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A short class with Sarah and her students. The other ways are under More.
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Today, we are going to explore compound interest. Can anyone tell me how compound interest differs from simple interest?
I think compound interest is calculated on the initial principal and also on the interest accumulated from previous periods?
Exactly! Compound interest builds upon itself, whereas simple interest is only calculated on the original principal. Let's visualize how that works.
Could you give us an example?
Of course! If you invest 100 dollars at an interest rate of 10% compounded annually, the amount after the first year is 110 dollars. But in the second year, you earn interest on the 110 dollars, not just the original 100 dollars.
So in the second year, we earn more than 10 dollars?
Correct! That's the beauty of compounding. Let's break down how we can derive a formula for calculating it.
To summarize, compound interest differs from simple interest as it accumulates on both the principal and accrued interest.
Now, let's derive the formula for compound interest. Suppose we have a principal amount P and a rate R% for n years. What should be our first step?
We calculate the interest for the first year, right?
Exactly! The interest earned in the first year is simply P × R/100. The total amount at the end of the first year is then A1 = P + (P × R/100). Let's write that down.
How do we continue for the second year?
Great question! The principal amount for the second year is now A1, which leads us to A2 = A1 + (A1 × R/100). Substituting A1 gives us the formula.
So the formula for the amount A can be written as A = P(1 + R/100)^n?
Right! This formula will help us calculate the total amount after n years.
To recap, the total amount after n years is derived from the expression A = P (1 + R/100)^n.
Next, let's apply the formula to calculate compound interest. Suppose P = 10000, R = 5%, and n = 2. Can anyone calculate the amount?
So A = 10000 × (1 + 5/100)^2. That means A = 10000 × (1.05)^2. I'll work it out.
That equals 10000 × 1.1025, which is about 11025.
Correct! And what is the compound interest then?
The compound interest is the total amount minus the principal, so 11025 - 10000 equals 1025.
Excellent! This method can be applied to any amount, rate, and time. Always remember to subtract the principal to find the compound interest.
To summarize, we apply the formula A = P(1 + R/100)^n to find the total amount, and CI = A - P gives us the compound interest.
For interest that compounds over time, remember the formula, it's not a crime!
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!
P-A-R: Principal Amount, Rate, Amount.
C.I. = Compound Interest: Calculate Including previous interest.
Term
What does CI stand for?
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Term
What is P in the formula A = P(1 + R/100)^n?
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The initial sum of money on which interest is calculated.
Interest calculated on the accumulated amount, including both the principal and previously earned interest.
The percentage at which interest is calculated, typically expressed annually.
The total amount of money accumulated after n years, including interest.
The number of years for which interest is compounded.
This section outlines a method for deducing a formula to calculate compound interest.
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.
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.