CTJan27 Online Year 7 - Compound Intrest

Questions for: What is Compound Interest?

Introduction: Simple vs. Compound Interest

Imagine you have some money saved. You can put this money in a bank, and the bank will pay you extra money for keeping it there. This extra money is called interest.

There are two main ways interest is calculated: Simple Interest and Compound Interest.

• Simple Interest: You earn interest ONLY on your initial amount of money. The amount of interest you earn each year is fixed.
• Compound Interest: This is more powerful! You earn interest on your initial amount, AND you also earn interest on the interest you've already been paid. It's like your interest starts earning its own interest.

Key Terms

To understand compound interest, you need to know these three terms:

1. Principal (P): This is the starting amount of money you invest or borrow. For example, if you put 1000 dollars in a bank account, your principal is 1000 dollars.
2. Interest Rate (R): This is the percentage of the principal that you earn as interest over a certain period (usually a year). It is often written as '% per annum' (p.a.), which means 'per year'. For calculations, we convert the percentage to a decimal (e.g., 10% = 0.10).
3. Time (T): This is the length of time the money is invested or borrowed for, usually in years.

How Does Compound Interest Work? An Example

Let's see how compound interest works step-by-step. Imagine you invest 1000 dollars (your Principal) in an account that pays 10% interest per year.

Year 1:

• You calculate the interest on the principal: 10% of 1000 dollars.
• Calculation: $1000 \times 0.10 = 100$ dollars.
• At the end of Year 1, you have your original principal plus the interest: $1000 + 100 = 1100$ dollars.

Year 2:

• Now, here's the magic of compounding! Instead of calculating interest on the original 1000 dollars, you calculate it on the new total of 1100 dollars.
• Calculation: 10% of 1100 dollars.
• $1100 \times 0.10 = 110$ dollars.
• At the end of Year 2, you have the total from Year 1 plus the new interest: $1100 + 110 = 1210$ dollars.

Comparison:

• With compound interest, you earned a total of $210$ dollars in two years ($100 + 110$).
• With simple interest, you would have only earned $100$ dollars each year on the original principal, for a total of $200$ dollars.

Compound interest helps your money grow faster because the amount your interest is calculated on increases each year.

Questions for: Calculating Interest Year-by-Year Manually

What is Compound Interest?

Imagine you put money in a savings account. The bank pays you interest, which is like a reward for saving with them.

With Simple Interest, the reward is always calculated based on the original amount you deposited (the principal).

With Compound Interest, the reward is calculated on your principal plus any interest you've already earned. This means your interest starts earning its own interest! This is why it's often called "interest on interest".

Key Terms

• Principal (P): The initial amount of money invested or borrowed.
• Rate (R): The percentage at which interest is charged, usually per year (per annum).
• Time (T): The duration for which the money is invested or borrowed, in years.
• Amount (A): The total money at the end of the period, which is the Principal + all the Interest earned.

How to Calculate Compound Interest Year-by-Year

We can calculate the final amount by finding the interest for each year one step at a time. The key is that the principal for the next year is the amount from the end of the previous year.

Example:
Calculate the compound interest on 1000 dollars for 2 years at a rate of 10% per annum.

Step 1: Calculate for the first year.

• Principal for Year 1: 1000 dollars
• Interest for Year 1: $10\% \text{ of } 1000 = \frac{10}{100} \times 1000 = 100$ dollars.
• Amount at the end of Year 1: Principal + Interest = $1000 + 100 = 1100$ dollars.

Step 2: Calculate for the second year.

• The amount from the end of Year 1 becomes the new principal for Year 2.
• Principal for Year 2: 1100 dollars
• Interest for Year 2: $10\% \text{ of } 1100 = \frac{10}{100} \times 1100 = 110$ dollars.
• Amount at the end of Year 2: Principal + Interest = $1100 + 110 = 1210$ dollars.

Step 3: Find the Total Compound Interest.

• Total Compound Interest = Final Amount - Original Principal
• Total Compound Interest = $1210 - 1000 = 210$ dollars.

Notice that the interest in the second year (110 dollars) was more than the first year (100 dollars) because you earned interest on the initial 1000 dollars and on the 100 dollars of interest from the first year.

Questions for: Introduction to the Compound Interest Formula

What is Compound Interest?

Imagine you put money in a savings account. The bank pays you 'interest', which is extra money as a thank you for keeping your money with them.

There are two main types of interest:

1. Simple Interest: You only earn interest on the initial amount you deposited. It's the same amount of interest every time.
2. Compound Interest: This is more powerful. You earn interest on your initial amount, and you also earn interest on the interest you've already accumulated. It's like your money starts earning its own money!

The Compound Interest Formula

To calculate the final amount of money you will have with compound interest, we use a special formula. It might look complicated at first, but each part has a specific meaning.

The formula is:

$A = P(1 + \frac{r}{n})^{nt}$

Understanding the Parts of the Formula

Let's break down each letter:

• A = Final Amount: This is the total amount of money you will have in the account at the end of the period. It includes your original money plus all the compound interest earned.

• P = Principal: This is the initial or starting amount of money. It's the money you first invest or deposit.

• r = Annual Interest Rate: This is the interest rate per year. CRITICAL: You must convert the percentage to a decimal before using it in the formula. For example, 5% becomes $0.05$, and 12% becomes $0.12$.

• n = Number of Compounding Periods per Year: This tells you how often the interest is calculated and added to your account each year.

  • Compounded annually: $n = 1$
  • Compounded semi-annually (twice a year): $n = 2$
  • Compounded quarterly (four times a year): $n = 4$
  • Compounded monthly: $n = 12$

• t = Time (in years): This is the total number of years the money is invested or borrowed for.

Example Calculation

Let's say you invest 1000 dollars for 2 years at an annual interest rate of 6%, and the interest is compounded semi-annually.

1. Identify the values:

   • $P = 1000$ (the initial amount)
   • $r = 6\% = 0.06$ (the rate as a decimal)
   • $n = 2$ (compounded semi-annually)
   • $t = 2$ (for two years)

2. Plug the values into the formula:

   • $A = 1000(1 + \frac{0.06}{2})^{2 \times 2}$

3. Solve the formula step-by-step:

   • First, the fraction: $0.06 \div 2 = 0.03$
   • Then, the bracket: $1 + 0.03 = 1.03$
   • Next, the exponent: $2 \times 2 = 4$
   • The equation becomes: $A = 1000(1.03)^4$
   • Calculate the power: $(1.03)^4 \approx 1.1255$
   • Final multiplication: $A = 1000 \times 1.1255 = 1125.5$

So, after 2 years, you would have 1125.50 dollars in your account.

Questions for: Applying the Formula for Annual Compounding

What is Interest?

Interest is the extra money you earn when you save or invest money, or the extra money you pay when you borrow money. There are two main types: Simple Interest and Compound Interest.

Simple vs. Compound Interest

Simple Interest is calculated only on the initial amount of money, which is called the principal. You earn the same amount of interest every year.

Compound Interest is more powerful. It is calculated on the principal amount plus any interest that has already been earned. This means you start earning interest on your interest. This is often called "interest on interest". When interest is calculated and added once per year, it is called annual compounding.

The Annual Compounding Formula

To calculate the total amount of money you will have after a certain number of years with annual compounding, we use a specific formula:

$A = P(1 + r)^t$

Let's break down what each letter means:

• A = the final Amount of money. This is the principal plus all the compound interest you've earned.
• P = the Principal. This is the initial amount of money you start with.
• r = the annual interest rate. This must be written as a decimal. To convert a percentage to a decimal, you divide by 100. For example, 5% becomes $5 \div 100 = 0.05$.
• t = the time in years. This is the number of years the money is invested or borrowed for.

How to Calculate Compound Interest (CI)

The formula gives you the total amount (A), not just the interest. To find only the interest earned, you subtract the original principal from the final amount:

Compound Interest (CI) = Final Amount (A) - Principal (P)

$CI = A - P$

Worked Example

Question: You invest 1000 dollars in a bank account that pays 4% interest compounded annually. How much money will you have in total after 3 years?

Solution:

1. Identify the variables:

   • Principal (P) = 1000 dollars
   • Rate (r) = 4% = $4 \div 100 = 0.04$
   • Time (t) = 3 years

2. Write down the formula:

   • $A = P(1 + r)^t$

3. Substitute the values into the formula:

   • $A = 1000(1 + 0.04)^3$

4. Solve the equation:

   • $A = 1000(1.04)^3$
   • $A = 1000(1.124864)$
   • $A = 1124.864$

5. State the answer:

   • After 3 years, you will have 1124.86 dollars (rounding to two decimal places for money).

If you wanted to find just the interest earned, you would do:
$CI = 1124.86 - 1000 = 124.86$ dollars.