COMMERCIAL ARITHMETIC (II)
In Commercial Arithmetic (I), we looked at currency exchange rates, profit, loss and commission. Now, we will explore simple and compound interest, appreciation, depreciation, hire purchase and income tax. Some of these concepts might be familiar to you, but we will examine different ways of understanding them.
Simple Interest
Interest is the price you pay to borrow money. When repaying the money, the interest that has accumulated over the time period of borrowing is paid on top of the original amount.
It is calculated using the expression: I=\frac{PRT}{100} Where I = Interest, P = Principal, R = Rate (in percentage) and T = Time.
The total amount is calculated as: \text{Amount} = Principal\ +\ Interest
In the rate of interest, p.a. (per annum) means every year, while p.m. (per month) means every month.
Example
A man borrowed Sh 12460 from a financial institution. The rate of simple interest was 12.5% p.a. After 6 months, he paid back Sh 8460. How much did he still owe the bank including interest?
I=\frac{PRT}{100} We substitute Sh 12460 for the principal, 12.5 for the rate and \frac{6}{12} for the time. I=\frac{12460\times12.5\times0.5}{100} I=Sh 778.75 Now, we get the overall amount by adding the principal and interest. A=12460+778.75=Sh 13238.75 The remaining amount is given by the total amount minus the amount that he paid. Remaining = 13238.75-8460= Sh 4778.75
Compound Interest
In compound interest, the interest is calculated after the first year and added to the principal. The amount from the first year is then used as the principal for the next year.
It is calculated using the formula: A = P\left( 1+\frac{r}{100} \right)^n Where A = final amount, P = principal, R = interest rate and n = number of interest periods.
To derive the formula for compound interest, we use the simple interest formula as we know that the simple interest (SI) for one year is equal to the compound interest for one year (when compounded annually).
Let Principal = P, Time = n and Rate = r.
\displaystyle SI=\frac{PrT}{100} , but since T = 1 year, \displaystyle SI=\frac{Pr}{100} .
The amount after the first year is calculated as: A = P + \frac{Pr}{100} A = P \left( 1 + \frac{r}{100} \right)=P_2 This amount is used as the principal for the second year, P_2 . A = P_2 + \frac{P_2r}{100} A = P_2 \left( 1+\frac{r}{100} \right) Substituting the original value of P_2 : A = P \left( 1 + \frac{r}{100} \right)\left( 1+\frac{r}{100} \right) A = P \left( 1 + \frac{r}{100} \right)^2 Similarly, if we proceed further to n years, we can see that: A = P\left( 1+\frac{r}{100} \right)^n
The above formula is only used when interest is compounded annually. Now we must ask ourselves, what happens when interest is compounded multiple times a year? You can pause for a moment and try to figure it out using the above formula as a guide.
When interest is compounded multiple times a year, we use a slight modification of the original formula: A=P \left( 1+\frac{r}{n} \right)^{nt}
Where
- A = amount
- P = principal
- r = rate of interest (expressed as a decimal)
- n = number of times interest is compounded per year
- t = time (in years)
In this formula, the rate is divided by the number of times the interest is compounded per year. For example, if our rate is 12% p.a. compounded quarterly (every 3 months), then our new rate will be \frac{12}{4} =3\%.
In addition, the part in brackets is raised to the power of the number of times the interest is compounded per year. For example, if it is compounded every 4 months for 4 years, then the power will be 3\times4=12 . There are 12 interest periods in this example.
Example 1
A man deposited Sh 16821 in a financial institution that gives an interest of 7.5% p.a. If the amount deposited accumulated to Sh 30000, find the number of years.
We can substitute these values into the compound interest formula to get: 30000= 16821\left( 1+\frac{7.5}{100} \right)^n Dividing both sides by 16821: 1.783= 1.075^n The key to this problem is to express both sides as logarithms to find n. \log(1.783)= \log(1.075^n) From the laws of logarithms, \log(M^p)=p\log(M). (If you do not understand this, read our article on further logarithms.) \log(1.783)= n\log(1.075) Dividing both sides by \log(1.075) : \boxed{n=8\ years}
Example 2
Find the rate per annum at which a certain amount of money triples after being invested for a period of 5 years compounded semi-annually.
A=P \left( 1+\frac{r}{n} \right)^{nt}Let the original amount of money be x . The amount will be triple this, i.e 3x . There are 2 interest periods per year, thus n = 2.
3x=x \left( 1+\frac{r}{2} \right)^{2\times5} Dividing both sides by x : 3= \left( 1+\frac{r}{2} \right)^{10} \sqrt[10]{3}= 1+\frac{r}{2} 1.116= 1+\frac{r}{2} 0.116= \frac{r}{2} r=0.2322 \boxed{ r=23.22\% }
Appreciation and Depreciation
Appreciation is the increase in the value of an item, such as land. The value increases every year by a certain percentage. The new value after calculating it for the first year is used as the principal in the calculations for the next year.
It is calculated using the same formula as that of compound interest: A = P\left( 1+\frac{r}{100} \right)^n
Depreciation is the decrease in the value of an item, such as a car. The value decreases every year by a certain percentage. It is calculated using the formula for appreciation, but with a subtraction sign instead of addition. A = P\left( 1-\frac{r}{100} \right)^n
Example
A bus costs Sh 400,000. It depreciates in value by 2% every month. Find the value at the end of 1 year.
A = P\left( 1-\frac{r}{100} \right)^n A = 400000\left( 1-\frac{2}{100} \right)^{12} A = 400000\left(0.98\right)^{12} A = \text{Sh}\ 313866.70
Hire Purchase
Hire purchase is a payment plan in which one pays for a product gradually instead of all at once. The person pays a certain amount then takes the product home, but continues to pay for it until the cost is fully paid. This is often preferred as someone may not have enough money to cover the full cost of a product. However, one has to pay interest on top of the original cash price.
There are a few terms that you will have to be familiar with in hire purchase:
- Down payment/Deposit – This is the amount paid before the goods are given to a person.
- Cash price – The amount of money that the product would cost if it was to be paid for all at once.
- Carrying charge – The interest charged above the cash price. It is calculated by finding the difference between the price if it was paid for through hire purchase and the cash price.
- Installment – This is a small amount of money that is paid monthly until the full price has been paid.
Now, the following is the most important part of this topic that people often miss. Take time to understand this part and you will find hire purchase to be very easy.
The difference between the deposit and the cash price is treated as a loan. This is because the customer is essentially borrowing money from the shop by paying through hire purchase. The shop would have had this money earlier, but the customer ‘borrows’ their money and pays it later. Thus, the shop is loaning the customer money.
Typically, the loan in hire purchase is charged compound interest at a certain rate, though in rare situations there may be simple interest.
When working out the rate of interest charged:
- The principal is the difference between the cash price and the deposit. This is because that is the amount that the customer is ‘borrowing’ from the shop.
- The amount is the total monthly installments paid. This is because that is the total amount of money that is ‘paid back’ for the loan.
In summary, we look at the principal of the loan as the amount of the cash price that isn’t paid in the deposit and we take the amount as the total monthly installments paid.
Example 1
Golana wants to buy a sewing machine on hire purchase terms. The cash price is Sh 7500. He can pay the cash price or make a down payment of Sh 2250 followed by 15 monthly installments of Sh 550 each.
(a) How much interest does she pay under the hire purchase?
The interest paid (or carrying charge) is equal to the amount paid under hire purchase minus the cash price.
The money paid under hire purchase is: \text{HP}=2250+ \left( 15\times550 \right)=\text{Sh}\ 10500 Thus, the carrying charge is: \text{CC}=10500-7500=\text{Sh}\ 3000
(b) Calculate the rate of compound interest charged per month.
The loan (principal) is the cash price minus the deposit. \text{Principal}=7500-2250=\text{Sh}\ 5250
The amount is equal to the total monthly installments. \text{Amount}=15\times550=\text{Sh}\ 8250 Now, we can plug these values into the compound interest formula to find the rate. A = P\left( 1+\frac{r}{100} \right)^n 8250 = 5250\left( 1+\frac{r}{100} \right)^{15} Dividing both sides by 5250: 1.5714 =\left( 1+\frac{r}{100} \right)^{15} \sqrt[15]{1.5714} = 1+\frac{r}{100} 1.03059=1+\frac{r}{100} 0.03059=\frac{r}{100} \boxed{ r=3.059\% }
Example 2
The cash price of a gas cooker is Ksh 20,000. A customer bought the cooker on hire purchase terms by paying a deposit of Ksh 10000 followed by 18 equal monthly installments of Ksh 900 each. Annual interest, compounded quarterly, was charged on the balance for the period of 18 months. Determine the rate of interest per annum.
The principal is equal to the difference between the cash price and deposit. \text{Principal}=20000-10000=\text{Sh}\ 10000 The amount is the total monthly installments, which is 18\times900=\text{Sh} 16200. Since the interest is compounded quarterly, it is compounded 4 times per year. Thus: A=P \left( 1+\frac{r}{n} \right)^{nt} Here, we substitute 1.5 for t, because 18 months is equal to one and a half years. 16200=10000 \left( 1+\frac{r}{4} \right)^{4\times1.5} Dividing both sides by 10000: 1.62= \left( 1+\frac{r}{4} \right)^{6} \sqrt[6]{1.62}= 1+\frac{r}{4} 1.08373= 1+\frac{r}{4} 0.08373= \frac{r}{4} r=0.3349 Remember that when using this formula, the rate is expressed as a decimal. Thus, to change it to a percentage: r=0.3349\times100 \boxed{r=33.49\% }
Income Tax
Income tax is a payment made by individuals to the government based on their taxable income.
There are a few terms that you need to be familiar with:
- Taxable income/Taxable amount – This is the portion of the gross income on which one is required to pay taxes.
- Band – The bracket in a table that is used to tax a certain range of money.
- Gross tax – The total tax without deductions.
- Relief – Money given by the taxing body to reduce the amount of tax people have to pay.
- Net tax – The gross tax minus the relief.
- Net salary – The net income minus tax and other deductions.
- Pay As You Earn (PAYE) – The net tax paid every month.
- Allowances – Money given to an employee to cover certain costs e.g. housing, medical costs
The taxable income is calculated as the sum of the salary and allowances.
If an employee is housed by the employer, the basic salary is increased by an imaginary 15%, minus the rent if any. This is for taxation purposes only and does not affect the actual basic salary of the employee. Note that if the employee is paid a house allowance but is not housed by the employer, the 15% is not added.
Example
The table below shows income tax rates in a certain year.
In the year, the monthly earnings of Larema were as follows:
- Basic salary – Ksh 64500
- House allowance – Ksh 12000
- Commuter allowance – Ksh 5000
Larema contributes 7.5% of his salary to a pension scheme. This contribution is exempted from taxation. He is entitled to a personal tax relief of Ksh 1408. Calculate:
(a) Larema’s monthly taxable income.
First, we must get the amount contributed to the pension scheme. It is 7.5% of the basic salary, i.e \displaystyle\frac{7.5}{100}\times 64500=\text{Sh}\ 4837.50
Thus the total taxable income will be: \begin{split}\text{Taxable income}&= 64500+12000+5000-4837.50\\&=\text{Sh}\ 76662.50\end{split}
(b) The tax payable by Larema that month.
When determining the amount of money in each tax bracket, we take the upper limit of the current band minus the upper limit of the previous band. We do not simply subtract the 2 numbers shown in the band. For example, in the band 12299 – 23885, the amount of money is 23885 – 12298 = 11587 . The table below shows the calculations for the tax payable for each band.
(c) Larema’s net pay that month.
The net pay is equal to the net income minus tax. Therefore: \text{Net pay} = 76662.50-15655.05= \text{Sh}\ 61007.45