Form 3 Mathematics

Chapter 9: Financial Mathematics 1

9.1. Introduction
9.2. Bank statement
9.3. Simple interest
9.4. Compound interest
9.5. Commission
9.6. Hire purchase
9.7. Summary
9.8. Further Reading
9.9. Test 9

9.1. Introduction

Everyone handles or deals with money everyday. In this chapter you are going to learn very important concepts on money. The knowledge you are going to acquire is going to be very useful in your everyday life. You are expected to understand every term which is going to be discussed so that you will be able to make correct calculations.

Objectives

After going through this chapter, you should be able to

Know what a bank statement is.
Interpret information found on a bank statement.
Calculate profit, loss, commission, discount, interests and instalments.
Solve problems in trading.

Key terms

Bank statement - A bank record of transactions made for the account holder by the bank over, usually a month.
Commission - The payment made to an agent for doing some work.
Compound interest - Non-fixed interest paid over a changing principal.
Deposit - The money that is put into the bank.
Depreciation - The decrease in value of goods over time.
Discount - The money deducted from marked price when you pay cash.
Hire purchase (HP) - Payment spread over an agreed period of time for an item which you cannot pay for cash.
Instalments - Part payment made to cover hire purchase.
Interest - The money you pay back when you borrow or is the money you get a after saving for a certain period.
Principal - The initial amount of money that was invested or borrowed.
Withdrawal - The money that is taken out of the bank.

Time

You should not spend more than 10 hours in this chapter.

Study skills

The key to mastery of mathematics is practice. You need to work out as many problems as possible about financial mathematics to have a better understanding of the subject.

9.2. Bank statement

A Bank statement shows records of all deposits and withdrawals made by the account holder over a period of time during the month of money kept in the bank. It also shows service charges to the bank for keeping the money in their bank. Table 9.1 shows a bank statement for Mr Jambe for the month of April 1 to 30 2022.

Table 9.1: Bank statement

Steward Bank Joina City Branch Current Account Statement USD Mr M. Jambe30 April 2022 121 Bishop Street
Date	Details	Withdrawals	Deposits	Balance
1 April	Balance brought forward			457.06
2 April	Cash		46.95	504.01
5 April	Cash	200.00		304.01
9 April	Cheque 0047	150.20		153.81
11 April	Cheque 63119		358.15	511.96
13 April	Bank charges	9.25		502.71
15 April	Cash		218.10	720.81
16 April	Cheque 0048	300.00		420.81
21 April	Stop Order insurance	90.50		330.31
24 April	Cheque 0049	116.20		214.11
27 April	Cash	50.00		164.11
28 April	Salary		465.46	629.57
30 April	Bank charges	15.92		613.65
	Total	932.07	1088.66

From the bank statement we can see that

all the money withdrawn and deposited is entered by dates and at the end of the month it is added to show the total amount withdrawn and deposited.
the balance is calculated at every transaction made and it is carried forward to the next month.
the bank has deposit forms and withdrawal forms which are used by customers to deposit and withdraw money.
customers are charged a certain fee for some transactions. Such charges are known as bank charges.
customers with Savings Accounts may receive interest for keeping their money in the bank. Those with Current Accounts are not eligible to getting interest.

You may attempt the following exercise.

Exercise 9.1: Bank statement

Questions

Use the bank statement in table 9.1 to answer the following questions.
1. What was the opening balance for April 2022?
2. What did Mr Jambe do to his account on 2 April 2022?
3. What happened to his balance on 2 April 2022?
4. How much was withdrawn on 16 April 2022?
5. What happened to the balance on 16 April 2022?
6. How much did Mr Jambe pay as service charges in April 2022?
7. How much money was on cheque number 63119?
8. How did the cheque in number 7 affect the balance on 11 April?
9. What do you understand by deposits and withdrawals?
10. How many deposits were made by Mr Jambe in April 2022?
11. If the amount which was withdrawn by cheque number 0048 was used to pay 6 people. How much did each person get if they were paid equal amount?

Answers

$457.06.
Deposit of $46.95.
Increased to $504.01.
$200.00.
Reduced to $420.81.
$25.17.
$358.15.
Increased to $511.96.
Deposit means putting money into the bank. Withdrawal means taking money out of the bank.
4.
$50.00.

You may attempt the following exercise.

Exercise 9.2: Bank statement preparation

Questions

Prepare a bank statement for the month of May 2022 using table 9.1 and the following information.
1. On 3 May, $65.50 was deposited.
2. On 7 May, a cheque of $100 was written to Mr Maposa and Mr Jambe withdraw $320 for diesel.
3. On 15 May, bank charges of $15 were made by the bank.
4. On 22 May, Mr Jambe’s salary of $465 appeared in the bank.
5. On 24 May, Mr Jambe got $500 from the bank to repair his car.
6. On 25 May, the bank cashed cheque number 0050 written by Mr Jambe.
7. On 31 May, the bank closed the columns of the bank statement.

Answers

Table 9.2: Bank statement

Steward Bank Joina City Branch Current Account Statement USD Mr M. Jambe31 May 2022 121 Bishop Street
Date	Details	Withdrawals	Deposits	Balance
1 May	Balance brought forward			613.65
3 May	Cash		65.50	679.15
7 May	Cheque	100.00		579.15
7 May	Cash	320.00		259.15
15 May	Bank charges	15.00		244.15
22 May	Salary		465.00	709.15
24 May	Cash	500.00		209.15
25 May	Cheque	50.00		159.15
31 May				159.15
	Total	985.00	530.00

9.3. Simple interest

Interest is money added by a bank to the sum deposited by a customer or money charged by a bank to a customer for borrowing from the bank. The money deposited is called principal. Interest is calculated on a fixed principal usually on yearly basis. If interest is calculated for five years and added to the principal, the sum is total amount.

The following terms are used when calculating interest.

Interest (I).
Principal (P).
Rate (R) – is usually given as a percentage (%).
Time (T) – is usually given in years.

The formula for calculating interest is

$ I = P \times R \times T $, or
$ I = {{P \times R \times T} \over {100}} $, if the rate is given as a percentage.

Example 1

Questions

Calculate the simple interest on $500 which was borrowed for 1 year at 10% per annum (p.a.)

Answers

$ I = {{P \times R \times T} \over {100}} $ P = $100, R = 10%, T = 1
$ I = {{500 \times 10 \times 1} \over {100}} $
$ I = $50 $

Example 2

Questions

Find the simple interest earned on
1. $350 deposited for 8 years at 4% per annum.
2. $40 deposited for 3.5 years at 5% per annum.
3. $150.40 deposited for 2 years 3 months at 12.5%.

Answers

$ I = {{P \times R \times T} \over {100}} $ P = $350, R = 4%, T = 8
$ I = {{350 \times 4 \times 8} \over {100}} $
$ I = $112 $
$ I = {{P \times R \times T} \over {100}} $ P = $40, R = 5%, T = 3.5
$ I = {{40 \times 5 \times 3.5} \over {100}} $
$ I = $7 $
$ I = {{P \times R \times T} \over {100}} $ P = $150.40, R = 12.5%, T = 2.25
$ I = {{150.40 \times 12.5 \times 2.25} \over {100}} $
$ I = $42.30 $

Example 3

Questions

How long will it take for a sum of
1. $400 invested at 6% per annum to earn interest of $48
2. $165 invested at 3% per annum to earn interest of $30.40
3. $250 invested at 8% per annum to earn interest of $60

Answers

$ I = {{P \times R \times T} \over {100}} $
$ T = {{100 \times I} \over {P \times R}} $
1. $ T = {{100 \times I} \over {P \times R}} $ I = $48, P = $400, R = 6%
  $ T = {{100 \times 48} \over {400 \times 6}} $
  $ T = 2 years $
2. $ T = {{100 \times I} \over {P \times R}} $ I = $30.40, P = $165, R = 3%
  $ T = {{100 \times 30.40} \over {165 \times 3}} $
  $ T = 6.14 $
  $ T = 6 years $
3. $ T = {{100 \times I} \over {P \times R}} $ I = $60, P = $250, R = 8%
  $ T = {{100 \times 60} \over {250 \times 8}} $
  $ T = 3 years $

Example 4

Questions

What rate per year must be paid for a principal of
1. $350 to earn interest of $60 in 2.5 years
2. $160 to earn interest of $45 in 3 years
3. $250 to earn interest of $120 in 7 years

Answers

$ I = {{P \times R \times T} \over {100}} $
$ R = {{100 \times I} \over {P \times T}} $
1. $ R = {{100 \times I} \over {P \times T}} $ I = $60, P = $350, T = 2.5 years
  $ R = {{100 \times 60} \over {350 \times 2.5}} $
  $ R = 6.86% $
2. $ R = {{100 \times I} \over {P \times T}} $ I = $45, P = $160, T = 3 years
  $ R = {{100 \times 45} \over {160 \times 3}} $
  $ R = 9.38% $
3. $ R = {{100 \times I} \over {P \times T}} $ I = $120, P = $250, T = 7 years
  $ R = {{100 \times 120} \over {250 \times 7}} $
  $ R = 6.86% $

9.4. Compound interest

In compound interest, not only is interest paid on the principal amount but interest is paid on the interest. This means it is compounded, that is, it is added to. This may sound complicated but the following example will make it clear.

Interpretation

Year 1: I = PR
Year 2: I = PR (This time principal will be principal for year 1 + interest for year 1)
Year 3: I = PR (This time principal will be principal for year 2 + interest for year 2)

Example 5

Questions

Calculate the compound interest earned on $300 at 10% per annum for 3 years.
Calculate the compound interest earned on $500 at 12.5% per annum for 4 years.

Answers

Year 1: P = $300, R = 10%
$ I = 300 \times {10 \over 100} $
$ I = $30 $

Year 2: P = $300 + $30 = $330, R = 10%
$ I = 330 \times {10 \over 100} $
$ I = $33 $

Year 3: P = $330 + $33 = $363, R = 10%
$ I = 363 \times {10 \over 100} $
$ I = $36.30 $

Therefore for the 3 years, the compound interest = $30 + $33 + $36.30 = $99.30.
Year 1: P = $500, R = 12.5%
$ I = 500 \times {12.5 \over 100} $
$ I = $62.50 $

Year 2: P = $500 + $62.50 = $562.50, R = 12.5%
$ I = 562.50 \times {12.5 \over 100} $
$ I = $70.31 $

Year 3: P = $562.50 + $70.31 = $632.81, R = 12.5%
$ I = 632.81 \times {12.5 \over 100} $
$ I = $79.10 $

Year 4: P = $632.81 + $79.10 = $711.91, R = 12.5%
$ I = 711.91 \times {12.5 \over 100} $
$ I = $88.99 $

Therefore for the 4 years, the compound interest = $62.50 + $70.31 + $79.10 + $88.99 = $300.90.

You may attempt the following exercise.

Exercise 9.3: Compound interest

Questions

Calculate compound interest on
1. $4000 at 10% for 2 years.
2. $800 at 9.5% for 3 years.
Peter saves $200 in an account which gives 6.5% per annum compound interest. Calculate
1. Total interest for 2 years.
2. Total amount in the account at the end of 2 years.
What is the value to which $700 will amount to in 3 years at 15 % per annum compound interest?
What is the
1. Simple interest on $800 at 10% per annum for 3 years?
2. Compound interest on $800 at 10% per annum for 3 years?

Answers

I = PR
1. Year 1: P = $4000, R = 10%
  $ I = 4000 \times {10 \over 100} $
  $ I = $400 $
  
  Year 2: P = $4000 + $400 = $4400, R = 10%
  $ I = 4400 \times {10 \over 100} $
  $ I = $440 $
  
  Therefore for the 2 years, the compound interest = $400 + $440 = $99.30.
2. Year 1: P = $800, R = 9.5%
  $ I = 800 \times {9.5 \over 100} $
  $ I = $76 $
  
  Year 2: P = $800 + $76 = $876, R = 9.5%
  $ I = 876 \times {9.5 \over 100} $
  $ I = $83.22 $
  
  Year 3: P = $876 + $83.22 = $959.22, R = 9.5%
  $ I = 959.22 \times {9.5 \over 100} $
  $ I = $91.13 $
  
  Therefore for the 3 years, the compound interest = $76 + $83.22 + $91.13 = $250.35.
I = PR
1. Year 1: P = $200, R = 6.5%
  $ I = 200 \times {6.5 \over 100} $
  $ I = $13 $
  
  Year 2: P = $200 + $13 = $213, R = 6.5%
  $ I = 213 \times {6.5 \over 100} $
  $ I = $13.85 $
  
  Therefore for the 2 years, the compound interest = $13 + $13.85 = $26.85.
2. Total amount = $200 + $26.85 = $226.85.
I = PR Year 1: P = $700, R = 15%
$ I = 700 \times {15 \over 100} $
$ I = $105 $

Year 2: P = $700 + $105 = $805, R = 15%
$ I = 805 \times {15 \over 100} $
$ I = $120.75 $

Year 3: P = $805 + $120.75 = $925.75, R = 15%
$ I = 925.75 \times {15 \over 100} $
$ I = $138.86 $

Therefore for the 3 years, the compound interest = $105 + $120.75 + $138.86 = $364.61.
Therefore $700 will amount to $700 + $364.61 = $1064.61 at 15% compound interest in 3 years.
1. Simple Interest (I) is
  $ I = {{P \times R \times T} \over {100}} $ P = $800, R = 10%, T = 3 years
  $ I = {{800 \times 10 \times 3} \over {100}} $
  $ I = $240 $
2. Compund Interest (I) is
  I = PR Year 1: P = $800, R = 10%
  $ I = 800 \times {10 \over 100} $
  $ I = $80 $
  
  Year 2: P = $800 + $80 = $880, R = 10%
  $ I = 880 \times {10 \over 100} $
  $ I = $88 $
  
  Year 3: P = $880 + $88 = $968, R = 10%
  $ I = 968 \times {10 \over 100} $
  $ I = $96.80 $
  
  Therefore for the 3 years, the compound interest = $80 + $88 + $96.80 = $264.80.

9.5. Commission

Commission is payment made to a worker relating to the number of sales the worker has made. Commission is paid to encourage people to work hard without being supervised. Commission is often paid to staff in the sales department. The more sales they make, the more money they are paid. This encourages the staff to sell as many products as possible. Ice cream vendors, insurance agents and bus drivers are among some of the people who may work on commission. Can you think of others? Commission is calculated as percentage of what one gets through the sale of goods.

Example 6

Questions

The bus driver gets $500 per month as salary and 5% of the total money she or he cashes in per month. Calculate
1. the commission she or he will receive if she or he cashes in $8,000 per month.
2. the total pay for that month.

Answers

Commission = 5% of $8,000
$ = {5 \over 100} \times 8000 $
= $400
Total pay = Salary + Commission
= $500 + $400
= $900

Example 7

Questions

From the previous example, if the driver cashes in $6,000 the following month, calculate
1. the commission.
2. the total take home salary.

Answers

Commission = 5% of $6,000
$ = {5 \over 100} \times 6000 $
= $300
Take home salary = Salary + Commission
= $500 + $300
= $800

Example 8

Questions

From the previous example, if during the third month the driver cashes in $10,000, calculate
1. the commission.
2. the month's take home salary.

Answers

Commission = 5% of $10,000
$ = {5 \over 100} \times 10000 $
= $500
Month's take home salary = Salary + Commission
= $500 + $500
= $1,000

Exercise 9.4: Commission

Questions

Chipo sells tickets for watching soccer, and she is paid $1 per every tickets she sells. How much will she get as commission after selling 2000 tickets?
A car sales lady get 2% in $1. Calculate her commission if she sold $60,525 worth of cars that month. What will be her total salary if her basic salary was $230?
A textbook cost $12.50. The author gets 15% per text book sold. In 2017, 10,000 copies of that book were sold. Calculate the author’s commission.
A man collects rent at a commission of 6.5%. In June 2018 he collected $12,860. Calculate
1. his commission.
2. the money he took home in June, if his monthly salary was $156.
A bank charges 3.5% commission on loans issued out. Calculate the commission the bank gets after issuing loans worth $100,000.
An electrical company worker gets 12 cents in every dollar as commission. One month the worker sold 4 stoves at $2,500 each and 2 television sets at $3,150 each. What amount did the worker get at the end of the month as commission? If the worker’s basic salary was $350 per month, what is the total amount of money taken home by the worker?

Answers

$2,000
Commission is $1,210.50.
Total salary is $1,440.50.
$18,750
1. $835.90
2. $991.90
$3,500
Commission is $1,956.
Total salary is $2,306.

9.6. Hire purchase

Hire purchase allows individuals or businesses to buy fixed assets over a long period of time while making monthly payments which include an interest charge. A person does not have to find a large sum of money to purchase an asset of choice. However, a cash deposit is paid at the start of the period and sometimes interest rates can be quite high which means that buying items through hire purchase is more expensive than paying cash. The interesting thing about hire purchase is that a person is given his/her item/asset of choice as soon as the processes for buying through hire purchase have been initiated and the customer’s eligibility to make monthly payments has been verified.

Example 9

Questions

A new television set costs $850 in cash, or a deposit of $180 is required and 12 monthly payments of $60. Calculate the hire purchase price and find the saving when bought through cash.

Answers

Total Instalments = 12 × $60 = $720
Hire purchase price = Deposit ($180) + Total instalments ($720) = $900
Saving, when bought through cash = Hire purchase price - Cash price
Saving, when bought through cash = $900 - $850 = $50

Example 10

Questions

The price of a car by hire purchase is $26,500. A 30% deposit is required. The remainder is spread over $ 1 {1 \over 2} $ years monthly instalments.
1. How much is the deposit?
2. What is the total amount to be paid through the instalments?
3. Calculate the monthly instalment.

Answers

Deposit = 30% of $26,500
$ = {30 \over 100} \times 26500 $
= $7,950
Total amount to be paid by instalments = $26,500 - $7,950 = $18,550.
Monthly instalment = Total instalment amount / Time to pay
Time to pay = $ 1 {1 \over 2} $ years = 18 months
Monthly instalment = $ {$18550 \over 18} $ = $103.56.

You may attempt the following exercise.

Exercise 9.5: Hire purchase

Questions

A man can buy a bicycle by either paying $930 cash or paying $85 monthly instalment for 13 months.
1. Calculate the cost of the bicycle through instalments.
2. What is the difference between cash price and the hire purchase price?
3. Express the difference as percentage.
A car costs $9,865 cash. Hire purchase requires a 15% deposit and 36 monthly instalments of $300.
1. How much does it cost on hire purchase.
2. How much is saved if bought for cash?
A refrigerator cost $745 cash, it can be bought by hire purchase but its price increases by 15%. If 10% is required as deposit and remainder is spread over 12 equal monthly instalments. Calculate the
1. Hire purchase price.
2. Deposit.
3. Monthly instalments.

Hint: The percentage is calculated on the original amount. Express increase or decrease as a fraction of the original amount then multiply by 100.

Answers

1. $1,105.
2. $175.
3. 18.8%.
1. $12,279.75.
2. $2,414.75.
1. $856.75.
2. $85.68.
3. $64.26.

9.7. Summary

In this chapter we learned that banks give services to both people and companies as their customers and/or clients. The customers are charged for the service they get from the bank. The bank also provides statements at the end of each month to its clients.

We also learned about simple interest and compound interest. In simple interest the principal is fixed, and in compound interest the principal changes. Hire purchase involves deposits and instalments. Hire purchase is more expensive than buying in cash.

9.8. Further Reading

Macrae, M. F., Madungwe, L. and Mutangadura, A. (2017). New General Mathematics Book 3. Pearson Capetown.
Macrae, M. F., Madungwe, L. and Mutangadura, A. (2017). New General Mathematics Book 4. Pearson Capetown.
Meyers, C., Graham, B., Dawe, L. (2004). Mathscape Working Mathematically. 9th Edition. MacMillan.
Pimentel, R. and Wall, T. (2011). International Mathematics. Hodder UK.
Rayner, D. (2005). Extended Mathematics. Oxford New York.

9.9. Test 9

Questions

The cash price of a table is $550 or a deposit of $110 is required. The remainder is increased by 15% and paid in 12 monthly instalments. Calculate
1. The total amount to be paid by instalments.
2. Each instalment to be paid monthly.
3. The percentage increase over cash price.
Calculate the simple interest on $600 which is saved for 2 years at 11% per annum. What is the total amount after 2 year?
Find the percentage rate per year if simple interest of $160.50 is paid on $4,570.00 invested for 6 months.
If $500 is deposited in the bank, how long does it take to become $62.00 if the simple interest rate is 6% per annum.
A bank charges $36 simple interest on a sum of money borrowed for 3 years. If the rate of interest is 12% per annum, find the sum of money.
Calculate the compound interest earned on $850, at $ 12 {1 \over 2} $ for 5 years.

Answers

1. $506.
2. $42.17.
3. $12%
$132, $732.
7%.
2 years.
$100.
$681.33.