# Commercial Arithmetic II - Mathematics Form 3 Notes

## 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, Principle is the sum of money borrowed or deposited P, Rate is the ratio of interest earned in a given period of time to the principle.
• 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 Formulae

Simple interest = principle x rate x time
100
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 1½years. Determine the future value of the investment at maturity.

Solution

I  = PRT
100
= sh. 16000 x 14  x 3
100   2
= sh 3360
Amount = P + I
= sh.16000 + sh 3360
= sh.19360

Example

Calculate the rate of interest if sh 4500 earns sh 500 after 1½years.

Solution

From the simple interest formulae
I =
PRT
100
R= 100 × I
P × T
P = sh 4500
I = sh 500
T = 1½
years

Therefore R = 100 x 500
4500 x 3/2
R = 7.4 %

Example

Esha invested a certain amount of money in a bank which paid 1 2% p.a. simple interest. After 5 years, his total savings were sh 5600.Determine the amount of money he invested initially.

Solution

Let the amount invested be sh P
T = 5 years
R = 12 % p.a.
A =sh 5600
But A = P + I
Therefore 5600 = P + P X
12 X 5
100
= P + 0.60 P
= 1.6 P

Therefore P = 5600
1.6
= sh 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 principle and therefore also earns interest.
• The interest earned is called compound interest. The period after which its compounded to the principle is called interest period.
• The compound interest maybe 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 1 st year and 2nd year will be same

i.e. = 2000 x 5 x 1/100 = Ksh. 100

Total simple interest for 2 years will be = 100 + 100 = ksh. 200

In Compound Interest (CI) the first year Interest will be same as of 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 x 5 x 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 +
)t
100
Where A = Amount
P = Principal
r = Rate % per annum
t = Time
A = P + CI
P (1 +
r )t = P + CI
100

### Types of Question

• 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 1

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 x 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 x 676/625
= ksh. 6760

Example

What will be the compound interest on ksh 31 250 at a rate of 4% per annum for 2 years?

Solution.

CI = P (1 + r/100)t  1
=31250 { (1 + 4/100)
2 1 }
=31250 (676/625 − 1 )
=31250 x 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
= 24200 x 100/121
= ksh. 20000

#### Type II

Example.

The time in which ksh. 15625 will amount to ksh. 17576 at 4% 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. 1 7576
A = P(1 + r/100)t
17576 = 15625 (1 + r/100)3
17576/1 5625 = (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. Remember
• When interest is compounded half yearly then Amount = P (1 + R/2)2t
100
I.e. in half yearly compound interest 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 1st, 2nd and 3rd 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 year compound half yearly.

Solution.

When interest is compounded half yearly
Then Amount = P [(1 +R/2)/100]2t
Amount = 5000 [(1 + 20/2)/100]3/2 × 2
= 5000 (1 + 10/100)3
=5000 x 1331/1000
= ksh 6655
CI = 6655 − 5000 = ksh. 1655

Example

Find compound interest 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 = 471 45 [(1 + 12/4)/100]½ x 4
= 47145 (1 + 3/100)2
= 47145 x 103/100 x 103/100
= ksh. 50016.13
CI = 50016.13 − 471 45
= ksh. 2871.13

Example

Find the compound interest on ksh. 1 8750 for 2 years when the rate of interest for 1st year is 4% and for 2nd year 8%.

Solution.

A = P (1 + R1/100) (1 + R1/100)
= 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 same rate is ksh. 50, find the sum and the rate.

Solution.

We will do this question by basic concept. Simple interest is same every year and there is no difference between SI and CI for 1 st year.

The difference arises in the 2nd year because interest of 1 st year is added in 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 x 100 = 8%

Shortcut:

Rate % = [(CI - SI)/(SI/2)] x 1 00
= [(2/50)/2] x 100

2/25 x 100
=8%
P = SI x 100/R x T = 50 x 100/8 x 2
= ksh. 312.50

Example

A sum of money lent CI amounts in 2 year 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 3rd year i.e. for 1 year.
Rate % = 441 x 100/8820 x 1
=5 %
Also A = P (1 + r/100)
t
8820 = P (1 + 5/100)2
= P (21/20)2
P = 8820 x 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 cost ksh 500 and every year it depreciates by 1 0% of its value at the beginning of that that year. What will its value be after value 4 years?

Solution

Value after the first year = sh (500 − 10 x 500)
100
= sh 450
Value after the second year = sh (450 − 10 x 450)
100
= sh 405
Value after the third year = sh (405 −
10 x 405)
100
= sh 364.50
Value after the fourth year = sh (364.50 −
10 x 364.50)
100
= sh 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 cost sh 400000.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 in the formula ;
A =sh.400000 (1 − 0.02
)12
=sh.400,000(0.98)12
= sh.313700

Example

The initial cost of a ranch is sh.5000, 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 =sh 5,000, 000(1 + 2/100)3
= sh. 5,000,000(1.02)3
= sh 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 x time)

Example

Achieng 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 sh 2250 and 15 monthly instalments of sh.550 each. How much interest does she pay under the instalment plan?

Solution

Total amount of instalments = sh 550 x 15 = sh 8250
Down payment (deposit) = sh 2250
Total payment = sh (8250 + 2250) = sh 10500
Amount of interest charged = sh (10500-7500)
= sh3000

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 by the table of rates given.

### Relief

• Every employee in kenya is entitled to an automatic personal tax relief of sh.12672 p.a (sh.1 056 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 sh.3 per pound subject to a maximum claim of sh .3000 per month.

Example

Mr. John earns a total of K£ 12300 p.a.Calculate how much tax he should pay per annum.Using the tax table below.

 Income tax K£ per annum Rate (sh per pound) 1 -5808 2 5809 - 1 1 280 3 1 1 289 - 1 6752 4 1 6753 - 22224 5 Excess over 22224 6

Solution

His salary lies between £ 1 and £1 2300.The highest tax band is therefore the third band.

For the first £5808, tax due is     sh 5808 x 2 = sh 11616
For the next £5472, tax due is    sh 5472 x 2 =  sh 16416
Remaining £1020, tax due          sh. 1020 x 4 = sh  4080 +
Total tax due                                                  sh 32112
Less personal relief of              sh.1056 x 12 = sh.12672 −
Sh 19440
Therefore payable p.a is sh.19400.

Example

Mr. Ogembo earns a basic salary of sh 15000 per month.in addition he gets a medical allowance of sh 2400 and a house allowance of sh 12000.Use the tax table above to calculate the tax he pays per year.

Solution

Taxable income per month = sh (15000 + 2400 + 12000) = sh.29400
Converting to K£ p.a = K£ 29400 x
12
20
= K£ 1 7640

Tax due
First £ 5808 = sh.5808 x 2 =       sh. 11616
Next £ 5472 = sh.5472 x 3 =      sh. 16416

Next £ 5472 = sh.5472 x 4 =      sh. 21888
Remaining £ 888 = sh.888 x 5 = sh    4440+
Total tax due                              sh  54360
Less personal relief                     sh  12672
Therefore, tax payable p.a          sh   41688

### PAYE

• In Kenya, every employer is required by the 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 purpose of tax calculation.
• If the tax payer is a director and is provided with a free house, then 1 5% of his salary is added to his salary before taxation.

Example

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

Solution

Basic salary       £ 9000

Housing £ 15  x 9000 = £1350
100
Less rent paid =
£1350 − £ 300 = £ 1050

Taxable income

Tax charged;
First £ 5808, the tax due is sh.5808 x 2           =    sh 11616
Remaining £ 4242, the tax due is sh 4242 x 3  =    sh 12726 +
sh 24342
Less personal relief                                              sh 12672
sh 11670

PAYE = sh 11670
12
= sh 972.50

Example

Mr. Odhiambo is a senior teacher on a monthly basic salary of Ksh. 1 6000.On top of his salary he gets a house allowance of sh 1 2000, a medical allowance of Ksh.3060 and a hardship 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.
 Income in £ per month Rate % 1 - 484 10 485 - 940 15 941 - 1 396 20 1 397 - 1 852 25 Excess over 1 852 30
2. In addition to PAYEE the following deductions are made on his pay every month
1. WCPS at 2% of basic salary
2. HHIF ksh.400
3. Co – operative shares and loan recovery Ksh 4800.

Solution

1. Taxable income = Ksh (16000 + 12000 + 3060 + 4635)
= ksh 35695
Converting to K£ = 35695
20
= K£ 1784.75

Tax charged is:

First £ 484 = £484 x
10 = £ 48.40
100
Next £ 456 = £456 x 15 = £ 68.40
100
Next £ 456 = £456 x 10 = £ 91 .20
100
Remaining £ 388 = £388 x 25 = £ 97.00.
100
Total tax due = £305.00
= sh 61 00
Insurance relief = sh 800 x 3 = sh 120
20
Personal relief = sh 1056
Total relief 120 + 1056 = sh 1176
Tax payable per month is sh 6100
less total relief                sh 1176
sh 4924
Therefore, PAYE is sh 4924.

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 PAYEE
2. Total deductions are
Sh ( 2 x 16000 + 400 + 4800 + 800 + 4924 ) = sh 11244
100
Net pay = sh (35695 – 11244)
= sh 24451

## Past KCSE Questions on the Topic.

1. A business woman opened an account by depositing Kshs. 12,000 in a bank on 1st July 1995. Each subsequent year, she deposited the same amount on 1st July. The bank offered her 9% per annum compound interest. Calculate the total amount in her account on 30
1. 30th June 1 996
2. 30th June 1 997
2. A construction company requires to transport 1 44 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 a site A, 49 km away.
1. Find how much he paid
2. Kimani spends Kshs 3,000 to transport every 8 tonnes of stones to site.
Calculate his total profit.
3. Achieng transported the remaining stones to sites 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 – 435436 – 870871 -13051306 – 1740Excess Over 1740 2 3 4 5 6
A company employee earn a monthly basic salary of Kshs 30,000 and is also given taxable allowances amounting to Kshs 1 0, 480.
1. Calculate the total income tax
2. The employee is entitled to a personal tax relief of Kshs 800 per month.
Determine the net tax.
3. 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 area as follows: there is to be down payment of 10% of the cash price and the rest of the money is to be paid through a loan at 1 0% per annum compound interest. A customer decided to buy the house through a loan.
1. Calculate the amount of money loaned to the customer.
2. The customer paid the loan in 3 year’s. Calculate the total amount paid for the house.
1. 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
1. Calculate the total amount of money the businessman paid to clear the loan in 1 ½ years.
2. The average income realized from the matatu per day was Kshs. 1 500. The matatu worked for 3 years at an average of 280 days year. Calculate the total income from the matatu.
3. During the three years, the value of the matatu depreciated at the rate of 1 6% 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 as 5% p.a or compound interest 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 values of P
7.
1. 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 358400. The interest earned each year is 12 800
Calculate
1. The amount of money which was deposited
2. The annual rate of interest that the bank paid
2. A computer whose marked price is Kshs 40,000 is sold at Kshs 56,000 on hire purchase terms
1. 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 (3mks)
2. Had Kioko bought the computer on cash terms he would have been allowed a discount of 1 2½ % on marked price. Calculate the difference between the cash price and the hire purchase price and express as a percentage of the cash price
3. Calculate the difference between the cash price and hire purchase price and express it 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 incomeIn ( Kshs) Tax rate percentage(%) in each shillings Under Kshs 9681 10% From Kshs 9681 but under 1 8801 15% From Kshs 1 8801 but 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 T.V set is Kshs 1 3, 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 1 994. Thereafter, every year, it appreciated by 1 0% of its previous years value find:
1. The value of the land at the start of 1 995
2. The value of the land at the end of 1 997
11. The table below shows Kenya tax rates in a certain year.
 Income K £ per annum Tax rates Kshs per K £ 1 - 451 2 2 451 3 - 9024 3 9025 - 1 3536 4 1 3537 - 1 8048 5 1 8049 - 22560 6 Over 22560 6.5
In that year Muhando earned a salary of Ksh. 1 651 0 per month. He was entitled to a monthly tax relief of Ksh. 960,
Calculate
1. Muhando annual salary in K £
2. 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 an 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 1 50,700. Find its value at the start of the three year period.
14. 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 1 5% yearly. Calculate the exact value of the water pump at the end of the fourth year

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