Commercial and Household Arithmetic

TEACHER’S GUIDE

SUBJECT : MATHEMATICS

TOPIC : COMMERCIAL AND HOUSEHOLD ARITHMETIC

SUBTOPIC : PROFIT AND LOSS, COMMISSION, INSURANCE, INTEREST AND DISCOUNT

CLASS : SENIOR ONE

TIME REQUIRED : 160 minutes (4 periods)

Brief Description of Unit

Every now and then we use items and services such as food, toothpaste, transport, telephone, water, electricity, fuel, and many others such as these in a bid to make our lives better and more fulfilling. These items or services are obtained in exchange for money; that is, we pay for each and every one of them.

And whenever money is spent there are quite a number of parties that stand to benefit from the transaction. The business owners may make a profit; the salespersons may earn a commission, while the customers may receive a discount.

If the business owners take out a loan from their bankers, then a sale means that they are able to pay interest on the loan, and should they opt to insure their businesses against the cruel acts of nature or mankind, then the insurance companies will be able to have a share of the profit through insurance premiums. Commercial and business arithmetic helps to show us what really happens in the world of commerce, the world to which you and I belong; it helps us appreciate the need to know how to count our money, both as we make it and as we bid farewell to it.

This unit will help the learners to understand the terms used and to apply this knowledge in their day-to-day life.

Objectives

By the end of this subtopic, the learners should be able to:

1. Distinguish between profit and loss
2. Define the term commission and calculate the commission due from a sale
3. Calculate the interest on a loan
4. Calculate the discount on a sale or purchase
5. Calculate insurance premiums

Job Related Life Skills

Demonstrate through activities the necessary job related life skills, namely:

1. Personal attributes – saving culture, Hard work, bargaining, Prudence, Courtesy
2. Communication – ability to read, write, listen and speak in appropriate ways for different audiences. Know and apply general and specialised vocabulary.
3. Problem solving – Budgeting decision making
4. Application of Numbers – Numerical skills.
5. Information skills – Record keeping

CONTENT AND CONCEPTS

Profit and loss

The most important desire for most businesses is to make more money than they invested. This difference between what has been realised and what was invested is referred to as profit.

Since business is about buying and selling of goods and services, we refer to money invested as buying price or cost price and that obtained as selling price.

Then Profit = Selling Price (S.P) – Cost Price (C.P)

Percentage Profit = Profit x 100

Cost Price

However, there are times when the money obtained (selling price) is less than the money that was invested (buying price). This undesirable difference (the negative profit) is referred to as a Loss. This is expressed as:

Loss = Cost Price (C.P) – Selling Price (S.P)

Percentage Loss = Loss x 100

Cost price

Activity one

Role-plays of Buying and Selling

Learners act as shopkeepers and customers.

Materials required

These may include - A bar of soap, matchboxes, exercise books, pens, pencils, geometry sets, (and any other readily available materials), money (notes and coins), paper bags and old newspapers for packing customers’ items.

Tasks to perform

1. The shopkeeper shall

• Devise means of attracting customers
• Convince customers to buy his/her merchandise
• Serve customers – offer genuine advice, pack products, give customers their balance
• Earnestly thank customers for doing business with him/her
• Ask customers to drop by next time and smile as he/she hands over the customer’s items.

2. The customer shall:

• Inquire whether the shopkeeper has certain items
• Ask for the price of each item
• Bargain for a discount

About three sets of students can do the role-play as the rest of the class observes. Give the learners time to comment on the qualities exhibited by the shopkeeper and why they think those qualities are necessary in life in general and in business in particular.

Some of the qualities expected to be exhibited by the shopkeeper are: communication skills, courtesy, numerical skills, interpersonal skills, etc.

Worked examples

The cost of manufacturing a school shirt is UGX 8,000. A manufacturer makes a profit of 20% while a wholesaler makes a profit of 25% on selling a shirt.

1. How much does the manufacturer charge for the shirt?
2. Find how much the customer pays for the school shirt.

Solution

1. Manufacturer’s profit = 20% x 8,000

= UGX 1,600

.^.. Manufacturer’s selling price = UGX 8,000 + UGX 1,600

= UGX 9, 600

2. Wholesaler’s profit = 25% x 9,600

= UGX 2,400

.^.. Wholesaler’s selling price = UGX 9,600 + UGX 2,400

= UGX 12,000.

The customer pays UGX 12, 000 for a school shirt.

UNEB 1987/2

Question.7 An article costing UGX 280,000 was sold making a profit of 20% of the selling price. Find the selling price.

Answer; UGX 336,000

UNEB 1988/2

Question.16 a) The cost of printing 4000 books was UGX 1.2 million. A bookshop bought all the books by paying all the printing costs plus 30% .The books were then retailed at UGX 500 per copy. Find the percentage profit made by the bookshop.

Answer: 28.2%

UNEB 1989/1

Question.9 Kapere bought 0.5km of wire from Uganda cables for UGX 25,000 and sold it at shs.75/= per metre find his percentage profit

Answer: 50%

UNEB

Question.16a) A shopkeeper had two similar television sets for sale marked UGX 75,000 each. He sold them to two men. Mr. Kato was allowed a 15% reduction after hard bargaining. Mr. D. High class bought at the marked price. The shopkeeper made a profit of 25% on his cost price in the sale to Kato. Find

1. How much the shopkeeper had paid for each television set?

Answer: UGX 51,000

ii) The profit from the sale to Mr. High class as a percentage of the cost price to the shopkeeper.

Answer: 47.1%

UNEB 1990/2

A shirt and a pair of trousers were each sold at UGX 6,000. The shirt was sold at a profit of 25% and the pair of trousers was sold at a loss of 20%. Find the percentage loss on both articles

UNEB 1997/2

Question.8 a blouse and skirts were each sold at UGX 12,420. On the blouse a profit of 15% was realised while on the skirt a loss of 12.5% was recorded. Calculate the percentage loss on both articles.

UNEB 1998/1

Question.3 Kato bought a car and sold it to tom at a loss of 25%. If his selling price was UGX 3.6million, find the cost price of the car.

Answer: UGX 4, 800,000

UNEB 2001/2

Question.6 Olga bought a motorcycle and sold it to Okello at a loss of 25%. If he sold it at UGX 1,200,000, find how much money Olga paid for it.

Answer: UGX 1,600,000

Interest

In everyday life people and companies borrow money from moneylenders in order to conduct their businesses and solve their problems. But the money that they borrow comes at a cost; all borrowers are required to pay back slightly more than what was borrowed, and this extra money paid is the cost of borrowing, or what is referred to as interest.

In brief, both the moneylenders and the borrowers refer to the money lent out or borrowed as principal (or a loan) and the extra money returned to the lender, the cost of the loan, as interest. This interest is usually a percentage (or fraction) of the principal and is referred to as the interest rate. When the interest is added to the principal then we obtain the amount to be returned to the moneylender.

There are two types of interest; simple interest and compound interest.

At this level, we shall only look at simple interest.

Simple interest

This is when we obtain interest on the principal only, no matter how long the period may be. It is a simple percentage of the principal in a given period, which may be expressed as:

Simple interest = Percentage of the principal in a given period

• r% of principal,

Where r is the interest rate given in unit period (unit means one, say one month or one year.)

Now if simple interest is denoted by I, principal by P and time period by T, then we obtain

I = PRT or I = P r T

100

Note that when we take T to be 1 year, any period less than 1 year is a proper fraction of 12 months (because there are 12 months in a year) and all periods above 1 year are either whole numbers or mixed fractions.

Worked example

1. A) Find the simple interest on UGX 1,800,000 for 4 years at 10% p.a

b) Find the amount after this time

Solution

a) Simple interest I = PRT

= UGX1,800,000 x 10 x 4

100

= UGX 720,000

b) Amount = Principal + interest

= UGX 1,800,000 + UGX 720,000

= UGX 2, 520,000

2. Find the rate at which UGX 300,000 needs to be deposited to earn a simple interest of UGX 12,000 in 10 months.

Solution

I = PRT

12,000 = 300,000 x R x 10/12 (changing 10 months to years)

R = 0.048

.^.. R = 4.8%

UNEB 1990/1

Mr. Mugabi put UGX 2,400 in his savings account at the bank. The bank’s simple interest rate was 5% p.a. Find the number of years he should leave the money in the bank in order to be able to receive a total sum of UGX 2,700.

Answer = 2½ years

UNEB 1992/1

A man borrowed UGX 200,000 from the bank at a simple interest rate of 2.5% per annum. He paid back the money in 24 equal monthly instalments over a period of two years. How much money did he pay every month?

Answer = UGX 8,750

UNEB 2005/1

Question.6 Calculate the simple interest on UGX 96,000 for 10momths at a rate of 8¹/₃% per annum.

Answer: UGX 6,666.67

Discounts

Everybody loves to buy the best products at a cheaper price. That is why we rush to buy products when the seller announces that they have reduced prices. The amount that the sellers remove from the original price is known as a discount.

(Insert photos showing discounts on window displays, billboards etc)

There are many reasons for offering a discount, which may include:

• Boosting sales so that more goods are sold, especially if the goods have been sold at a lower pace.
• Encouraging bulk purchases whereby those who buy more goods pay less for each one
• Encouraging cash purchases so that those who buy goods pay in cash.
• Encouraging prompt payments so that those who buy goods on credit pay in time.

There are also several types of discounts, depending on the aim of the salesperson offering it.

I) A cash discount is offered when goods are paid for in cash

ii) A trade discount is offered when someone in the same line of business buys goods.

iii) A credit discount is offered to buyers who pay for the goods that were taken on credit before a certain period expires.

We also have quantity discounts, which are given to the customers who buy in bulk and sales discounts, given to every customer who buys something, among others.

Methods of calculating discounts

We can calculate the discounts to be given or received using any of the following methods

(a) When a single percentage (ratio) is used

Here the sales person offers a single discount rate for all purchases that meet a certain condition; for example, buying goods exceeding a certain amount or paying for them in cash.

We use two formulae; one for determining the discount (amount) and the other for calculating the discount rate (percentage). But some problems will only be solved using algebra, especially when only one of the variables in each formula is given.

Discount offered = original price – New price

And Percentage discount = Discount offered x 100%

Original price

Example

Women stationers bought books from TK Publishers worth Shs.550, 000 but were asked to pay only Shs.505, 000

a) What was the trade discount given to them?

b) Calculate the percentage discount they received.

Solution

a) Original price is Shs.550, 000

New price is Shs.505, 000

Discount offered = Shs.550, 000 – Shs.505, 000

= Shs.45, 000

b) Percentage discount = Discount offered/Original price x 100%

= 45,000/550,000 x 100%

= 8%

(b) When more than one percentage is used

We may also be given a number of discount rates corresponding to different quantities of the goods bought.

For example, a salesman may offer a 2% discount on goods below UGX 2,000,000 and 5% discount on goods above UGX 2,000,000. so if Ahmad buys goods worth UGX 4,500,000 here is how we calculate his total discount.

Step 1: Find the discount corresponding to the first discount rate

2/100 x 2,000,000/- = 40, 000/-

Step 2: Find the discount corresponding to the second discount rate i.e. amount above 2,000,000/- is

= Shs4, 500,000 - 2,000,000/-

= shs2, 500,000/-

Discount = 5/100 x 2,500,000/-

= 125, 000/-

Step 3: Then add the discounts to obtain the total

I.e. Total discount = 40, 000/- + 125,000/-

= 165, 000/-

If a general percentage discount is required, the next step is to calculate it using the total above

Percentage discount = Discount offered/Original price x 100%

= 165,000/4,500,000 x 100%

= 3.67%

Sample Question from UNEB past papers

UNEB 1991/1

Question.7 A man bought a shirt at 20% discount. If he paid UGX 2,000, find the original price of the shirt.

Answer: UGX 2,500

UNEB 1994/1

Question.8 a) in a showroom the price of a car is given as UGX 5,800,000. During sale a discount of 15% is allowed. How much does a customer pay for a car?

Answer: UGX 4, 930,000

UNEB 1999/2

Question.3 The price of a car in a showroom is UGX 8.4 million. A 8.5% cash discount is allowed when a customer pays cash for the car. Determine how much a customer pays by paying cash.

Answer: UGX 7,686,000

UNEB 2001/1

Question.7 Find the discount on a bicycle priced UGX 64,000 but sold off at a discount of 7½ %. How much was paid for it?

Answer: UGX 59,200

UNEB 2004/2

Question.11b) a cooking oil factory offers a trade discount of 2% to its customers. It also offers a 1% cash discount to any customer who pays cash for the oil bought. If the factory price for a 20 litre jerrycan of cooking oil is UGX 30,000. Find the amount of money a customer saves by paying cash for 100 jerrycans of the oil.

Answer: UGX 29,400

(c) Using ready reckoners

We can also calculate discounts by using a table containing values for each discount rate that the company may offer. Such tables are known as ready reckoners. To reckon means to calculate. So a ready reckoner is a table of computed values that are ready to use.

Consider the table below

Discount rate

Price (Shs) 2% 5% 7% 10% 12%

50,000 1,000 2,500 3,500 5,000 6,000

70,000 1,400 3,500 4,900 7,000 8,400

90,000 1,800 4,500 6,300 9,000 10,800

120,000 2,400 6,000 8,400 12,000 14,400

150,000 3,000 7,500 10,500 15,000 18,000

A ready reckoner has the discount rates in the first row and the original prices in the first column. The rest are the respective discounts. Example 1 shows how to use the ready reckoner.

Example 1

If a salesman decides to offer a 2% discount on an item in his warehouse, then an item marked at 50,000/-

a) Find the discount from the ready reckoner

b) Find the new price

Solution

a) UGX 50,000 receives a discount of UGX 1,000 at 2% (refer to the table)

b) Its new price will be UGX 50,000 - UGX 1,000 = UGX 49,000.

Example2

If an item costing 150,000/- is to receive a discount of 7%,

Find i) the discount using the ready reckoner

ii) The new price

Solution:

i) Discount = UGX 10,500

ii) Its new price will be UGX 150,000 - UGX 10,500 = UGX 139,500.

Activity 2

1. Using the ready reckoner above find the discount given for an item marked at

i) 90,000/= at 12% ii) 70,000/- at 2%

iii) 120,000/- at 5% iv) 100,000/- at 7%

2. Copy and complete the ready reckoner below:

PRICE/shs	DISCOUNT RATES
	3%	8½%	16%	20%	30%
50,000	1,500
70,000			11,200
90,000					27,000
120,000				24,000
150,000		12,750

Commission

An agent may be hired to sell goods on behalf of the company with the agreement that he/she will receive a part of the selling price. But sometimes a business manager may also wish to motivate his workers to sell more goods by offering them a part of the selling price on top of their salaries. This fraction or part of the selling price is known as a commission. It is sometimes expressed as a percentage of the selling price; for example, a commission of 5% of the selling price.

Like discounts, commissions can also be calculated using any of the methods we saw above.

Example

Apollo sold factory equipment worth 4 million shillings at a commission of 4 %. How much money did he earn?

Solution

Commission = selling price x Percentage Commission

= UGX 4,000,000 x 4/100

= UGX 160,000

Example

A commission agent earned a commission of UGX 160,000 when he sold goods worth UGX 2,000,000. What was his percentage commission?

Solution

Percentage commission = Commission/Price x 100 %

= 160,000/2,000,000 x 100 %

= 8%

UNEB 1989/2

Question.9 As part of his pay a land agent is normally given commission which is a percentage of the sales price. The rates are as follows:

Sales price	commission
First 1million shillings	3%
Anything above 1 million	13/4%

Find his total commission on a farm, which is sold for shs.3.5 million shillings

The answer is UGX 73,750

Ready reckoners for commissions

This is a table that gives different commissions for different sales at different rates.

Sales (Shs) 5% 7% 10% 12% 15%

500 25 35 50 60 75

1000 50 70 100 120 150

1500 75 105 150 180 225

2000 100 140 200 240 300

2500 125 175 250 300 375

3000 150 200 300 360 450

Example

Use a table of ready reckoners to calculate the commission on shs.700 at 5%.

Solution

In the table, 700 do not directly exist. So we split it into values, which can be read from the table at 5% commission.

So shs.700 = 500 + 200

Commission on 500 at 5% = Shs. 25, and commission on 200 at 5% = Shs. 10 (not on table, but it is the same as the commission on 2000/200

= Shs. 100/10

= shs.10

Therefore, total commission on Shs.700 = 25 + 10

= Shs. 3

INSURANCE

According to Dr. Olli-Pekka Ruukskanen, C.E.O, Uganda Insurers Association, the main benefit of insurance is to increase a person’s welfare by taking away the uncertainty about the consequences of adverse future events.

Insurance, therefore, is about protecting oneself, the family and property in case of unfortunate eventualities for instance risks, which involve fire, theft and others such as health, death and many others such as these.

On the side of insurance companies, taking on other people’s risks makes business sense because the majority of risks that are insured against may not occur, so the company can still make profits.

A written contract between the individual/business and an insurance company is called an insurance policy. This agreement spells out clearly what risk is being protected. The individual/business agrees to pay an amount of money once or regularly for the insurance policy. This money is called the premium.

8.6.2 Types of insurance policies

An insurance policy entirely depends on the risk an individual/business wants to insure against.

Examples include:

 Fire policy  Theft and burglary

 Third party risk policy  Motor vehicle policy

 Fidelity guarantee  Marine insurance policy, etc

 Life insurance policy

It is not so important to understand all the related terminologies, what matters is knowing the fact that insurance can cover everything you own, including your wealth and your health.

Activity 3

1. Ask the learners to name at least five Insurance companies in their country.
2. In groups of 5 – 8 members, learners may visit any insurance company of their choice to find out the following;

1. The actual meaning of the term Insurance
2. The different terms used in insurance
3. The types of insurance policies undertaken
4. The relationship between insurance companies and the business community
5. How insurance companies compensate those who suffer the insured risks
6. How insurance companies come up with premiums or make profits

2. The importance of someone insuring his/her life, business or property
3. The challenges involved in the insurance business

Caution

Ensure that each member in a group writes down or records his/her findings, which they should discuss together when they return to school and come up with a general report to be presented to the rest class.

Use the dialogue method for groups to present their findings to the rest of the class in 5-10 minutes

Task 2

Alternatively, you can invite a resource person from any insurance company to your class to discuss pertinent issues about insurance.

Learners should take notes of any relevant points and you should allow them to freely interact with this resource person.

Example

Mukwano has a life insurance valued at UGX 5,000,000. She pays an annual premium of 18% of the cover value. Calculate her annual premium.

Solution

Annual premium = 18% x 5,000,000 = UGX 900,000.

Example 2

Blessed Alice insures all the goods in her shop to the value of UGX 7,800,000 so that, in the event of any loss, she may recover the value of her goods. Given that the premium is 24% of the total cover, find the amount of money due when her goods are stolen.

Solution

Amount of money = cover value + premium

= 7,800,000 + (24% x 7,800,000)

= UGX 9,672,000

Activity 4

Read the story below and answer the questions that follow

Mr. and Mrs. Kiwanuka have 12 acres of land that they use as follows:

Land Use	Acreage
Banana plantation	3
Coffee plantation	4
Pasture	2
Home and farm structures	1
Rested land	2

They have three children; Cynthia (20 years old), Charles (17 years old) and John (14 years old). During the December holidays the children were assigned the responsibility to manage the above farm activities. Cynthia is placed in charge of the banana plantation; Charles takes charge of the coffee plantation while John looks after the 4 Friesian cows.

The three are supposed to make daily records of whatever takes place on the farm. A summary of the two-month holiday is given here below:

Maintenance Costs Produce/Yield

a) Banana plantation Shs 475,000 Banana 1,280 bunches

b) Coffee plantation Shs 380,000 Dry coffee 5,800 kg

c) Dairy feeds & drugs Shs 532,000 Milk 4,857 litres

Cow dung 3 tonnes

Urine from 4 cows 2,520 litres

The family sells the farm produce as follows:

Bunch of matooke Shs 12,500

1 litre of milk Shs 900

1 kg of dry coffee Shs 850

Jessica and Jack Grocery Store buy all the banana and milk produced by the farm, while Hillary Coffee Processors buys all the coffee from the farm.

Jack and Jessica have a clean, spacious stall where they sell each bunch of matooke at Shs 16,000 and each litre of milk at Shs 1,100. Due to the great customer care that they have, their customers keep coming back as well as recommending them to their friends. This has helped them to build a beautiful house and put their children in school.

Hillary processes the coffee by removing the husks, sorting the beans and packaging them for export. For every 100 kg of dry coffee he gets 78 kg of processed coffee, which he sells at Shs 1,350 per kilogram. Then he sells off the coffee husks to farmers at Shs 150 per kg.

Questions

1. How much money did Kiwanuka’s family make from the sale of
   1. Banana produce?
   2. Coffee produce?
   3. Milk?
2. What was the total expenditure on the farm during the two months?
3. (i) Which of the projects had the highest expenses?

(ii) Did this project generate any profit or loss? Give reasons for your answer.

4. Jessica and Jack Grocery Store hired a lorry driver to transport 600 bunches of matooke to their stall. On its way the lorry got off the road and overturned due to over-speeding. 120 bunches were totally destroyed and the rest were taken to the stall two days later. Since they were about to ripen, Jessica advised Jack to reduce the unit price of the bananas to Shs 13,000. Still they paid Shs 270,000 to the lorry driver as the cost of transport. Luckily, Jack had foreseen the likely risks of his business and had taken out a comprehensive insurance policy. The lorry driver was not as lucky because he had not insured his vehicle. He ended up selling his Shs 38,000,000 truck as scrap at Shs 2,850,000.
   1. Calculate the percentage loss made by the grocery
   2. How much did the insurance company pay to the grocery store as compensation for the loss?
   3. What was the lorry driver’s loss?
   4. What type of insurance would have saved him this loss?
5. Usually Mr. and Mrs. Kiwanuka give each child a commission of 2% of the net profit earned from the produce of the project, which he/she was supervising. How much commission did each child get?
6. The family saved 30% of the profit from all the produce and kept it with a bank at an interest rate of 14% per annum. Calculate the interest obtained after three years.
7. Hillary Coffee Processors sold all the processed coffee to an exporter at a discount of 3%
   1. How many kilograms did he get after processing?
   2. Calculate the discount the exporter got
   3. How much did Hillary Coffee Processors obtain from the sale of coffee husks?
   4. If Hillary incurred processing costs worth Shs 540,000, calculate his percentage profit.
8. (i) Which other products can earn income for Mr. and Mrs. Kiwanuka from their farm?

(ii) Which type of customers need the products in (i) above?

9. Of what use are coffee husks?
10. What could Mr. and Mrs. Kiwanuka do with the land that has been under rest for the next two years? Give reasons for your answer.