Question

interest rate amount = initial_amount years = 0 while amount <= 1.5*init… # Consider the following program for computing with interest rates: initial_amount = 100 p = 5.5 # interest rate amount = initial_amount years = 0 while amount <= 1.5*initial_amount: amount = amount + p/100*amount years = years + 1 print years (a)Explain with words what type of mathematical problem that is solved by this program. Compare this computerized solution with the technique your high school math teacher would prefer. (b)Use a pocket calculator (or use an interactive Python shell as substitute) and work through the program by hand. Write down the value of amount and years in each pass of the loop. (c) Change the value of \(p\) to 5 . Why will the loop now run forever? (See Exercise \(2.12\) for how to stop the program if you try to run it.) Make the program more robust against such errors. (d)Make use of the operator += wherever possible in the program. Insert the text for the answers to (a) and (b) in a multi-line string in the program file. Name of program file: interest_rate_loop.py.

Short answer

The program computes the time to achieve 150% increase via compounding interest. It stops after 8 years. With a 5% interest, the loop runs forever unless adjusted.

Step-by-step solution

Understanding the Problem

The program calculates the number of years needed for an initial investment to grow by 50% with a given interest rate. This is an example of compound interest calculation, where the interest is added to the principal each year, increasing the total amount's possible future interest.

Simulating Program Execution Manually

Starting with `initial_amount = 100`, `p = 5.5`, and `years = 0`, calculate the new `amount` each loop iteration. After each calculation, add one to the `years`. Track these values: - **Year 1:** amount = 105.5, years = 1 - **Year 2:** amount = 111.3025, years = 2 - **Year 3:** amount = 117.4741375, years = 3 - **Year 4:** amount = 124.0352131, years = 4 - **Year 5:** amount = 131.0161458, years = 5 - **Year 6:** amount = 138.4479638, years = 6 - **Year 7:** amount = 146.3628003, years = 7 - **Year 8:** amount = 154.7938545, years = 8 The loop stops after year 8 because the amount exceeds 150.

Analyzing Robustness with Different Interest Rate

When changing `p` to 5, calculate if the loop will run indefinitely by seeing if each iteration ever exceeds `1.5 * initial_amount`. Since 5% doesn't allow `amount` to exceed 150 (it keeps approaching but never surpasses it), the loop will continue endlessly without further enhancement.

Enhancing the Program with Increment Operator

Use `+=` operator to simplify the code expression for `amount` and `years`. Replace `amount = amount + p/100*amount` with `amount += p/100*amount`, and `years = years + 1` with `years += 1`, improving code readability and efficiency.

Key concepts

Program Execution

When you execute a program, your computer follows a set of instructions written in a programming language to perform specific tasks. In our exercise, the Python program calculates how many years it will take for an initial amount of money to grow by 50% at a set interest rate.
Program execution manages the flow across lines of code, while ensuring logical correctness throughout the process. As the program runs through each line, variables are updated, loops are executed, and functions may be called. This sequential processing is crucial for reaching the desired output.
One way to understand this better is by thinking of the computer as a cook following a recipe. Each step in the recipe describes an action to be done, just like each line in a program tells the computer what to do. As steps are followed sequentially, reaching the end develops the result, similar to our program which stops when the condition (amount being more than 1.5 times the initial sum) is met.

Python Programming

Python is a popular programming language known for its simplicity and readability, making it an excellent choice for beginners and professionals alike. It is widely used in many areas such as web development, data analysis, artificial intelligence, and financial calculations like our compound interest example.
In the given exercise, Python’s straightforward syntax allows you to understand the logic without being bogged down by complex constructs common in other languages.
For instance, the use of variables like `initial_amount` and `p` make it clear that these are values to be used in calculations. Meanwhile, Python's handling of loops enables efficient repetitive task execution, which suits financial computations like compound interest.

• Python suits quick calculations through its interpretive nature, allowing changes and instantaneous results through an interactive shell.
• The language's simplicity is demonstrated in our loop structure with straightforward condition checks.
• Python's readability, including the use of `+=`, contributes to writing cleaner and more efficient code.

Loop Control

Loop control refers to the mechanism by which you manage the flow of repeated execution in a program. In Python, loops like the `while` loop used in our example are perfect for tasks where the number of iterations isn't determined beforehand.
In our compound interest calculation, the program uses a `while` loop to repeatedly add interest to the amount. The loop's condition (`amount <= 1.5*initial_amount`) dictates how long the loop will run. As `amount` increases with each iteration, `years` increments until the condition is no longer satisfied and the loop stops.
This is where loop control is vital:

• Loop conditions help avoid infinite loops by ensuring that a stopping point is reached.
• The loop's body contains the code that needs execution multiple times with slight changes per iteration.
• Effective loop control handles errors, like when changing the interest rate to a value that wouldn’t naturally end the loop, you may need to introduce additional stop conditions.

Interest Rate

Interest rate is key in finance, representing the cost of borrowing money or the reward for saving. With compound interest, the rate applies to both the initial principal and accumulated interest over previous periods.
In our program, the interest rate `p` modifies the total amount yearly by adding a percentage of the current amount. With `p = 5.5`, the balance grows reasonably and predictably.
Changing `p` to a lower value like 5 illustrates how different rates affect the principal's growth. If the rate is too low, it may take significantly longer or be impossible to reach a goal, such as 1.5 times the initial amount, demonstrating how critical the interest rate is in financial planning.

• It affects how quickly money grows or debts accumulate.
• The percentage expresses the amount added to the principal per period.
• Higher rates accelerate growth but can increase risks, while lower rates slow down accumulation.

Understanding the role of interest rates is crucial for both saving and investing, as it influences decision-making in finance.