Problem Solving with Financial Contexts

TL;DR

You'll learn to use algebra to solve real-world money problems, focusing on budgeting, interest, and profit. Breaking down these problems helps you turn words into equations you can solve. Understanding the basic math concepts behind finance makes managing money much clearer.

1. The Mental Model

Think of financial problems as puzzles where the pieces are numbers and the rules are math operations. Your job is to translate the story into an algebraic equation, find the missing piece, and then check if your answer makes sense in the money world.

2. The Core Material

When you're faced with a financial problem, you're usually trying to find an unknown value related to money. This could be how much money you'll have, how much something costs, or how long it will take to save up.

Understanding Key Terms

Photo by RDNE Stock project on Pexels

Before diving into problems, let's nail down some common financial terms you'll encounter:

• Principal (P): The initial amount of money deposited or borrowed.
• Interest (I): The extra money earned on an investment or paid on a loan.
• Interest Rate (r): The percentage charged or earned on the principal over a specific period, usually per year. Always convert percentages to decimals for calculations (e.g., 5% = 0.05).
• Time (t): The duration, in years, for which the money is invested or borrowed.
• Amount (A): The total money after interest is added (Principal + Interest).
• Income: Money coming in.
• Expenses: Money going out.
• Profit: Income minus expenses.

Simple Interest

Photo by https://kaboompics.com/ on Pexels

Simple interest is calculated only on the initial principal amount. It's a straightforward way to calculate interest.

The formula for simple interest is:
I = P * r * t

And the total amount (A) you'll have after earning simple interest is:
A = P + I or A = P * (1 + r * t)

Budgeting and Balances

Photo by Kindel Media on Pexels

Budgeting involves tracking income and expenses to see where your money goes. A common problem might ask you to find a remaining balance or how much you can afford to spend.

Balance = Total Income - Total Expenses

Profit and Loss

Photo by Nataliya Vaitkevich on Pexels

Businesses care about profit. It's simply the money left over after all costs are paid.

Profit = Revenue (or Sales) - Costs

Or, if costs are higher than revenue, you have a loss.

graph TD
    Start["Problem Statement"] --> Identify_Knowns["Identify Known Values (P, r, t, Income, Expenses, etc.)"]
    Identify_Knowns --> Identify_Unknown["Identify What You Need To Find (I, A, Balance, Profit)"]
    Identify_Unknown --> Choose_Formula["Choose the Correct Formula (Simple Interest, Budget, Profit)"]
    Choose_Formula --> Substitute_Values["Substitute Known Values into the Formula"]
    Substitute_Values --> Solve_Equation["Solve the Algebraic Equation"]
    Solve_Equation --> Check_Answer["Check if Answer Makes Sense in Context"]
    Check_Answer --> End["Solution"]

Setting up Equations

The trickiest part is often translating the problem into an equation. Look for keywords:

• "Total," "sum," "combined" usually mean addition.
• "Difference," "less than," "remain" usually mean subtraction.
• "Of," "per," "each" often imply multiplication.
• "Per," "divided equally" often imply division.

Example: "You invest $1000 at a 3% simple interest rate for 2 years. How much interest will you earn?"

• Knowns: P = $1000, r = 0.03, t = 2 years.
• Unknown: I.
• Formula: I = P * r * t
• Equation: I = 1000 * 0.03 * 2

3. Worked Example

Let's say you're saving for a new laptop that costs $850. You currently have $300 saved. You can save an additional $70 per week from your part-time job. How many weeks will it take you to save enough money for the laptop?

1. Identify Knowns:

   • Target Cost = $850
   • Current Savings = $300
   • Weekly Savings = $70

2. Identify Unknown:

   • Number of weeks (let's call it w)

3. Set up the Equation:
   Your current savings plus your weekly savings multiplied by the number of weeks must equal the target cost.

   Current Savings + (Weekly Savings * Number of Weeks) = Target Cost
   $300 + ($70 * w) = $850

4. Solve the Equation:

   • Subtract $300 from both sides:
     $70 * w = $850 - $300
     $70 * w = $550
   • Divide both sides by $70:
     w = $550 / $70
     w ≈ 7.857

5. Check the Answer:
   Since you can only save in whole weeks, you'd need to save for 8 full weeks to have enough money. In 7 weeks, you'd have $300 + (7 * $70) = $300 + $490 = $790, which isn't enough. In 8 weeks, you'd have $300 + (8 * $70) = $300 + $560 = $860, which is enough.

So, it will take you 8 weeks to save enough for the laptop.

4. Key Takeaways

• Always identify the known values and what you need to find in a financial problem.
• Convert percentages to decimals (e.g., 5% to 0.05) before using them in formulas.
• Simple interest is calculated only on the initial principal: I = P * r * t.
• Budgeting helps you track money: Balance = Income - Expenses.
• Profit is what's left after costs: Profit = Revenue - Costs.
• Break down word problems into smaller, manageable parts to build your equation.

Common Mistakes to Avoid:

• Forgetting to convert interest rates from percentages to decimals.
• Mixing up annual rates with monthly or weekly periods without adjusting.
• Not answering the specific question asked (e.g., finding interest when the total amount was requested).
• Ignoring units (like years, months, or weeks) and ensuring they're consistent.

5. Now Try It

You want to buy a new smartphone for $600. You have $150 saved already. Your parents said they would give you