Indicies - Explanation, Examples and Laws of Indicies

 

Understanding Compound Interest: When Money Grows on Itself.

In my last post, I explained the concept of simple interest. Simple interest rewards you only on your original amount (the principal). But in real life - with most bank savings, loans, and investments, the interest itself earns more interest over time. That’s called Compound Interest.

---

What is Compound Interest?

Compound Interest (C.I.) is the interest calculated on both the principal and the accumulated interest from previous periods. In other words, your money earns interest on your previous interest.

---

Formula for Compound Interest

The total amount after compounding is given by:

$$A = P \left(1 + \frac{R}{100}\right)^T$$

Where:

• $A$ = Total Amount after $T$ years

• $P$ = Principal (original amount)

• $R$ = Rate of interest per annum (in %)

• $T$= Time (in years)

Then,

$$\text$$

---

Worked Examples

Example 1

Find the compound interest on GHS 10,000 for 2 years at 10% per annum.

$$A=10000(1+10100)2=10000(1.1)2=10000×1.21=12100 A = 10000 \left(1 + \frac{10}{100}\right)^2 = 10000 (1.1)^2 = 10000 \times 1.21 = 12100$$ $$C.I. = 12100 - 10000 = 2100$$

Compound Interest = GHS 2,100

---

Example 2

Find the amount and compound interest on GHS 5,000 for 3 years at 8% per annum.

$$A = 5000 \left(1 + \frac{8}{100}\right)^3 = 5000 (1.08)^3 = 5000 \times 1.2597 = 6298.5$$ $$C.I. = 6298.5 - 5000 = 1298.5$$

Compound Interest = GHS 1,298.50
Total Amount =  5,000+ 1298.5= GHS 6,298.50

---

Example 3

Find the compound interest on GHS 2,000 for 4 years at 5% per annum.

$$A = 2000 \left(1 + \frac{5}{100}\right)^4 = 2000 (1.05)^4 = 2000 \times 1.2155 = 2431$$ $$C.I. = 2431 - 2000 = 431$$

Compound Interest = GHS 431

---

Example 4

Find the compound interest on GHS 8,000 for 1 year 6 months at 10% per annum compounded yearly.
Since 1 year 6 months = 1.5 years,

$$A = 8000 (1.1)^{1.5}$$ $$A = 8000 \times 1.1487$$

$$$$

$$= 9189.6$$$$C.I. = 9189.6 - 8000 = 1189.6$$

Compound Interest = GHS 1,189.60

---

Example 5

Find the amount on GHS 1,000 after 2 years if interest is compounded half-yearly at 8% per annum.
When compounded half-yearly:

$$R = \frac{8}{2} = 4\%, \quad T = 2 \times 2 = 4$$ $$A = 1000 (1.04)^4 = 1000 \times 1.1699 = 1169.9$$$$C.I. = 169.9$$

Compound Interest = GHS 169.90

---

Example 6

Find the compound interest on GHS 5,000 for 3 years at 10% per annum compounded quarterly.
When compounded quarterly:

$$R = \frac{10}{4} = 2.5\%, \quad T = 3 \times 4 = 12$$ $$A = 5000 (1.025)^{12} = 5000 \times 1.3449 = 6724.5$$

$$\mathrm{<br>}$$

$$C.I. = 6724.5 - 5000 = 1724.5$$

Compound Interest = GHS 1,724.50

---

Example 7

Find the principal that amounts to GHS 3,628 in 3 years at 10% per annum compound interest.

$$A = P(1.1)^3 = 1.331P$$ $$3628 = 1.331P$$ $$P = \frac{3628}{1.331} = 2726.1$$

Principal = GHS 2,726.10

---

Example 8

At what rate will GHS 1,250 amount to GHS 1,331 in 2 years at compound interest?

$$1331 = 1250(1 + \frac{R}{100})^2$$ $$\frac{1331}{1250} = (1 + \frac{R}{100})^2$$ $$1.0648 = (1 + \frac{R}{100})^2$$ $$1 + \frac{R}{100} = 1.032$$ $$R = 3.2\%$$

---

Example 9

The population of a town is 10,000 and it increases at 5% per year. Find the population after 3 years.

$$A = 10000(1.05)^3 = 10000 \times 1.1576 = 11576$$

Population after 3 years = 11,576

---

Key Difference Between Simple and Compound Interest

The main difference between simple and compound interest lies in how the interest is calculated and how the money grows over time

1.Calculation: Simple Interest is calculated only on the original principal (the initial amount of money invested or borrowed).

Compound Interest, on the other hand, is calculated on both the principal and the accumulated interest from previous periods.

2.Growth Pattern: In Simple Interest, the total amount grows linearly — by the same amount each year.

In Compound Interest, the total amount grows exponentially, meaning it increases by larger amounts each year because interest is added to interest.

Example: With Simple Interest, if you earn ₵100 each year on ₵1,000, your total interest increases by ₵100 yearly.

With Compound Interest, the interest you earn each year increases because each year’s interest is added to the principal before calculating the next year’s interest.

---

                 Assignment 

Try solving these to test your understanding :

1. Find the compound interest on GHS 6,000 for 3 years at 5% per annum.

2. What will GHS 8,000 amount to after 2 years at 10% per annum compound interest?

3. Find the rate at which GHS 2,000 will amount to GHS 2,420 in 2 years.

4. Find the compound interest on GHS 4,000 for 2½ years at 8% per annum compounded yearly.

5. Find the principal that amounts to GHS 6,655 in 3 years at 5% compound interest.

6. If GHS 5,000 is invested at 12% per annum compounded half-yearly, find the amount after 2 years.

7. A sum of money grows from GHS 3,000 to GHS 3,969 in 3 years. Find the rate of compound interest.

8. The population of a city increases by 4% annually. If it is 50,000 now, find its population after 3 years.

9. Find the difference between the compound interest and simple interest on GHS 10,000 for 2 years at 10% per annum.

10. A man borrows GHS 20,000 at 8% per annum compounded yearly. Find the total amount after 3 years.