1. Simple interest and compound interest of installment payment
Example 1. A person borrows1000000 yuan from a bank, and repays it in five equal installments with the interest rate of R.
(1) According to compound interest (the current interest is included in the next principal interest), how many ten thousand yuan should be repaid in each period?
(2) According to the simple interest calculation (the interest of the current period is not included in the interest of the principal of the next year), how many ten thousand yuan should be repaid in each period?
Solution: (1) Method 1: Assuming that X million yuan must be repaid in each installment, the amount owed after the first installment expires is:
The amount owed after the expiration of the second installment is:
The amount owed after the expiration of the fifth installment is:
As it is paid off in five installments, that is.
Method 2: Assume that each installment has to pay X million yuan, and the first installment has to pay X million yuan, which is equivalent to settlement.
Ten thousand yuan;
In the second installment, RMB 10,000 is returned, which is equivalent to the amount at the time of settlement.
Ten thousand yuan;
……
In the fifth installment, it will still be X million, and it will still be X million at the time of checkout.
Because the sum of the five periods = the sum of the principal and interest of the ten thousand yuan deposited in the bank, that is
therefore
Method 3: If each installment repays X million yuan, then
The first repayment of X million yuan is equivalent to a loan.
Ten thousand yuan;
The second repayment of X million yuan is equivalent to a loan.
Ten thousand yuan;
X million yuan repaid in the fifth installment is equivalent to a loan.
Ten thousand yuan.
By the known:
(2) assuming that each period must pay X million yuan, then
The first payment is X million yuan, which is equivalent to the amount at the time of checkout.
Ten thousand yuan;
The second is to pay X million yuan, which is equivalent to checkout.
Ten thousand yuan;
……
In the fifth installment, it will still be X million, and it will still be X million at the time of checkout.
Because the sum of five periods =a ten thousand yuan deposited in the bank for five periods, that is,
As can be seen from the above calculation, if a bank loan is paid off in equal installments with the interest rate of R, and each installment must be paid back by X yuan, then
(compound interest form);
A simple form of interest.
Deposit money in the bank
Example 2. If a classmate has an allowance of 5 yuan per month and deposits it in the bank at the end of the month, the monthly interest rate will be compounded by 0.2%, and the annual interest rate will be compounded by 6%. Ask him how much interest he will pay each month after three years (put it in one place).
Analysis: First, analyze the sum of deposit principal and interest in each year (unit: yuan). Separated by month
The first month:;
The second month:;
……
Twelfth month: 5.
Then, the sum of the annual principal and interest is:
One yuan in the first year is converted into annual interest, calculated by compound interest, and the sum of principal and interest due after two years is:
One yuan in the second year. Similarly, the sum of principal and interest due after one year is:
One yuan in the third year, because it has been completely withdrawn, there is no interest on the deposit in this year, and the principal and interest after three years are:
Solution: Let the sum of principal and interest of each deposit year be A, then
Let the principal and interest withdrawn after three years be y.
A: After three years, the principal and interest will be *** 193 yuan.
3. Income comparison of installment payment
Example 3. In order to promote the sales of goods, a store has two payment methods for customers to choose, that is, to buy a home appliance. The value of household appliances is 2 150 yuan.
The first payment method: pay RMB 150 yuan on the day of purchase, and then pay it to 200 yuan on the first day of each month, plus interest on arrears. The monthly interest is compound interest 1%.
Second payment method: pay 150 yuan on the day of purchase, and then pay once a month, for a total of 10 months. The monthly payment is unchanged, and the monthly interest is calculated according to compound interest, and the monthly interest is 1%.
Try to compare the two payment methods, and calculate the monthly payment and total payment for the purchase of this household appliance.
Solution: The first payment method: If you pay 150 yuan at the time of purchase, you will owe 2,000 yuan, and you need to pay in 10 times as required, then later.
The first time processing:
(yuan)
The second payable:
……
N th Payable:
The amount of each payment constitutes a series in sequence, the first item is 220, and the tolerance is arithmetic progression.
The sum of 10 payments is:
Therefore, the actual * * * payment is 2260 yuan.
The second payment method: 65,438+050 yuan is paid at the time of purchase, and the remaining 65,438+00 months' value-added is:
Let's repay X yuan every month, then the sum of the monthly repayment plus the interest generated during the last repayment is: this constitutes a geometric series, and its sum is:
The monthly payment is 2 1 1.2 yuan, 10 lump sum payment is 2 1 12 yuan, and the actual payment is 2262 yuan, which is more affordable than the former.
Example 4. At a job fair, Company A and Company B set their own salary standards: Company A promised a monthly salary of 1500 yuan in the first year, and the monthly salary increased by 230 yuan over the previous year; Company B promises to pay 2000 yuan in the first year, and the monthly salary will increase by 5% on the basis of the previous year. Suppose someone was admitted to both Company A and Company B at the beginning of the year. Excuse me:
(1) If this person has worked in Company A or Company B for N consecutive years, what is his monthly salary in the N-th year?
(2) The person intends to work continuously in a company for 65,438+00 years, and the application standard is only the total salary (excluding other factors). Which company should this person choose and why?
Solution: (1) Let's assume that the monthly salary of this person in Company A and Company B in the nth year is respectively. It is known that arithmetic progression with 1500 as the first term and a tolerance of 230 is a geometric series with 2000 as the first term. Then,
(2) If the person has worked in Company A for 10 years, his total wage income is ().
If this person has worked in Company B continuously for 10 years, his total wage income is:
Because the total income in company A is higher, this person should choose company A. ..
Simple interest and compound interest of installment payment; The problem of bank deposits and loans and the comparison of income from installment payment are closely related to people's lives. When solving this kind of problem, we should carefully analyze and straighten out the relationship between various quantities in the problem, and then use the mathematical knowledge we have learned to solve the problem and make correct and reasonable answers.
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