Commercial arithmetic: compound interest — KCSE Mathematics

To compare total amounts accrued under simple and compound interest, we first define the formulas:

• Simple Interest (SI):
  [ SI = P \times r \times t ]
  Where:

  • P = Principal amount
  • r = Rate of interest (as a decimal)
  • t = Time (in years)

• Compound Interest (CI):
  [ A = P \times (1 + r)^t ]
  Where A is the total amount after time t.

Example:

Let’s say we have:

• Principal (P) = 10,000 Ksh
• Rate (r) = 5% (0.05 as a decimal)
• Time (t) = 3 years

1. Calculate Simple Interest:
   [ SI = 10,000 \times 0.05 \times 3 = 1,500 \text{ Ksh} ]
   Total Amount = Principal + SI = 10,000 + 1,500 = 11,500 Ksh

2. Calculate Compound Interest:
   [ A = 10,000 \times (1 + 0.05)^3 = 10,000 \times 1.157625 = 11,576.25 \text{ Ksh} ]

Conclusion:

• Total amount under Simple Interest = 11,500 Ksh
• Total amount under Compound Interest = 11,576.25 Ksh

Thus, compound interest yields a higher total amount than simple interest.

In commercial arithmetic, compound interest can also be applied to depreciation. The formula used is:

A = P(1 – R/100)ⁿ
Where:

• A = the amount after depreciation
• P = the initial value (principal)
• R = the rate of depreciation per period
• n = the number of periods

To solve depreciation problems, substitute the values into the formula. Remember, in depreciation, the value decreases over time.

Example: If a car is purchased for Ksh 1,000,000 and depreciates at a rate of 15% per year, find its value after 3 years.

• Given:
  • P = 1,000,000
  • R = 15
  • n = 3

Solution:
A = 1,000,000(1 – 15/100)³
A = 1,000,000(0.85)³
A = 1,000,000(0.614125)
A ≈ Ksh 614,125
Thus, the value of the car after 3 years is approximately Ksh 614,125.

Compound interest is the interest calculated on the initial principal and also on the accumulated interest from previous periods. It can be computed using the formula:

A = P(1 + r/n)^(nt)
Where:

• A = the amount of money accumulated after n years, including interest.
• P = the principal amount (initial investment).
• r = annual interest rate (decimal).
• n = number of times that interest is compounded per year.
• t = number of years the money is invested or borrowed.

To find the compound interest (CI), use the formula:
CI = A - P.

For example, if you invest KSh 10,000 at an annual interest rate of 5%, compounded annually for 3 years, you would calculate:

• A = 10000(1 + 0.05/1)^(1*3)
• A = 10000(1.05)^3 = 10000 * 1.157625 = KSh 11576.25
• CI = 11576.25 - 10000 = KSh 1576.25.
  Thus, the compound interest earned is KSh 1576.25.