Commercial Arithmetic (II)
Numbers · Commercial Arithmetic (II)
Syllabus tag: KCSE | Mathematics | Form 3 | Topic 6 Commercial Arithmetic (II)
Lesson objectives
By the end of this topic, you should be able to:
- Define principal, rate and time in relation to interest.
- Calculate compound interest and appreciation or depreciation.
- Calculate hire purchase costs and instalment payments.
- Calculate income tax, given tax bands and reliefs.
Commercial Arithmetic (II)
Form 1 covered simple interest. This topic handles interest that compounds, and the tax system that applies to income.
a) Principal, rate and time
The principal is the sum borrowed or invested.
The rate is the percentage charged per period, usually per year.
The time is the number of periods.
b) Compound interest
With compound interest, each period's interest is added to the principal. The next period then earns interest on a larger sum.
A = P(1 + r/100)ⁿ, where A is the amount and n the number of periods.
The interest is A − P. The formula gives the amount, not the interest, and confusing the two is the standard error.
c) When compounding is not yearly
If interest compounds half-yearly, halve the rate and double the periods.
A rate of 12% per year compounded half-yearly means 6% for 2n periods.
Quarterly means a quarter of the rate and four times the periods.
More frequent compounding gives a slightly larger amount, since interest starts earning sooner.
d) Appreciation and depreciation
Appreciation uses the same formula. Land and buildings usually appreciate.
Depreciation uses A = P(1 − r/100)ⁿ. Vehicles and machinery depreciate.
Take a vehicle worth KES 800 000 depreciating at 15% for 3 years. It is worth 800 000 × 0.85³, about KES 491 300.
e) Hire purchase
Hire purchase price = deposit + (instalment × number of instalments).
It always exceeds the cash price. The difference is the carrying charge.
To find the interest rate charged, treat the carrying charge as interest on the balance after the deposit.
f) Income tax
Income tax is charged in bands. Each slice of income is taxed at its own rate, not the whole income at the top rate.
That is the point most often misunderstood. Moving into a higher band taxes only the part above the threshold.
Taxable income is gross income plus taxable benefits, such as housing.
Work through the bands in order, taxing each slice at its own rate, then add them.
g) Tax relief
Relief is a fixed amount subtracted from the tax due, not from the income.
Net tax = gross tax − relief.
Personal relief applies to every taxpayer. Insurance relief may apply on top.
If the relief exceeds the gross tax, the tax payable is zero, never negative.
h) Where this is used
Comparing loan and savings offers. Judging a hire purchase deal. Understanding a payslip. Valuing assets over time.
Words to know
- Compound interest -- interest calculated on the principal together with accumulated interest.
- Appreciation -- an increase in value over time.
- Depreciation -- a loss in value over time, applied to the reducing balance.
- Hire purchase -- buying by deposit and instalments, at a price above the cash price.
- Personal relief -- an amount subtracted from computed tax.
:::checkpoint Check yourself
- Find the amount when KES 20 000 is invested at 10% compound for 2 years.
- How much of that is interest?
- A machine worth KES 100 000 depreciates at 20% a year. Find its value after 3 years.
- Why is moving into a higher tax band not as costly as it first appears? :::
Bridge to practice
The exercises begin with compound interest and compounding periods, move through appreciation, depreciation and hire purchase, and finish with income tax. For every interest question, underline whether the amount or the interest has been asked for before you begin.