4.12 Hire Purchase and Installment scheme


Let us assume that when you grow up you want to set up a factory manufacturing scientific instruments such as thermometers … For that you need to buy machines to manufacture these. Assume that an important machine costs Rs 1, 00,000 and you have only few thousands rupees for taking care of other expenses and you need money to buy this machine. You have several alternatives to borrow the required money. One option could be to take a loan from a Bank.  Another option is to approach a company which provides the loan through ‘Hire Purchase Scheme”. Let us say you have approached a company called ‘Easyfinance Ltd” for ‘Hire Purchase scheme’. Like in case of Banks you also need to enter into agreement called ‘Hire Purchase Agreement’ with them. You (borrower) are called Hirer,  ‘Easy finance Ltd’ (the company given loan) is called  Vendor(seller)

The scheme works as follows:


1. The Hirer has to make an initial payment say in your case Rs 10,000. This is called ‘down payment’ (This gives some confidence to the Vendor that you are serious about your business)

2. Hirer has to agree that he will pay the balance amount in installments periodically (could be every month or once in 3 months or every year. Let us say you agree to pay   ‘n’ number of installments each of Rs 10,000.

3. Till Hirer (you) pays the last installment, the equipment (in your case Machine) can not be sold to any one.

4. Though Hirer could be using the equipment for his business(in your case  machine  for manufacturing), till the  final payment is made the equipment is considered  as  given on loan to Hirer by Vendor( Hirer can only use but can not sell, damage, destroy…)

5. Assume that Hirer is not able to pay the complete amount (fails to pay few installments) then:

-Vendor can take the equipment back from Hirer and sell or give the equipment to somebody.

-whatever amount paid by Hirer is treated as rent for the period under the agreement

- Whatever amount the Hirer had paid till then to the Vendor is not returned the Hirer.

- Thus, Hirer lost all the money paid. Also he can not own the equipment.


Since Vendor is giving money as loan to the Hirer, Installment amount includes interest. Thus Installment amount has two parts-Major part towards the loan and a small part towards Interest.


In the above case :

Loan amount taken = 1,00,000

Total installment amount = Installment amount *number of Installments= 10,000*12 = 1,20,000

Total amount paid to Vendor = Down Payment + Total installment amount = 10,000+1,20,000= 1,30,000

In effect Hirer has paid Vendor 30,000 over on a loan of 1,00,000.. Thus 30,000 can be treated as total Interest.


Installment Scheme


4.12 Example:

Let us assume that your family wants to purchase a new TV costing Rs.30,000 and your family is ready to pay the amount in monthly Installments in stead of  paying full amount at one time. This method of paying in installments is called ‘Installment scheme’.  This type of scheme is offered by Vendors(TV Shop Owner).

In this scheme the buyer makes an initial payment called down payment and agrees to pay the specified amount as demanded by Vendor (TV Shop Owner) in equal monthly installments thereafter. In this scheme, the moment the initial payment is made to vendor, TV set will be owned by your family even though full payment has not been made to the vendor.

In case your family does not make full payment then the Vendor (TV Shop owner) can not come to your home and take the equipment (TV) away. He has to go to court to settle the matter. (For this reason, normally Vendors (TV Shop Owner) insist on your family handing over post dated Cheques.  This is because, if a cheque is dishonored (bouncing) it becomes a criminal case and issuer of Cheque can be sent to Jail.


Since TV shop owner has not collected the full amount at the time of your purchase, and has given TV worth Rs.30, 000 on loan he needs to collect the interest amount. This Interest part is included in the Installment Amount.

The formula for calculating rate of interest under Installment Scheme is as follows


R% = 2400*E/ N[(N+1)*I -2*E]



R : Rate of interest

E : Extra Amount paid over and above the price of Equipment/Product (Total amount paid to the Vendor- Price of Product)

I : Installment amount

N : number of installments


Assume that in the above Example, your family makes a down payment of Rs.1000 and agrees to pay 35 installment amounts of Rs 1000 each. Let us calculate rate of Interest.


Price of TV = Rs.30,000

Down payment = Rs 1,000

Total Installment amount paid = Installment Amount* Number of Installments = 1000*35 = Rs. 35,000

Total amount paid to Vendor = Down Payment + Total Installment Amount paid = 1000+35,000 = Rs 36,000

Extra Amount paid = Total amount paid to Vendor - Price of TV = 36,000-30,000 = 6000


Now let us find out Rate of interest as per the formula by substituting

E= 6000

I =1000



R = 2400*6000/ 35(36*1000 -2*6000) = 2400*6000/ 35*24000 = 17.14%


Activity: Compare Installment schemes of several TV Shop owners after visiting many of them.   This is how you can use mathematics in your daily life and give advice to your friends and relatives and save money for them and get gifts from them.


4.12 Summary of learning




Points learnt


Hire purchase scheme, Installment scheme


R = Rate of interest, E = Extra Amount paid, I =Installment amount, N = number of installments. Then

R = 2400*E/ n[(n+1)*I -2*E]



Additional Points:



Repayment of Loans in installments:


If the loan period is few months then the simple interest is charged on the loan, under installment scheme. However if the period is few years, compound interest is charged either compounding every month or every quarter or every half year or every year. In such cases each monthly installment includes the compounded interest also. Since monthly installment amount is fixed for all the installments, the installment is called EMI (Equated Monthly Installment). In case of housing loans the interest is compounded monthly. In all cases EMI is calculated using the formula:

EMI = L*( E /100) {(1+ (E/100)) N/ [(1+ (E/100)) N-1]}


L = Loan amount

N = Number of installments

E = Effective Interest rate (converted) charged for the compounding period

Effective rate of interest (E) is calculated as follows:

(For example:

Let the rate of interest per annum be 16%.

If the interest rate is compounded every year then E=16, if it is compounded every half year then E =16/2=8, if it is compounded every quarter then E=16/4=4 and if it is compounded every month then E =16/12=4/3.


4.12. Problem 1: A person borrowed some money on compound interest and returned it in three years in equal installments. If the rate of interest is 15%pa and annual installment is Rs 4,86,680, find the sum borrowed?



Let L be the sum borrowed.

N = 3

E = 15

Since EMI = 4,86,680

We have

486680 = L*(15/100) {(1+ (15/100)) 3/ [(1+ (15/100)) 3-1]}

= 0.437976962L (Use Calculator or log tables)

L = 11,11,200


4.12. Problem 2: A housing society charges Rs 16,00,000 cash or Rs 5,85,500 cash down payment and three half yearly installments for a flat. If the society charges 16% pa compounded half yearly, calculate the value of each installment. Also find the total interest charged.





L = 16,00,000-5,85,500=10,14,500

E = 16/2 = 8

N = 3

EMI = 1014500*(8/100) {(1+ (8/100))3/ [(1+ (8/100))3-1]}

= 3,93,660

Total amount paid = 585500+393660*3 = 17,66,480

 Total interest paid = total amount paid – loan amount = 17,66,480-16,00,000 = Rs 1,66,480


4.12. Problem 3: If a borrower takes a loan of Rupees 10 Lakhs @ 7.5% PA for a period of 15 years find the EMI.





L = 10, 00,000

N = 180

E = 7.5/12 =0.625

 E/100 = 0.00625

(1+ (E/100))N = 3.069452

EMI = 1000000*0.00625*(3.069452/2.069452)

= Rs 9,270