Mathematics Form 1-4 : CHAPTER FOURTY SIX – COMMERCIAL ARITHMETIC II

Specific Objectives

By the end of the topic, the learner should be able to:

• Define principal, rate, and time in relation to interest;
• Calculate simple interest using the simple interest formula;
• Calculate compound interest using the step-by-step method;
• Derive the compound interest formula;
• Apply the compound interest formula for calculating interest;
• Define appreciation and depreciation;
• Use the compound interest formula to calculate appreciation and depreciation;
• Calculate hire purchase;
• Calculate income tax given the income tax bands.

Content

• Principal, rate, and time
• Simple interest
• Compound interest using step-by-step method
• Derivation of compound interest formula
• Calculations using the compound interest formula
• Appreciation and depreciation
• Calculation of appreciation and depreciation using the compound interest formula
• Hire purchase
• Income tax

Simple Interest

Interest is the money charged for the use of borrowed money for a specific period of time. If money is borrowed or deposited, it earns interest. The principal is the sum of money borrowed or deposited (P). The rate is the ratio of interest earned in a given period of time to the principal.

The rate is expressed as a percentage of the principal per annum (P.A). When interest is calculated using only the initial principal at a given rate and time, it is called simple interest (I).

Simple Interest Formula

Simple interest =

Example

Franny invests Ksh 16,000 in a savings account. She earns a simple interest rate of 14%, paid annually on her investment. She intends to hold the investment for 1 year. Determine the future value of the investment at maturity.

Solution

I = 16000 × 14 × 1 / 100 = Ksh 2240

Amount = P + I = 16000 + 2240 = Ksh 18240

Example

Calculate the rate of interest if Ksh 4500 earns Ksh 500 after 1 year.

Solution

From the simple interest formula:

I =

Rearranged to find R:

R =

Given:

• P = Ksh 4500
• I = Ksh 500
• T = 1 year

Therefore, R = (500 × 100) / (4500 × 1) = 11.11%

Example

Esha invested a certain amount of money in a bank which paid 12% p.a. simple interest. After 5 years, his total savings were Ksh 5600. Determine the amount of money he invested initially.

Solution

Let the amount invested be Ksh P

T = 5 years

R = 12% p.a.

A = Ksh 5600

But A = P + I = P + = P + 0.60 P = 1.6 P

Therefore, P = 5600 / 1.6 = Ksh 3500

Compound Interest

Suppose you deposit money into a financial institution; it earns interest in a specified period of time. Instead of the interest being paid to the owner, it may be added to (compounded with) the principal and therefore also earns interest. The interest earned is called compound interest. The period after which it is compounded to the principal is called the interest period.

The compound interest may be calculated annually, semi-annually, quarterly, monthly, etc. If the rate of compound interest is R% p.a and the interest is calculated n times per year, then the rate of interest per period is R/n%.

Example

Moyo lent Ksh 2000 at interest of 5% per annum for 2 years. First, we know that simple interest for the 1^st year and 2^nd year will be the same:

Simple interest for 1 year = 2000 × 5 × 1 / 100 = Ksh 100

Total simple interest for 2 years = 100 + 100 = Ksh 200

In compound interest (CI), the first year interest will be the same as simple interest (SI), i.e., Ksh 100. But year II interest is calculated on P + SI of 1^st year, i.e., on Ksh 2000 + Ksh 100 = Ksh 2100.

So, year II interest in compound interest becomes:

= 2100 × 5 × 1 / 100 = Ksh 105

So it is Ksh 5 more than the simple interest. This increase is due to the fact that SI is added to the principal and this Ksh 105 is also added in the principal if we have to find the compound interest after 3 years.

Direct formula in case of compound interest is:

A = P (1 + R/100)^t

Where:

• A = Amount
• P = Principal
• R = Rate % per annum
• t = Time (years)

A = P + CI

P (1 + R/100)^t = P + CI

Types of Questions

• Type I: To find CI and Amount
• Type II: To find rate, principal, or time
• Type III: When difference between CI and SI is given
• Type IV: When interest is calculated half yearly or quarterly etc.
• Type V: When both rate and principal have to be found

Type I

Example

Find the amount of Ksh 1000 in 2 years at 10% per annum compound interest.

Solution

A = P (1 + r/100)^t

= 1000 (1 + 10/100)²

= 1000 × 121/100 = Ksh 1210

Example

Find the amount of Ksh 6250 in 2 years at 4% per annum compound interest.

Solution

A = P (1 + r/100)^t

= 6250 (1 + 4/100)²

= 6250 × 676/625 = Ksh 6760

Example

What will be the compound interest on Ksh 31250 at a rate of 4% per annum for 2 years?

Solution

CI = P (1 + r/100)^t – P

= 31250 { (1 + 4/100)² – 1 }

= 31250 (676/625 – 1)

= 31250 × 51/625 = Ksh 2550

Example

A sum amounts to Ksh 24200 in 2 years at 10% per annum compound interest. Find the sum.

Solution

A = P (1 + r/100)^t

24200 = P (1 + 10/100)²

= P (11/10)²

P = 24200 × 100/121 = Ksh 20000

Type II

Example

The time in which Ksh 15625 will amount to Ksh 17576 at compound interest is?

Solution

A = P (1 + r/100)^t

17576 = 15625 (1 + 4/100)^t

17576 / 15625 = (26/25)^t

(26/25)^t = (26/25)³

t = 3 years

Example

The rate percent if compound interest of Ksh 15625 for 3 years is Ksh 1951.

Solution

A = P + CI = 15625 + 1951 = Ksh 17576

A = P (1 + r/100)^t

17576 = 15625 (1 + r/100)³

17576 / 15625 = (1 + r/100)³

(26/25)³ = (1 + r/100)³

26/25 = 1 + r/100

26/25 – 1 = r/100

1/25 = r/100

r = 4%

Type IV

1. When interest is compounded half yearly, then Amount = P (1 + R/2/100)^2t. That is, in half yearly compound interest, the rate is halved and time is doubled.
2. When interest is compounded quarterly, then rate is made ¼ and time is made 4 times. Then A = P (1 + R/4/100)^4t.
3. When rate of interest is R1%, R2%, and R3% for 1^st, 2^nd, and 3^rd year respectively, then A = P (1 + R1/100)(1 + R2/100)(1 + R3/100).

Example

Find the compound interest on Ksh 5000 at 20% per annum for 1.5 years compounded half yearly.

Solution

When interest is compounded half yearly:

Amount = P (1 + R/2/100)^2t

Amount = 5000 (1 + 20/2/100)³

= 5000 (1 + 10/100)³

= 5000 × 1331/1000 = Ksh 6655

CI = 6655 – 5000 = Ksh 1655

Example

Find compound interest on Ksh 47145 at 12% per annum for 6 months, compounded quarterly.

Solution

As interest is compounded quarterly:

A = P (1 + R/4/100)^4t

A = 47145 (1 + 12/4/100)²

= 47145 (1 + 3/100)²

= 47145 × 103/100 × 103/100 = Ksh 50016.13

CI = 50016.13 – 47145 = Ksh 2871.13

Example

Find the compound interest on Ksh 18750 for 2 years when the rate of interest for 1^st year is 4% and for 2^nd year 8%.

Solution

A = P (1 + R1/100)(1 + R2/100)

= 18750 × 104/100 × 108/100 = Ksh 21060

CI = 21060 – 18750 = Ksh 2310

Type V

Example

The compound interest on a certain sum for two years is Ksh 52 and simple interest for the same period at the same rate is Ksh 50. Find the sum and the rate.

Solution

Simple interest is the same every year and there is no difference between SI and CI for the 1^st year. The difference arises in the 2^nd year because interest of 1^st year is added to the principal and interest is now charged on principal + simple interest of 1^st year.

So in this question:

2 year SI = Ksh 50

1 year SI = Ksh 25

Now CI for 1^st year = 52 – 25 = Ksh 27

This additional interest 27 – 25 = Ksh 2 is due to the fact that 1^st year SI i.e. Ksh 25 is added in principal. It means that additional Ksh 2 interest is charged on Ksh 25.

Rate % = (2 / 25) × 100 = 8%

Shortcut:

Rate % =

= [(2 / 50) / 2] × 100 = (2 / 25) × 100 = 8%

P = SI × 100 / (R × T) = 50 × 100 / (8 × 2) = Ksh 312.50

Example

A sum of money lent CI amounts in 2 years to Ksh 8820 and in 3 years to Ksh 9261. Find the sum and rate %.

Solution

Amount after 3 years = Ksh 9261

Amount after 2 years = Ksh 8820

By subtracting last year’s interest Ksh 441, it is clear that this Ksh 441 is SI on Ksh 8820 from 2^nd to 3^rd year, i.e., for 1 year.

Rate % = (441 × 100) / (8820 × 1) = 5%

Also A = P (1 + r/100)^t

8820 = P (1 + 5/100)² = P (21/20)²

P = 8820 × 400 / 441 = Ksh 8000

Appreciation and Depreciation

Appreciation is the gain of value of an asset while depreciation is the loss of value of an asset.

Example

An iron box costs Ksh 500 and every year it depreciates by 10% of its value at the beginning of that year. What will its value be after 4 years?

Solution

Value after the first year = 500 – 0.10 × 500 = Ksh 450

Value after the second year = 450 – 0.10 × 450 = Ksh 405

Value after the third year = 405 – 0.10 × 405 = Ksh 364.50

Value after the fourth year = 364.50 – 0.10 × 364.50 = Ksh 328.05

In general, if P is the initial value of an asset, A the value after depreciation for n periods, and r the rate of depreciation per period:

A = P (1 – r/100)ⁿ

Example

A minibus costs Ksh 400,000. Due to wear and tear, it depreciates in value by 2% every month. Find its value after one year.

Solution

A = P (1 – r/100)ⁿ

Substituting P = 400,000, r = 2, and n = 12:

A = 400,000 × (1 – 0.02)¹² = 400,000 × (0.98)¹² ≈ Ksh 313,700

Example

The initial cost of a ranch is Ksh 5,000,000. At the end of each year, the land value increases by 2%. What will be the value of the ranch at the end of 3 years?

Solution

The value of the ranch after 3 years = 5,000,000 × (1 + 0.02)³ ≈ Ksh 5,306,040

Hire Purchase

Method of buying goods and services by instalments. The interest charged for buying goods or services on credit is called carrying charge.

Hire purchase = Deposit + (instalments × time)

Example

Aching wants to buy a sewing machine on hire purchase. It has a cash price of Ksh 7500. She can pay a cash price or make a down payment of Ksh 2250 and 15 monthly instalments of Ksh 550 each. How much interest does she pay under the instalment plan?

Solution

Total amount of instalments = 550 × 15 = Ksh 8250

Down payment (deposit) = Ksh 2250

Total payment = 8250 + 2250 = Ksh 10500

Amount of interest charged = 10500 – 7500 = Ksh 3000

Note: Always use the above formula to find other variables.

Income Tax

Taxes on personal income is income tax. Gross income is the total amount of money due to the individual at the end of the month or the year.

Gross income = salary + allowances / benefits

Taxable income is the amount on which tax is levied. This is the gross income less any special benefits on which taxes are not levied. Such benefits include refunds for expenses incurred while one is on official duty.

In order to calculate the income tax that one has to pay, we convert the taxable income into Kenya pounds (K£) per annum or per month as dictated by the table of rates given.

Relief

• Every employee in Kenya is entitled to an automatic personal tax relief of Ksh 12,672 p.a (Ksh 1056 per month).
• An employee with a life insurance policy on his life, that of his wife or child, may make a tax claim on the premiums paid towards the policy at Ksh 3 per pound subject to a maximum claim of Ksh 3000 per month.

Example

Mr. John earns a total of K£ 12,300 p.a. Calculate how much tax he should pay per annum using the tax table below.

Solution

His salary lies between £1 and £12,300. The highest tax band is therefore the third band.

For the first £5808, tax due is 5808 × 2 = Ksh 11,616

For the next £5472, tax due is 5472 × 3 = Ksh 16,416

Remaining £1020, tax due is 1020 × 4 = Ksh 4080

Total tax due = 11616 + 16416 + 4080 = Ksh 32,112

Less personal relief of 1056 × 12 = Ksh 12,672

Net tax payable p.a = 32112 – 12672 = Ksh 19,440

Example

Mr. Ogembo earns a basic salary of Ksh 15,000 per month. In addition, he gets a medical allowance of Ksh 2400 and a house allowance of Ksh 12,000. Use the tax table above to calculate the tax he pays per year.

Solution

Taxable income per month = 15000 + 2400 + 12000 = Ksh 29,400

Converting to K£ p.a = 29400 × 12 / 1000 = K£ 352.8 (Assuming conversion factor)

Tax due:

First £5808 = 5808 × 2 = Ksh 11,616

Next £5472 = 5472 × 3 = Ksh 16,416

Next £5472 = 5472 × 4 = Ksh 21,888

Remaining £888 = 888 × 5 = Ksh 4,440

Total tax due = 11616 + 16416 + 21888 + 4440 = Ksh 54,360

Less personal relief = Ksh 12,672

Net tax payable p.a = 54360 – 12672 = Ksh 41,688

PAYE

In Kenya, every employer is required by law to deduct income tax from the monthly earnings of his employees every month and to remit the money to the income tax department. This system is called Pay As You Earn (PAYE).

Housing

If an employee is provided with a house by the employer (either freely or for a nominal rent), then 15% of his salary is added to his salary (less rent paid) for the purpose of tax calculation. If the taxpayer is a director and is provided with a free house, then 15% of his salary is added to his salary before taxation.

Example

Mr. Omondi, who is a civil servant, lives in government house and pays a rent of Ksh 500 per month. If his salary is £9000 p.a, calculate how much PAYE he remits monthly.

Solution

Basic salary = £9000

Housing = 15% of £9000 = £1350

Less rent paid = £500 × 12 = £6000 (Assuming monthly rent)

Taxable income = £9000 + £1350 – £6000 = £4350 (Assuming annualized)

Tax charged:

First £5808, tax due is 5808 × 2 = Ksh 11,616

Remaining £4242, tax due is 4242 × 3 = Ksh 12,726

Total tax due = 11616 + 12726 = Ksh 24,342

Less personal relief = Ksh 12,672

Net PAYE = 24342 – 12672 = Ksh 11,670

Monthly PAYE = 11670 / 12 ≈ Ksh 972.50

Example

Mr. Odhiambo is a senior teacher on a monthly basic salary of Ksh 16,000. On top of his salary, he gets a house allowance of Ksh 12,000, a medical allowance of Ksh 3060, and a hardship allowance of Ksh 4635. He has a life insurance policy for which he pays Ksh 800 per month and claims insurance relief.

1. Use the tax table below to calculate his PAYE.

2. In addition to PAYE, the following deductions are made on his pay every month:

• WCPS at 2% of basic salary
• HHIF Ksh 400
• Co-operative shares and loan recovery Ksh 4800

Solution

Taxable income = 16000 + 12000 + 3060 + 4635 = Ksh 35695

Converting to K£ (assuming 1 K£ = 1000 Ksh): 35695 / 1000 = £35.695

Tax charged is:

First £484 = £484 × 10% = £48.40

Next £456 = £456 × 15% = £68.40

Next £456 = £456 × 20% = £91.20

Remaining £388 = £388 × 25% = £97.00

Total tax due = £48.40 + £68.40 + £91.20 + £97.00 = £305.00 = Ksh 6100

Insurance relief = 800 × 3 = Ksh 2400 (Assuming 3% relief)

Personal relief = Ksh 1056

Total relief = 1056 + 2400 = Ksh 3456

Tax payable per month = 6100 – 3456 = Ksh 2644

Note: For the calculation of PAYE, taxable income is rounded down or truncated to the nearest whole number. If an employee’s due tax is less than the relief allocated, then that employee is exempted from PAYE.

Total deductions

Sh (WCPS + HHIF + Co-operative shares and loan recovery) = 2% of 16000 + 400 + 4800 = 320 + 400 + 4800 = Ksh 5520

Net pay = 35695 – 5520 = Ksh 30175

End of topic

Past KCSE Questions on the Topic

1. A businesswoman opened an account by depositing Kshs. 12,000 in a bank on 1^st July 1995. Each subsequent year, she deposited the same amount on 1^st July. The bank offered her 9% per annum compound interest. Calculate the total amount in her account on:

   • 30^th June 1996
   • 30^th June 1997

2. A construction company requires to transport 144 tonnes of stones to sites A and B. The company pays Kshs 24,000 to transport 48 tonnes of stone for every 28 km. Kimani transported 96 tonnes to site A, 49 km away.

   • Find how much he paid
   • Kimani spends Kshs 3,000 to transport every 8 tonnes of stones to site. Calculate his total profit.
   • Achieng transported the remaining stones to site B, 84 km away. If she made 44% profit, find her transport cost.

3. The table shows income tax rates:

   Monthly taxable pay	Rate of tax Kshs in 1 K£
   1 – 435	2%
   436 – 870	3%
   871-1305	4%
   1306 – 1740	5%
   Excess Over 1740	6%

   A company employee earns a monthly basic salary of Kshs 30,000 and is also given taxable allowances amounting to Kshs 10,480.

   • Calculate the total income tax
   • The employee is entitled to a personal tax relief of Kshs 800 per month. Determine the net tax.
   • If the employee received a 50% increase in his total income, calculate the corresponding percentage increase on the income tax.

4. A house is to be sold either on cash basis or through a loan. The cash price is Kshs 750,000. The loan conditions are as follows: there is to be a down payment of 10% of the cash price and the rest of the money is to be paid through a loan at 10% per annum compound interest.

   A customer decided to buy the house through a loan.

   • (i) Calculate the amount of money loaned to the customer.
   • (ii) The customer paid the loan in 3 years. Calculate the total amount paid for the house.
   • (b) Find how long the customer would have taken to fully pay for the house if she paid a total of Kshs 891,750.

5. A businessman obtained a loan of Kshs 450,000 from a bank to buy a matatu valued at the same amount. The bank charges interest at 24% per annum compound quarterly.

   • Calculate the total amount of money the businessman paid to clear the loan in 1 ½ years.
   • The average income realized from the matatu per day was Kshs 1500. The matatu worked for 3 years at an average of 280 days per year. Calculate the total income from the matatu.
   • During the three years, the value of the matatu depreciated at the rate of 16% per annum. If the businessman sold the matatu at its new value, calculate the total profit he realized by the end of three years.

6. A bank either pays simple interest at 5% p.a or compound interest at 5% p.a on deposits. Nekesa deposited Kshs P in the bank for two years on simple interest terms. If she had deposited the same amount for two years on compound interest terms, she would have earned Kshs 210 more.

   Calculate without using Mathematics Tables, the value of P.

7. (a) A certain sum of money is deposited in a bank that pays simple interest at a certain rate. After 5 years the total amount of money in an account is Kshs 358,400. The interest earned each year is 12,800.

   • The amount of money which was deposited (2 marks)
   • The annual rate of interest that the bank paid (2 marks)

   (b) A computer whose marked price is Kshs 40,000 is sold at Kshs 56,000 on hire purchase terms.

   • Kioko bought the computer on hire purchase terms. He paid a deposit of 25% of the hire purchase price and cleared the balance by equal monthly installments of Kshs 2625. Calculate the number of installments (3 marks)
   • Had Kioko bought the computer on cash terms he would have been allowed a discount of 12 ½ % on marked price. Calculate the difference between the cash price and the hire purchase price and express as a percentage of the cash price.

8. The table below is a part of tax table for monthly income for the year 2004:

   Monthly taxable income (Kshs)	Tax rate percentage (%) in each shilling
   Under Kshs 9681	10%
   From Kshs 9681 but under 18801	15%
   From Kshs 18801 but under 27921	20%

   In the tax year 2004, the tax of Kerubo’s monthly income was Kshs 1916. Calculate Kerubo’s monthly income.

9. The cash price of a TV set is Kshs 13,800. A customer opts to buy the set on hire purchase terms by paying a deposit of Kshs 2280. If simple interest of 20% p.a is charged on the balance and the customer is required to repay by 24 equal monthly installments. Calculate the amount of each installment.

10. A plot of land valued at Ksh 50,000 at the start of 1994. Thereafter, every year, it appreciated by 10% of its previous year’s value. Find:

    • The value of the land at the start of 1995
    • The value of the land at the end of 1997

11. The table below shows Kenya tax rates in a certain year:

    Income K£ per annum	Tax rates Kshs per K£
    1 – 4512	2
    4513 – 9024	3
    9025 – 13536	4
    13537 – 18048	5
    18049 – 22560	6
    Over 22560	6.5

    In that year Muhando earned a salary of Ksh 16,510 per month. He was entitled to a monthly tax relief of Ksh 960. Calculate:

    • Muhando’s annual salary in K£
    • The monthly tax paid by Muhando in Ksh

12. A tailor intends to buy a sewing machine which costs Ksh 48,000. He borrows the money from a bank. The loan has to be repaid at the end of the second year. The bank charges interest at the rate of 24% per annum compounded half yearly. Calculate the total amount payable to the bank.

13. The average rate of depreciation in value of a water pump is 9% per annum. After three complete years its value was Ksh 150,700. Find its value at the start of the three-year period.

14. 1. A water pump costs Ksh 21,600 when new. At the end of the first year its value depreciates by 25%. The depreciation at the end of the second year is 20% and thereafter the rate of depreciation is 15% yearly. Calculate the exact value of the water pump at the end of the fourth year.