Grade 11 Simple Interest Comprehensive Study Guide

Fundamental Concepts of Interest

• Definition of Interest: Interest is an amount that a person gets or pays on top of the original investment or loan. From a borrower's perspective, it is the money paid for the use of money. It can also function as a mechanism for imposing a penalty on a borrower for failing to pay a matured financial obligation at a specific time.

• Parties Involved in Interest Transactions:

  • Lender or Creditor: Refers to the party lending money or extending credit. This party expects the money to earn income from the transaction. For example, in a bank deposit, the depositor is considered the creditor.
  • Borrower or Debtor: Refers to the party using the money or credit. This party expects future expenses at the cost of using the capital. For example, in a bank deposit, the bank is considered the debtor because it is obliged to pay interest to the depositor.

Core Elements of Interest Calculation

Interest is computed using three primary elements:

• Principal ($P$): The amount of money extended for credit or the amount of money deposited in a bank for safekeeping.
• Interest Rate ($r$): The charged amount for using the money over a certain period. It is commonly expressed as a percentage per year ($\% \text{ per year}$) but must be converted to decimal form for calculations. Unless specified otherwise, the rate is assumed to be annual.
• Time ($t$): The period covered from the moment the principal is borrowed until its due date. The reference point for time is typically $1 \text{ year}$ or $12 \text{ months}$.
• Maturity Date: The specific due date for the payment of the principal amount.

Simple Interest Theory and Formulas

• Simple Interest: Refers to interest that is computed only on the original principal during the entire period or duration of borrowing.

• Simple Interest Formula:     $I = Prt$     Where:

  • $I$ = Interest amount
  • $P$ = Principal amount
  • $r$ = Simple interest rate (in decimal)
  • $t$ = Time (written in years)

• Maturity Value (Future Value): This is the sum of the principal and the interest accumulated. It represents the total amount to be paid or received at the end of the term.

• Maturity Value Formulas:

  • $M = P + I$
  • $M = P(1 + rt)$

• Derivation of Maturity Value:

  1. Start with the definition: $M = P + I$
  2. Substitute the interest formula ($I = Prt$): $M = P + (Prt)$
  3. Apply the distributive property (factor out $P$): $M = P(1 + rt)$

Step-by-Step Calculation Examples

Example 1: Basic Simple Interest and Maturity Value

Scenario: On April 1, 2017, Angela borrowed $P200\,000$ from Prime Lending at $2\%$ interest payable in $1 \text{ year}$.

Calculation for Interest:

• $P = 200\,000$
• $r = 0.02$
• $t = 1$
• $I = 200\,000 \times 0.02 \times 1$
• $I = 4\,000$
• Observation: Angela must pay an additional $P4\,000$ as interest.

Calculation for Maturity Value:

• $M = P + I$
• $M = 200\,000 + 4\,000$
• $M = 204\,000$
• Observation: Angela must pay a total of $P204\,000$ after one year.

Example 2: Non-Integer Time Periods

Scenario: Russel borrowed $P15\,000$ payable after $2 \text{ years and } 9 \text{ months}$ with a simple interest rate of $8\%$.

Step 1: Convert Time to Years Interest is stated as a yearly rate, so time must be converted:

• $2 \text{ years and } 9 \text{ months} = 2 \frac{9}{12} \text{ years}$
• $2 \frac{3}{4} \text{ years} = 2.75 \text{ years}$

Step 2: Find Maturity Value

• $P = 15\,000$
• $r = 0.08$
• $t = 2.75$
• $M = 15\,000[1 + (0.08)(2.75)]$
• $M = 15\,000(1 + 0.22)$
• $M = 15\,000(1.22)$
• $M = 18\,300$
• Conclusion: Russel needs to repay $P18\,300$.

Example 3: Comparison of Investment Portfolios

Scenario: JM has $P45\,000$ and must choose between two annual-rate options:

• Option A: $5\%$ rate for a $2\text{-year}$ portfolio.
• Option B: $4\%$ rate for a $21\text{-month}$ portfolio.

Example 4: Business Financing

Scenario: Michael borrowed $P450\,000$ at $3\%$ simple interest. How much will he pay after $5 \text{ years}$?

• $P = 450\,000$
• $r = 0.03$
• $t = 5$
• $M = 450\,000[1 + (0.03)(5)]$
• $M = 450\,000(1.15)$
• $M = 517\,500$

Practice Problems

1. Maya: Deposited $P18\,000$ at $4\%$ interest for $3 \text{ years}$.

   • $I = 18\,000 \times 0.04 \times 3 = P2\,160$

2. Anthony: Paid deep surcharge of $P10\,500$ on a $P70\,000$ loan after $5 \text{ years}$. Seek the interest rate.

   • $10\,500 = 70\,000 \times r \times 5$
   • $10\,500 = 350\,000r$
   • $r = 0.03 \text{ or } 3\%$

3. Sam: Invested $P200\,000$ at $8\%$ annual rate. Total money after $6 \text{ months}$ ($0.5 \text{ years}$)?

   • $M = 200\,000[1 + (0.08)(0.5)] = 200\,000(1.04) = P208\,000$

4. Annie: Wants to earn $P8\,000$ at $6.4\%$ yearly over $2 \text{ years}$. Required deposit?

   • $8\,000 = P \times 0.064 \times 2$
   • $8\,000 = 0.128P$
   • $P = P62\,500$

5. Nina: Earned $P4\,500$ from $P75\,000$ over $18 \text{ months}$ ($1.5 \text{ years}$). Interest rate?

   • $4\,500 = 75\,000 \times r \times 1.5$
   • $4\,500 = 112\,500r$
   • $r = 0.04 \text{ or } 4\%$

6. Pauline: Investment of $P24\,000$ grows to $P28\,500$ at $15\%$. Time needed?

   • $I = 28\,500 - 24\,000 = 4\,500$
   • $4\,500 = 24\,000 \times 0.15 \times t$
   • $4\,500 = 3\,600t$
   • $t = 1.25 \text{ years}$ (or $1 \text{ year and } 3 \text{ months}$)

7. Jasmine: Loan fee of $P312.50$ after $3 \text{ months}$ ($0.25 \text{ years}$) at $10\%$. Borrowed amount?

   • $312.50 = P \times 0.10 \times 0.25$
   • $312.50 = 0.025P$
   • $P = P12\,500$