Deducing a Formula for Compound Interest

7.5 Deducing a Formula for Compound Interest

Description

Quick Overview

This section outlines a method for deducing a formula to calculate compound interest.

Standard

The section explains how to derive a formula for compound interest using specific examples and mathematical deductions. It emphasizes understanding the principles behind the calculation of compound interest compared to simple interest.

Detailed

Deducing a Formula for Compound Interest

In this section, we delve into the process of deriving a formula for compound interest, highlighting the differences with simple interest. The discussion begins with Zubeda inquiring about an easier way to determine compound interest. The teacher introduces the concept of calculating compound interest on a sum compounded annually at a certain rate.

If we consider a principal sum (P) subjected to a rate of interest (R%), the derivation involves calculating the interest accrued each year. By observing that the amount to be paid at the end of the first year (A1) includes the principal and the interest from that year, we can express this as:

A1 = P + (P × R/100)

Following this, the amount at the end of the second year builds on the first year's amount:

A2 = A1 + (A1 × R/100)

This leads to the formulation which effectively compounds the interest based on the previous year's total (P)
:

A = P (1 + R/100)^n

The section presents practical examples showing how to compute the compound interest using this newly derived formula, reinforcing the relationship between the principal, rate, and the duration in years. Notably, it concludes by demonstrating how to calculate both the total amount to be paid and the compound interest after a specified duration.

Key Concepts

  • Compound interest builds on itself, unlike simple interest.

  • The formula for calculating compound interest is A = P(1 + R/100)^n.

  • CI is calculated by subtracting the principal from the total amount.

Memory Aids

🎵 Rhymes Time

  • For interest that compounds over time, remember the formula, it's not a crime!

📖 Fascinating Stories

  • Imagine investing $100 at 5%. After year one, you earn $5. In year two, you earn more than a dollar, as it's on the total including your prior dollar!

🧠 Other Memory Gems

  • P-A-R: Principal Amount, Rate, Amount.

🎯 Super Acronyms

C.I. = Compound Interest

  • Calculate Including previous interest.

Examples

  • Example: If the principal is 10000 with a rate of 5% for 2 years, the formula gives A = 10000 × (1 + 0.05)^2 = 11025; thus, CI = 11025 - 10000 = 1025.

  • Example: A principal of 5000 at a rate of 8% for 3 years results in A = 5000 × (1.08)^3 ≈ 6300, leading to CI = 6300 - 5000 = 1300.

Glossary of Terms

  • Term: Principal (P)

    Definition:

    The initial sum of money on which interest is calculated.

  • Term: Compound Interest (CI)

    Definition:

    Interest calculated on the accumulated amount, including both the principal and previously earned interest.

  • Term: Rate (R)

    Definition:

    The percentage at which interest is calculated, typically expressed annually.

  • Term: Amount (A)

    Definition:

    The total amount of money accumulated after n years, including interest.

  • Term: n years

    Definition:

    The number of years for which interest is compounded.