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 390km2. 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 390km2 = 85 x 390 = 331.5 km2
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:
1st year; Principal = ₦100 000, Rate = 6%, Time = 1 year (We take 1 year at a time)
1st year; I = PRT = 100 000 x 6 x 1 = ₦6000
100 100
Amount at end of 1st year = Principal + 1st year Interest = ₦100000 + ₦6000 = ₦106 000
2nd 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 2nd year + Interest for 2nd year
= ₦106000 + ₦6360 = ₦112 360
Compound Interest Formula
When a sum of money P, is invested for a Time T (in years) at R% per annum, the total amount A, after T years is given by the formula
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) 31/3% (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
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