a chance to explore bisection search. This problem set has
three ps1b.pyproblems. You should save your code for the first problem as and the third problem as ps1c.pynd your code! , and make sure to hand in all t ps1a.pyhree, t fiheles. se Dcondon' pt forroblegem at tos include comments to help us understa
Collaboration
Before You Start: Read the Style Guide

Read the style guide sections 1, 2, and 3.
Part A: House Hunting

You have graduated from MIT and now have a great job! You move to the San Francisco Bay Area and decide that you want to start saving to buy a house. As housing prices are very high in the Bay Area, you realize you are going to have to save for several years before you can afford to make the down payment on a house. In Part A, we are going to determine how long it will take you to save enough money to make the down payment given the following assumptions:

1. Call the cost of your dream home total_cost.
2. Call the portion of the cost needed for a down payment portion_down_payment . F or simplicity, assume that portion_down_payment = 0.25 (25%).
3. Call the amount that you have saved thus far current_savings. You st art with a current savings of $0.
4. at the end of each month, you rAssume that you invest your current savings wieceive an additionasely,l with an acurrent_savings*r/12nnual return of funds to p r ( in other wut intords,o your savings – the 12 is because r is an a nnual rate) . Assume that your investments earn a return of r = 0.04 (4%).
5. Assume your annual salary is annual_salary.
6. Assume you are going to dedicate a certain amount of your salary each month to saving for the down payment. Call that portion_saved. This va riable should be in decimal form (i.e. 0.1 for 10%).
7. At the end of each month, your savings will be increased by the return on your investment, plus a percentage of your monthly salary (annual salary / 12).

Write a program to calculate how many months it will take you to save up enough money for a down payment. You will want your main variables to be floats, so you should cast user inputs to floats. Your program should ask the user to enter the following variables:
1. The starting annual salary (annual_salary)
2. The portion of salary to be saved (portion_saved)
3. The cost of your dream home (total_cost)

Hints

To help you get started, here is a rough outline of the stages you should probably follow in writing your code:
● Retrieve user input. Look at input() if you need help with getting user input. For this problem set, you can assume that users will enter valid input (e.g. they won’t enter a string when you expect an int)
● Initialize some state variables. You should decide what information you need. Be careful about values that represent annual amounts and those that represent monthly amounts.
Try different inputs and see how long it takes to save for a down paprogram print results in the format shown in the test cases beymentlow. . Please make your

Test Case 1
>>>
Enter your annual salary: 12000 0
Enter the percent of your salary to save, as a decimal: .10
Enter the cost of your dream home: 1 000000 Number of months: 183
>>>

Test Case 2
>>>
Enter your annual salary: 80000
Enter the percent of your salary to save, as a decimal: .15 Enter the cost of your dream home: 5 00000
Number of months: 105
>>>

Part B: Saving, with a raise

Background

In Part A, we unrealistically assumed that your salary didn’t change. But you are an MIT graduate, and clearly you are going to be worth more to your company over time! So we are going to build on your solution to Part A by factoring in a raise every six months.

your program to include the folIn ps1b.py, copy your solution to Part A ( lowing as we are going to reuse much of that machinery). Modify
1. Have the user input a semith -annual salary raise semi_annual_raise ( as a decimal percentage)th
2. month, the 18After the 6 month, incr th month, and so on. ease your salary by that percentage. Do the same after the 12

Write a program to calculate how many months it will take you save up enough money for a down
required down payment percenpayment. LIke before, assume that your investments earn a rtage is 0.25 (or 25%). Have the user eeturn of nter = 0 r t.04he fol (orlowing 4%) a vndariables: the
1. The starting annual salary (annual_salary)
2. The percentage of salary to be saved (portion_saved)
3. The cost of your dream home (total_cost)
4. The semi­annual salary raise (semi_annual_raise)
Hints

To help you get started, here is a rough outline of the stages you should probably follow in writing your code:
● Retrieve user input.
● Initialize some state variables. You should decide what information you need. Be sure to be careful about values that represent annual amounts and those that represent monthly amounts.
● Be careful about when you increase your salary – this should only happen after t he 6th, 1 2th, 1 8th month, and so on.
Try different inputs and see how quickly or slowly you can save enough for a down payment. Please make your program print results in the format shown in the test cases below.

Test Case 1
>>>
Enter your starting annual salary: 1 20000
Enter the percent of your salary to save, as a decimal: .05
Enter the cost of your dream home: 500000
Enter the semi­annual raise, as a decimal: . 03
Number of months: 142
>>>

Test Case 2
>>>
Enter your starting annual salary: 8 0000
Enter the percent of your salary to save, as a decimal: .1
Enter the cost of your dream home: 800000
Enter the semi­annual raise, as a decimal: . 03
Number of months: 159
>>>

Test Case 3
>>>
Enter your starting annual salary: 7 5000
Enter the percent of your salary to save, as a decimal: .05
Enter the cost of your dream home: 1 500000
Enter the semi­annual raise, as a decimal: . 05
Number of months: 261
>>>


Part C: Finding the right amount to save away

In Part B, you had a chance to explore how both the percentage of your salary that you save each month and your annual raise affect how long it takes you to save for a down payment. This is nice, but suppose you want to set a particular goal, e.g. to be able to afford the down payment in three years. How much should you save each month to achieve this? In this problem, you are going to write a program to answer that question. To simplify things, assume:
1. Your semi­annual raise is .07 (7%)
2. Your investments have an annual return of 0.04 (4%)
3. The down payment is 0.25 (25%) of the cost of the house 4. The cost of the house that you are saving for is $1M.

You are now going to try to find the best rate of savings to achieve a down payment on a $1M house in
36 months. Since hitting this exactly is a challenge, we simply want your savings to be within $100 of the required down payment.

In ps1c.py , write a program to ca bisection sear chrch to fi to help you do this effi lculate the best savings rnish. You should be able to rciently.ate, as a function of y You shouleuse some of td keep tourrackhe st cod of tartingehe you w sa numblary.rotee r of
You should use
steps it takes your bisections sea for part B in this problem.

going to worry about the difference between 7.041% and 7.039%). This means we can search for an
integer( between 0 and 10000 (using i nteger division), and then convert it to a d a fter 3ecima6 montl phs.ercentage By using using float division) to use when we are calculating the current_savings
this range, there are only a finite number of numbers that we are searching over, as opposed to the infinite number of decimals between 0 and 1. This range will help prevent infinite loops. The reason we use 0 to 10000 is to account for two additional decimal places in the range 0% to 100%. Your code should print out a decimal (e.g. 0.0704 for 7.04%).


Hints

● Watch out for integer division when calculating if a percentage saved is appropriate and when calculating final decimal percentage savings rate.
● Remember to reset the appropriate variable(s) to their initial values for each iteration of bisection search.
Test Case 1
>>>
Enter the starting salary: 1500 00
Best savings rate: 0.4411
Steps in bisection search: 12
>>>

Test Case 2
>>>
Enter the starting salary: 3000 00
Best savings rate: 0.2206
Steps in bisection search: 9
>>>

Test Case 3
>>>
Enter the starting salary: 1000 0
It is not possible to pay the down payment in three years.
>>>



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