Commercial Arithmetic

The range of topics that we call commercial arithmetic is used mainly in business transactions and they include rates, ratios, proportions, instalments and percentages. The knowledge of percentage is very important when we are dealing with simple interest, discounts and commission.

Ratios and Rates

Ratios

This is a numerical way of comparing quantities of the same kind. To express quantities in ratios, they must be expressed in the same units. It is usually expected that ratios are expressed in the simplest form possible.

**Example 1:** A radio costs ₦2500 and a television set costs ₦3000. What is the ratio of the costs of radio to television?

Solution:

Radio: Television

2500: 3000

If we divide both numbers by 500, the ratio of radio to television will become 5: 6

So Radio: Television = 2500: 3000 = 5:6

**Example 2:** The cost price of a table ₦5000 was reduced by ₦1500. Find the ratio of the old price to the new price.

Solution:

Old price = ₦5000; New price = old price – ₦1500 = ₦5000 – ₦1500 = ₦ 3500

Old price: New price = ₦5000: ₦3500

Divide both numbers by their HCF (500) gives 10: 7

Then the ratio of old price to new price = 10: 7

TRY THESE:

(1) Three children divide ₦1 050 between them in the ratio 6: 7: 8. What is the size of the largest share? [fusion_builder_container hundred_percent=”yes” overflow=”visible”][fusion_builder_row][fusion_builder_column type=”1_1″ background_position=”left top” background_color=”” border_size=”” border_color=”” border_style=”solid” spacing=”yes” background_image=”” background_repeat=”no-repeat” padding=”” margin_top=”0px” margin_bottom=”0px” class=”” id=”” animation_type=”” animation_speed=”0.3″ animation_direction=”left” hide_on_mobile=”no” center_content=”no” min_height=”none”][₦400]

(2) The ages of a parent and child are in the ration 8:3. If the child’s age now is 12, what will be the ratio of their ages in 4 years’ time? [9: 4]

Rates

The connection and/or exchange between quantities of different types can be established using a rate. For example, different countries use different currency denominations so when someone in Nigeria (₦) wants to buy goods from someone in USA ($) they need to know the rate of exchange between dollars ($) and naira (₦) for them to do the business.

**Example 1:** If $1 = ₦157.39, how much will a man pay in naira (₦) for a good that costs $360.

Solution: Using the rate of exchange $360 in naira = $ 360 x (equivalent of $1 in naira)

= 360 x 157.39 = ₦56 660.40

**Example 2:** A tank full of water lasts 15 weeks if 3 litres a day are used. How long will it last if 10 litres a day are used?

Solution: At rate of 3 litres/ day, water lasts for 15 weeks

At a rate of 10 litres/day, the water will last for 3 x 15 = 4.5 weeks

10

An increase in the usage per day brought about a decrease in the number of days of use.

TRY THESE

(1) A car goes 60 km in 48 minutes. Find his speed in km/h [75km/h]

(2) If a sack of grain can feed 80 chickens for 18 days. How many days will it last 120 chickens? [12]

Percentages

A percentage is a hundredth part of a quantity. This means that 5% of an item is the same as “5 parts out of hundred”. Percentages are useful in many calculations of commercial arithmetic.

**Example 1: ** What percentage of 2 is 5?

Solution: Let the percentage be Y

Y% of 2 is 5 ===> Y/100 of 2 = 5

Y x 2 = 5 ===> 2Y = 500

100

Y = 250. Therefore 5 is 250% of 2

**Example 2:** The original area of a farm is 390km^{2}. If the famer sells 15% of the land, what area would be left?

Solution:

Percentage sold = 15%

Percentage remaining = (100 – 15) % = 85%

Area of land remaining = 85% of 390km^{2} = 85 x 390 = 331.5 km^{2}

100

TRY THESE:

1. A casting was made of copper and zinc. 65% of the casting is copper and there are 147g of zinc. What is the mass of the casting? [420g]

2. Out of a possible 120 marks, a student scores 78 marks. Express his marks as a percentage. [65%]

Simple Interest and Compound Interest

Interests

When money is given out for the purpose of business, the borrower is often expected to add an extra amount to the money to be paid when he pays the lender. This extra amount is called interest and is calculated as a percentage of the money lent. However the mode of the calculation can be one of two types which are the (a) Simple Interest and (b) Compound Interest

(a) Simple Interest

There is a formula used to determine how much extra money is added to each person’s money and formula is called the Simple Interest formula.

S.I = Principal X Rate X Time; principal = amount you save, Rate= Percentage that bank wants to add to your money

100

**Example:** Find the simple interest on ₦7000 saved at a rate of 6% per annum for 2 years

Solution:

S.I = P x R X T = 7000 x 6 X 2 = ₦840

100 100

**Example 2:** A man saved ₦15000 for two years at simple interest. If the money amounted to ₦15 600 at the end of the two years, what is the rate of interest?

Solution:

Principal (P) = ₦15000; Total Amount = Principal (P) + Interest (I) = ₦15600

I = Amount – Principal = ₦15600 – ₦15000 = ₦600

From I = = P x R X T = 15000 x R X 2 = ₦600

100 100

600 x 100 = 30000R; R = 60000 = 2%

30000

(b) Compound Interest: This is when interest to be paid on a principal is calculated at regular intervals and added to the principal at regular intervals. This means that the principal grows as the interest is added.

**Example 1:** A man borrows a sum of ₦100 000 at 6% per annum compound interest. How much will he pay after 2 years if the money is compounded annually?

Solution:

1^{st} year; Principal = ₦100 000, Rate = 6%, Time = 1 year (We take 1 year at a time)

1^{st} year; I = PRT = 100 000 x 6 x 1 = ₦6000

100 100

Amount at end of 1^{st} year = Principal + 1^{st} year Interest = ₦100000 + ₦6000 = ₦106 000

2^{nd} year: Principal = ₦106000, Rate = 6%, Time = 1

= I = PRT = 106 000 x 6 x 1 = ₦ 6360

100 100

Total amount at the end of the 2 years = Principal for 2^{nd} year + Interest for 2^{nd} year

= ₦106000 + ₦6360 = ₦112 360

Compound Interest Formula

When a sum of money ** P**, is invested for a Time

*T**(in years) at*

**% per annum, the total amount**

*R***, after**

*A***years is given by the formula**

*T***A = P (1 + ^{R}/100)^{ T}**

Example 1: The sum of ₦100 000 is invested at 6% per annum compound interest. Find the amount after 2 years.

Solution: We want to find A, we know that P = ₦100 000, T = 2 years and R = 6% so we put these values in the formula

**A = P (1 + ^{R}/100)^{ T }===> **A = ₦100 000 (1 +

^{6}/100)

^{ 2}

= 100 000 (1 +0.06)^{ 2} = 100 000 (1.06)^{ 2}

= ₦112, 360

NOTE: You can see from the examples where we used the compound interest formula and where we treated it on a yearly basis that the answers obtained are the same. However, when a specific interval is given, it is IMPORTANT to use the method of intervals.

TRY THESE:

(1) Give the amount (to the nearest naira) which ₦600 will amount in 3 years at 11% per annum compound interest. [₦821]

(2) A saver deposits ₦240 000 in a bank which pays simple interest at 9½% per annum half yearly. Find the amount in the bank at the end of 1 year [₦263 342]

[Hint: Calculate at intervals of ½ years]

Profits and Loss

In our business transactions, we pay for goods and services and get paid for goods we produce or services we render. When the amount a person makes from a business transaction is more than the money he put in the transaction, such a person is said to make profit or gain. For someone who buys an item and then sells it later, if he sells the item at an amount (selling price) that is greater than the amount at which he bought the item (cost price), then the person has made profit.

Profit = Selling price – Cost Price

Percentage Profit = Profit x 100% = Selling Price – Cost Price

Cost Price Cost Price

However if the person sells the item an amount that is less than the amount at which he bought the item, the person is said to have suffered a loss.

Loss = Cost Price – Selling Price.

Percentage Loss = Loss x 100% = Cost Price – Selling Price

Cost Price Cost Price

Example1: A trader bought a television set for $100 and then sells it for $110. Calculate (i) the profit (ii) The percentage profit

Solution:

(i) Profit = Selling Price – Cost Price = $ (110 – 100) = $10

(ii) Percentage profit = Profit x 100% = $10 x 100% = 10%

Cost Price $100

Example 2: A man made a loss of 20% when he sold a radio for ₦120. Find (a) The cost price (b) The loss

Solution: Let the cost price be y

% loss = cost price – selling price x 100% è 20% = y – 120 x100 è Y

Cross multiplying will give us 20y = 100y -12000

Gather like terms 12000 = 80y; y = ₦150

(b) Loss = Cost Price – Selling Price = ₦150 – ₦120 = ₦ 30

Discount, Commission & Hire Purchase (Instalments)

Discounts

This is a reduction in the cost price of an item to encourage bulk purchase or cash payment. Discounts are also given at special periods such as festival periods in other to encourage people to buy large volumes of the products.

Discounts are measured as a percentage of the original worth of the item.

**Example 1:** A trader gives a 15% discount on purchase of a dozen of books which is sold for ₦800 each. How much will the man pay for the dozen?

Solution:

1 book = ₦800 therefore a dozen book = ₦800 x 12 = ₦9600

Discount = 15% of ₦9600 = 15 x 9600 = ₦1440

100

Therefore the amount paid will be ₦9600 – ₦1440 = ₦8160

**Example 2:** After a discount of 5% a man pays ₦6460 for a shirt. What is the actual worth of the church?

Solution:

Let the worth of the shirt be ₦Y

Percentage discount is 5%, so amount paid is (100 – 5) % = 95%

===> ₦6460 = 95% of Y ===> 6460 = 0.95 x Y

===> 6460 ÷ 0.95 = Y

Y = ₦6800

Commission:

This is money paid to someone for selling goods or for marketing another person’s services. It is calculated based on the amount of goods sold or the worth of the services marketed and this is done to make these sales agents put more effort because the higher the amount of goods you sell, the higher the commission.

Example: A sales agent receives a 5% on every computer set he sells. If he sold 2 sets of computer at ₦60000 each how much will he collect as commission?

Solution:

Amount of computer = ₦60000; Number of units sold = 2

Commission on 1 computer set = 5% of amount = 5% of ₦60000 = ₦3000

Total commission = Commission for 1 set number of sets = ₦3,000 x 2 = ₦6 000

Hire Purchase**:**

There are times when we want to buy an item that costs a huge amount. When we cannot pay for such at once we can pay in small parts. This part payment is called INSTALMENTS or HIRE PURCHASE The total of these part payments can be higher than the *one – off* payment in cash.

For example if a secondary school student wants to buy an item that costs ₦100 but he can only afford to pay ₦20 at any particular pay the student can negotiate with the seller than he will pay the complete money in 5 days. The ₦20 he pays each day is called an instalment. An instalment payment makes it easier and more comfortable for people to buy expensive items.

Instalment payment is also called hire purchase.

**Example 1:** The hire purchase price of a computer is ₦84 000. 25% of this amount is paid in deposit while the rest is spread out over 12 equal monthly instalments.

(a) Calculate the amount of the deposit

(b) How much is the monthly instalments?

**Solution:**

(a) Amount of deposit = 25% of total amount = 25% of ₦84,000 = ₦21,000

(b) Monthly instalments = Remaining amount ÷ 12

= (₦84 000 – ₦21 000) ÷ 12 = ₦63 000 ÷ 12 = ₦5 250

Example 2: To buy a pair of shoes, a lady can either make a one – off payment of ₦48 000 or she can pay in 6 weekly instalments of ₦9000.

(a) Find the total cost of paying for the item in instalments.

(b) Calculate the difference between the one – off payment and total cost in instalments.

**Solution:**

(a) Cost of paying the instalments = Amount of weekly instalment x Number of weeks

₦9000 x 6 = ₦54 000

(b) Difference between one – off payment and instalments = ₦54 000 – ₦48 000 = ₦6 000

Tests and Exercises

1. Express 495g as a percentage of 16.5kg (a) 3% (b) 3^{1}/3% (c) 15% (d) 30%

**Guideline:**

Make sure that both quantities are in same units. 16.5kg = 16.5 x 1000g = 16500

495g as a percentage of 16.5kg = 495 x 100% = 49500 = 3%. Answer is Option A

16500 16500

2. X sold an article to Y at a profit of 20%. Y then sold it to Z at a loss of 20% of what it cost him. What is the ratio final price: original price? (a) 25:36 (b) 5: 7 (c) 4: 5 (d) 24: 25 (e) 1:1

**Guideline:**

If the original amount is P, then X sells at P + (20% of P) = 1.2P

Y sells at 1.2P – (20% of 1.2P) = 1.2P – 0.24P = 0.96P (Final price)

Final price: original Price = 0.96P: P = 24: 25. Answer is Option D

3. Amina buys 100 oranges at 20 for ₦30 and another 200 oranges at 4 for ₦10. If she sells all the oranges at ₦3 each, what was her profit? (a) ₦150 (b) ₦250 (c) ₦500 (d) ₦650

**Guideline:**

100 oranges at 20 for ₦30 = ₦1.50/orange; Cost = 100 x ₦1.50 = ₦150

200 oranges at 4 for ₦10 = ₦2.50/orange; Cost = 200 x ₦2.50 = ₦500

Total cost price = ₦ (150 + 500) = ₦650

Amina sold at ₦3/orange, therefore ₦3 x 300 = ₦900

Profit = Selling price – Cost price = ₦900 – ₦650 = ₦250. Answer is Option B

4. A headmaster contributes 7% of his income into a fund and his wife contributes 4% of her income. If the husband earns ₦5,500 per annum and the wife earns ₦4,000 per annum, find the sum of their contribution to the fund. (a) ₦1 045 (b) ₦605 (c) ₦545 (d) ₦490

**Guideline:**

Man’s contribution = 7% of ₦5 500 = 7/100 x ₦5 500 = ₦385

Wife’s contribution = 4% of ₦4 000 = 4/100 x ₦4 000 = ₦160

Total contribution = ₦ (385 + 160) = ₦545. Answer is Option C

5. If ₦2 500 amounted to ₦3 500 in 4 years at simple interest, find the rate at which the interest was charged. (a) 5% (b) 7½% (c) 8% (d) 10%

**Guideline:**

Interest = Amount – Principal = ₦3500 – ₦2500 = ₦1000

P = ₦2500, T = 4% and R =?

Using the formula S.I = PRT ===> 1000 = 2500 x R x 4

100

= 1000 x 100 = 1000R; 10000 ÷ 1000 = R

R = 10%. Answer is Option D

For more on classwork notes, click here

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