Chapter 3: Transactions in the home and office

In this chapter you will learn how to do calculations that you are likely to use in your everyday life. You will learn how to work with household expenditure such as electricity bills, water rates and a family budget. You will also learn to do commercial calculations such as determining profit, loss, interest, discount and commission.

3.1 Commercial transactions

Profit and loss

A trader is someone who buys and sells goods or services. The cost price is the price that the trader pays when she buys an item. It also includes any other money she spends on the item before she sells it. The selling price is the price at which the trader sells an item to a customer. The difference between the cost price and the selling price is the profit or loss. If the selling price is more than the cost price, the trader makes a profit. If the selling price is less than the cost price, the trader makes a loss.

cost price The cost price is the price that a trader pays when she buys an item, and includes any other money she spends on the item before she sells it.

selling price The selling price of an item is the price at which a trader sells the item to a customer.

profit When the selling price of an item is more than its cost price, a trader makes a profit on the item.

loss When the selling price of an item is less than its cost price, a trader makes a loss on the item.

We can express profit or loss as an amount of money or as a percentage:

$\text{profit/loss} = \text{selling price} - \text{cost price}$ $\% \text{ profit/loss} = \dfrac{\text{profit/loss}}{\text{cost price}} \times 100$

Worked example 3.1: Calculating percentage profit or loss

A trader buys fifty 1-litre cartons of drinking yoghurt from a factory. He pays ₦550 per one 1-litre carton. On the way to the storeroom, eight cartons burst open and the yoghurt spills out. The trader sells the remaining cartons at ₦600 each. Calculate his percentage profit or loss.

Step 1: Calculate the total cost price.
\[50\times ₦550 = ₦27,500\]
Step 2: Calculate the total selling price.

The trader could only sell $50-8=42$ cartons.
\[42 \times ₦600 = ₦25,200\]
Step 3: Calculate the profit or loss and express it as an amount of money.
\begin{align}\text{profit/loss} &= \text{selling price} - \text{cost price} \\ &= ₦25,200 - ₦27,500 \\ &= -\, ₦2,300 \end{align}
The minus sign shows that the trader made a loss. This is because the selling price is less than the cost price.
\[\text{loss} = ₦2,300\]
Step 4: Calculate the percentage profit or loss. Round to two decimal places where necessary.
\begin{align} \% \text{ loss} &= \frac{\text{loss}}{\text{cost price}} \times 100 \\ &= \frac{ ₦2,300}{ ₦27,500} \times 100 \\ &= 8.36 \% \end{align}

If we have the percentage profit or loss and either the cost price or the selling price, we can calculate the other amount. One way to do this is as follows:

Let the cost price represent $100 \%$ .
The profit/loss is given as a percentage.
Let the selling price represent $(100 \pm \text{profit/loss}) \%$ .

Then you can use the following formulae:

$\text{cost price} = \text{selling price} \times \dfrac{100}{100+\%\text{ profit}}$
OR
$\text{cost price} = \text{selling price} \times \dfrac{100}{100-\%\text{ loss}}$
$\text{selling price} = \text{cost price} \times \dfrac{100+\%\text{ profit}}{100}$
OR
$\text{selling price} = \text{cost price} \times \dfrac{100-\%\text{ loss}}{100}$

Worked example 3.2: Calculating cost price from percentage profit or loss

A retailer buys twenty pairs of women's sneakers from a manufacturer. One of the pairs gets damaged as the truck with the shoes is unloaded. The retailer decides to sell this pair at a lowered price of ₦6,800. The loss is 20%. Calculate the cost price of this pair of sneakers.

Step 1: Determine the percentage that represents the amount you must calculate.

In this case, you must calculate the cost price. The cost price represents 100%.
Step 2: Determine the percentage that represents the amount you have.

You have the selling price. A loss was made on the item, so the selling price represents $100-\%\text{ loss}=100-20=80\%$ .
Step 3: Multiply the amount you have by the ratio of the percentages you determined in Steps 1 and 2.
\begin{align} \text{cost price} &= \text{selling price} \times \frac{100}{100-\%\text{ loss}} \\ &= ₦6,800 \times\frac{100}{80} \\ &= ₦8,500 \end{align}

Always round to two decimal places in your calculations in this chapter, unless otherwise instructed.

A washing machine is bought for ₦18,500 by a trader and then sold for ₦26,000. Calculate the percentage profit.

\begin{align} \text{profit} &= \text{selling price} - \text{cost price} \\ &= ₦26,000 - ₦18,500 \\ &= ₦7,500 \end{align} \begin{align} \% \text{ profit} &= \frac{\text{profit}}{\text{cost price}} \times 100 \\ &= \frac{ ₦7,500}{ ₦18,500} \times 100 \\ &= 40.5\% \end{align}
A microwave that was bought for ₦18,400 is used for a year and then sold at a second hand store for ₦16,000. Calculate the percentage loss.

\begin{align}\text{loss} &= \text{selling price} - \text{cost price} \\ &= ₦16,000 - ₦18,400 \\ &= -\, ₦2,400 \end{align} \begin{align} \% \text{ loss} &= \frac{\text{loss}}{\text{cost price}} \times 100 \\ &= \frac{ ₦2,400}{ ₦18,400} \times 100 \\ &= 13.04 \% \end{align}
The owner of a corner shop buys 50 kg of parboiled rice at ₦14,000 from a supplier. She makes up fifty 1-kilogram packets and sells each of them at ₦350. Calculate her percentage profit.

\begin{align} \text{profit} &= \text{selling price} - \text{cost price} \\ &= ( ₦350 \times50) - ₦14,000 \\ &= ₦3,500 \end{align} \begin{align} \% \text{ profit} &= \frac{\text{profit}}{\text{cost price}} \times 100 \\ &= \frac{ ₦3,500}{ ₦14,000} \times 100 \\ &= 25 \% \end{align}
A retailer buys three Android tablets from a supplier at ₦18,000 each. Two of them are damaged before he can sell them. He sells the undamaged tablet at ₦22,500 and the two damaged tablets at ₦15,500 each. Calculate his percentage profit or loss.

\begin{align} \text{profit/loss} &= \text{selling price} - \text{cost price} \\ &= ( ₦22,500 + 2\times ₦15,500) - (3\times ₦18,000) \\ &= ₦53,500 - ₦54,000\\ &= -\, ₦500 \end{align} \begin{align} \% \text{ loss} &= \frac{\text{loss}}{\text{cost price}} \times 100 \\ &= \frac{ ₦500}{ ₦54,000} \times 100 \\ &= 0.93 \% \end{align}
A shop owner buys vegetable oil from a supplier at ₦720 per litre. If she wants to make a profit of 15%, what must her selling price be?

\begin{align} \text{selling price} &= \text{cost price} \times \frac{100+\%\text{ profit}}{100} \\ &= ₦720 \times \frac{115}{100} \\ &= ₦828 \end{align}
A shop owner buys jars of coconut oil at ₦2,500 each from a supplier. While the jars are being transported, the lids of the containers are scratched. The shop owner decides to sell the jars at a loss of 6% each. What is the selling price per jar?

\begin{align} \text{selling price} &= \text{cost price} \times \frac{100-\%\text{ loss}}{100} \\ &= ₦2,500 \times \frac{94}{100} \\ &= ₦2,350 \end{align}
An supplier of air conditioners makes a profit of 20% on each unit sold at ₦100,500. What is the cost price per unit?

\begin{align} \text{cost price} &= \text{selling price} \times \dfrac{100}{100+\%\text{ profit}} \\ &= ₦100,500 \times \frac{100}{120} \\ &= ₦83,750 \end{align}
In the storeroom of a grocery store, the labels of some milk tins got wet by mistake and fell off. The manager decides to sell the tins without the labels at loss of 30%. If she sells each tin at ₦980, what was the cost price per tin?

\begin{align} \text{cost price} &= \text{selling price} \times \dfrac{100}{100-\%\text{ loss}} \\ &= ₦980 \times \frac{100}{70} \\ &= ₦1,400 \end{align}
A furniture trader sells a sofa bed for ₦78,750 at a profit of 5%. Determine what the selling price must be for the trader to make a profit of 8%.

\begin{align} \text{cost price} &= \text{selling price} \times \dfrac{100}{100+\%\text{ profit}} \\ &= ₦78,750 \times \frac{100}{105} \\ &= ₦75,000 \end{align} \begin{align} \text{selling price} &= \text{cost price} \times \frac{100+\%\text{ profit}}{100} \\ &= ₦75,000 \times \frac{108}{100} \\ &= ₦81,000 \end{align}
A store owner buys a pack of soft drinks that come in 35 cl bottles. The bottles cost ₦75 each and she sells all of them for ₦80 each. She makes a profit of ₦100 altogether. Calculate the percentage profit she made.

$\text{profit per bottle}= &#8358;80- &#8358;75 = &#8358;5$ $\therefore \text{number of bottles}=\frac{ &#8358;100}{ &#8358;5}=20$ $\therefore \text{cost price}= &#8358;75 \times20 = &#8358;1,500$ \begin{align} \% \text{ profit} &= \frac{\text{profit}}{\text{cost price}} \times 100 \\ &= \frac{ ₦100}{ ₦1,500} \times 100 \\ &= 6.67 \% \end{align}

Interest

Interest is extra money you pay on a loan, or money you earn on an investment. A loan is an amount of money you borrow, that you must normally pay back with interest. An investment is an amount of money you give to a business or bank in the hope of getting more money back.

interest Interest is extra money you pay on a loan, or money you earn on an investment.

loan A loan is an amount of money you borrow, that you must normally pay back with interest.

investment An investment is an amount of money you give to a business or bank in the hope of getting more money back.

When you take out a loan or make an investment, the original amount of money you borrow or invest is called the principal amount ( $P$ ). Interest that is calculated as a percentage of the principal amount is called simple interest ( $SI$ ). When we work with simple interest, the interest rate ( $i$ ) is the percentage of the principal amount that you pay or earn per specified time period. The number of time periods ( $n$ ) is the number of months or years over which your loan or investment runs. The actual amount ( $A$ ) is the actual amount of money you pay back or get out. It is the principal amount plus the simple interest.

We calculate simple interest as follows:

$SI=P \times i \times n$

We calculate the actual amount as follows:

\begin{align} A &= P+SI \\ \therefore A&=P + P\times i \times n \\ \text{or } A&=P(1+in) \end{align}

You will learn about another type of interest, called compound interest, in later grades.

principal amount ( $P$ ) The principal amount is the original amount of money borrowed when a loan is taken out, or paid in when an investment is made.

simple interest ( $SI$ ) Interest that is calculated as a percentage of the principal amount is called simple interest.

interest rate ( $i$ ) For simple interest, the interest rate on the loan or investment is the percentage of the principal amount that you pay or earn per specified time period.

number of time periods ( $n$ ) The number of time periods is the number of months or years over which a loan or an investment runs.

actual amount ( $A$ ) The actual amount is the actual amount of money you pay back on a loan or get out from an investment. It is the principal amount plus the simple interest.

Worked example 3.3: Calculating simple interest and actual amount

A person invests ₦50,000 in a fixed deposit account with an interest rate of 8% per annum. Calculate the interest earned over 5 years, as well as the balance of the account at the end of the 5 years.

The phrase "per annum" means per year.

Step 1: Make a list of the symbols $A$ , $P$ , $SI$ , $i$ , and $n$ . Insert the information you have next to each symbol. Write a question mark next to the symbol(s) you have to calculate.
\begin{align} A&=\,? \\ P&= ₦50,000 \\ SI &=\, ? \\ i&=8 \% \\ n&=5 \end{align}
Step 2: Write down the formula or formulae that you will need to use.

For simple interest: $SI=P \times i \times n$

For the actual amount: $A=P+SI$
Step 3: Substitute the values you have and do the calculations.

Remember that a percentage is a fraction: $8\% = \dfrac{8}{100}$

The simple interest is:
\begin{align} SI&=P \times i \times n \\ &= ₦50,000 \times 8\% \times5 \\ &= ₦250,000 \times 8\% \\ &= ₦250,000 \times \frac{8}{100} \\ &= ₦20,000 \end{align}
The actual amount is:
\begin{align} A&=P + SI \\ &= ₦50,000 + ₦20,000 \\ &= ₦70,000 \end{align}

Worked example 3.4: Calculating principal amount

A person invested an amount at 5% per annum for 3 years. At the end of the 3 years, the value of the investment was ₦17,250. Calculate the principal amount that was invested.

Step 1: Make a list of the symbols $A$ , $P$ , $i$ , and $n$ . Insert the information you have next to each symbol. Write a question mark next to the symbol you have to calculate.
\begin{align} A&= ₦17,250 \\ P&=\,? \\ i&=5 \% \\ n&=3 \end{align}
Step 2: Write down the formula that you will need to use.

If you have to calculate $P$ , use this version:
$A=P(1+in)$
Step 3: Make the symbol you have to calculate the subject of the formula.
\begin{align} A&=P(1+in) \\ P&=\frac{A}{1+in} \\ \end{align}
Step 4: Substitute the values you have and calculate the answer.

Remember that the number 1 is equal to 100%.
\begin{align} A&=P(1+in) \\ P&=\frac{A}{1+in} \\ &=\frac{ ₦17,250}{1+5\%\times 3} \\ &=\frac{ ₦17,250}{100\%+15\%} \\ &=\frac{ ₦17,250}{\frac{115}{100}} \\ &= ₦17,250 \times \frac{100}{115} \\ &= ₦15,000 \end{align}

Worked example 3.5: Calculating interest rate

A person took a loan of ₦120,000. She paid back ₦144,000 over 4 years. Calculate the interest rate.

Step 1: Make a list of the symbols $A$ , $P$ , $i$ , and $n$ . Insert the information you have next to each symbol. Write a question mark next to the symbol you have to calculate.
\begin{align} A&= ₦144,000 \\ P&= ₦120,000 \\ i&=\,? \\ n&=4 \end{align}
Step 2: Write down the applicable formula.

If you have to calculate $i$ , use this version:
$A=P+Pin$
Step 3: Make the symbol you have to calculate the subject of the formula.
\begin{align} A&=P+Pin \\ A-P&=Pin \\ i&=\frac{A-P}{Pn} \end{align}
Step 4: Substitute the values you have and calculate the answer.

To get a percentage, find a fraction with 100 as the denominator.
\begin{align} A&=P+Pin \\ A-P&=Pin \\ i&=\frac{A-P}{Pn} \\ &=\frac{ ₦144,000- ₦120,000}{ ₦120,000\times 4} \\ &=\frac{ ₦24,000}{ ₦480,000} \\ &=\frac{1}{20}\\ &=\frac{5}{100} \\ &=5\% \end{align}

Worked example 3.6: Calculating number of time periods

A person borrowed ₦20,000 from his friend. He paid back the money monthly. The friends agreed on a monthly interest rate of 2%. The total amount paid back was ₦27,200. Calculate how many months it took to pay the borrowed amount and the interest.

Step 1: Make a list of the symbols $A$ , $P$ , $i$ , and $n$ . Insert the information you have next to each symbol. Write a question mark next to the symbol you have to calculate.
\begin{align} A&= ₦27,200 \\ P&= ₦20,000 \\ i&=2\% \\ n&=\,? \end{align}
Step 2: Write down the applicable formula.

If you have to calculate $n$ , use this version:
$A=P+Pin$
Step 3: Make the symbol you have to calculate the subject of the formula.
\begin{align} A&=P+Pin \\ A-P&=Pin \\ n&=\frac{A-P}{Pi} \end{align}
Step 4: Substitute the values you have and calculate the answer.
\begin{align} A&=P+Pin \\ A-P&=Pin \\ n&=\frac{A-P}{Pi} \\ &=\frac{ ₦27,200- ₦20,000}{ ₦20,000\times \frac{2}{100}} \\ &=\frac{ ₦7,200}{ ₦400} \\ &=18\text{ months} \end{align}

It is always helpful to list the symbols and to insert the information you have next to each symbol. This helps you to identify what you need to find, and to choose the correct formula.

Calculate the interest due on a loan of ₦60,000 to be repaid over 6 years at an interest rate of 8% per annum.

\begin{align} SI&=P \times i \times n \\ &= ₦60,000 \times 8\% \times6 \\ &= ₦360,000 \times \frac{8}{100} \\ &= ₦28,800 \end{align}
Calculate the total amount of money paid back over 5 years on a loan of ₦25,000 with an interest rate of 15% per annum.

\begin{align} A&=P+Pin \\ &= ₦25,000 + ₦25,000 \times 15\% \times5 \\ &= ₦25,000 + ₦125,000 \times \frac{15}{100} \\ &= ₦25,000+ ₦18,750 \\ &= ₦43,750 \end{align}
A person invested ₦20,000 for 3 years at a bank that offers an interest rate of 5% per annum. At the end of the 3 years, she withdrew the full amount and invested it with another bank for 2 years. The new bank's interest rate is 6% per annum. Calculate the simple interest she earned in the 5 years.

First bank:
\begin{align} SI&=P \times i \times n \\ &= ₦20,000 \times 5\% \times3 \\ &= ₦60,000 \times \frac{5}{100} \\ &= ₦3,000 \end{align}
She withdrew $&#8358;20,000+ &#8358;3,000= &#8358;23,000$ .

Second bank:
\begin{align} SI&=P \times i \times n \\ &= ₦23,000 \times 6\% \times2 \\ &= ₦46,000 \times \frac{6}{100} \\ &= ₦2,760 \end{align}
She earned interest of $&#8358;3,000+ &#8358;2,760= &#8358;5,760$ .
If a bank offers an interest rate of 10% per annum, how long must you invest ₦15,000 if you want it to increase to ₦25,500?

\begin{align} A&=P+Pin \\ A-P&=Pin \\ n&=\frac{A-P}{Pi} \\ &=\frac{ ₦25,500- ₦15,000}{ ₦15,000\times \frac{10}{100}} \\ &=\frac{ ₦10,500}{ ₦1,500} \\ &=7\text{ years} \end{align}
If the total repayment on a loan of ₦80,000 with an interest rate of 5% per annum was ₦102,000, how long did it take to repay the loan?

\begin{align} A&=P+Pin \\ A-P&=Pin \\ n&=\frac{A-P}{Pi} \\ &=\frac{ ₦102,000- ₦80,000}{ ₦80,000\times \frac{5}{100}} \\ &=\frac{ ₦22,000}{ ₦4,000} \\ &=\frac{11}{2} \\ &=5\frac{1}{2}\text{ years} \end{align}
Calculate the annual interest rate offered by a bank where a deposit of ₦75,000 increases to ₦87,000 in 4 years.

The phrase "annual interest rate" means the interest rate per year.

\begin{align} A&=P+Pin \\ A-P&=Pin \\ i&=\frac{A-P}{Pn} \\ &=\frac{ ₦87,000- ₦75,000}{ ₦75,000\times 4} \\ &=\frac{ ₦12,000}{ ₦300,000} \\ &=\frac{2}{50}\\ &=\frac{4}{100} \\ &=4\% \end{align}
The total repayments on a loan of ₦105,000 add up to ₦142,800 after 3 years. Calculate the annual interest rate on the loan.

\begin{align} A&=P+Pin \\ A-P&=Pin \\ i&=\frac{A-P}{Pn} \\ &=\frac{ ₦142,800- ₦105,000}{ ₦105,000\times 3} \\ &=\frac{ ₦37,800}{ ₦315,000} \\ &=\frac{3}{25}\\ &=\frac{12}{100} \\ &=12\% \end{align}
The total repayments on a loan add up to ₦325,600 after 4 years. If the interest rate is 12% per annum, calculate the original amount of money borrowed.

\begin{align} A&=P(1+in) \\ P&=\frac{A}{1+in} \\ &=\frac{ ₦325,600}{1+12\%\times 4} \\ &=\frac{ ₦325,600}{100\%+48\%} \\ &=\frac{ ₦325,600}{\frac{148}{100}} \\ &= ₦325,600 \times \frac{100}{148} \\ &= ₦220,000 \end{align}
A person invested an amount of money at a bank for 6 years. The bank's interest rate is 4% per annum. The interest accumulated after 6 years is ₦3,000. Calculate the total value of the investment at the end of the 6 years.

Look out for when you need to do the calculation in two parts. To find the total value of the investment, you need to work out the initial investment ( $P$ ) and add the interest given.

\begin{align} SI&=P \times i \times n \\ P&= \frac{SI}{in} \\ &= \frac{ ₦3,000}{4\% \times 6} \\ &= \frac{ ₦3,000}{\frac{24}{100}} \\ &= ₦3,000\times \frac{100}{24} \\ &= ₦12,500 \end{align} \begin{align} A&=P+SI \\ &= ₦12,500+ ₦3,000 \\ &= ₦15,500 \end{align}
Suppose your bank offers an interest rate of 8% per annum on investments. Calculate how much money you must add to an investment of ₦10,000 if you want to increase the value of your investment to ₦49,000 in 5 years.

\begin{align} A&=P(1+in) \\ P&=\frac{A}{1+in} \\ &=\frac{ ₦49,000}{1+8\%\times 5} \\ &=\frac{ ₦49,000}{100\%+40\%} \\ &=\frac{ ₦49,000}{\frac{140}{100}} \\ &= ₦49,000 \times \frac{100}{140} \\ &= ₦35,000 \end{align}
You must add:
$&#8358;35,000- &#8358;10,000= &#8358;25,000$

Discount

A discount is a reduction in the marked price of an item, so that the customer pays less than the normal price for the item. Traders often give a discount when customers pay cash instead of buying on credit. They may also give discount when customers buy a large number of a specific item.

discount A discount is a reduction in the marked price of an item, so that the customer pays less than the normal price.

We can express discount as an amount of money or as a percentage:

$\text{discount} = \text{normal price} - \text{discounted price}$ $\% \text{ discount} = \dfrac{\text{discount}}{\text{normal price}} \times 100$

Worked example 3.7: Calculating percentage discount

A supermarket has too much stock of a certain brand of breakfast cereal. The manager decides to sell the 350 g boxes that are normally priced at ₦1,800 for ₦1,656. Calculate the percentage discount per box.

Step 1: Calculate the discount and express it as an amount of money.
\begin{align} \text{discount} &= \text{normal price} - \text{discounted price} \\ &= ₦1,800- ₦1,656 \\ &= ₦144 \end{align}
Step 2: Calculate the percentage discount. Round to two decimal digits if necessary.
\begin{align} \% \text{ discount} &= \frac{\text{discount}}{\text{normal price}} \times 100 \\ &= \frac{ ₦144}{ ₦1,800} \times 100 \\ &= 8 \% \end{align}

One way to do calculations with discount is as follows:

Use $100 \%$ to represent the normal price of the item.
Express the discount as a percentage.
Use $(100 - \text{discount}) \%$ to represent the discounted price.

Then you can use the following formulae:

$\text{normal price} = \text{discounted price} \times \dfrac{100}{100-\%\text{ discount}}$ $\text{discounted price} = \text{normal price} \times \dfrac{100-\%\text{ discount}}{100}$

Worked example 3.8: Calculating normal price from percentage discount

A retailer offers 5% discount if a customer buys more than 20 soft drinks in 33 cl cans. A customer buys 24 cans and pays ₦3,420. Calculate the normal price per can.

Step 1: Determine the percentage that represents the amount you must calculate.

In this case, you must calculate the normal price. The normal price represents $100 \%$ .
Step 2: Determine the percentage that represents the amount you have.

You have the discounted price. It represents $100-\%\text{ discount}=100-5=95\%$ .
Step 3: Multiply the amount you have by the ratio of the percentages you determined in Steps 1 and 2.
\begin{align} \text{normal price} &= \text{discounted price} \times \dfrac{100}{100-\%\text{ discount}} \\ &= ₦3,420 \times\frac{100}{95} \\ &= ₦3,600 \end{align}
Step 4: Calculate the price per unit.
$\text{normal price per can}=\frac{ &#8358;3,600}{24}= &#8358;150$

At a clearance sale, a dress that normally costs ₦6,000 is sold for ₦3,540. Calculate the percentage discount offered on the dress.

\begin{align} \text{discount} &= \text{normal price} - \text{discounted price} \\ &= ₦6,000- ₦3,540 \\ &= ₦2,460 \end{align} \begin{align} \% \text{ discount} &= \frac{\text{discount}}{\text{normal price}} \times 100\\ &= \frac{ ₦2,460}{ ₦6,000} \times 100 \\ &=41\% \end{align}
An electronics store offers a 12% discount if their customers pay cash. Calculate what you will pay for a television set priced at ₦58,000 if you pay cash.

\begin{align} \text{discounted price} &= \text{normal price} \times \frac{100-\%\text{ discount}}{100} \\ &= ₦58,000 \times \frac{88}{100} \\ &= ₦51,040 \end{align}
An online store offers a 45% discount on all their headphones. If a pair of gaming headphones is sold at ₦6,875, what is its normal price?

\begin{align} \text{normal price} &= \text{discounted price} \times \frac{100}{100-\%\text{ discount}} \\ &= ₦6,875 \times \frac{100}{55} \\ &= ₦12,500 \end{align}
A clothing store has a special offer on a certain brand of sandals, priced at ₦3,000 a pair. If you buy two pairs, you get a 30% discount on the second pair. The same store offers 5% discount if you pay cash. Calculate the average price per pair of sandals if you buy two pairs and pay cash.

The cost of the second pair is:
\begin{align} \text{discounted price} &= \text{normal price} \times \frac{100-\%\text{ discount}}{100} \\ &= ₦3,000 \times \frac{70}{100} \\ &= ₦2,100 \end{align}
The cost of both pairs is:
$&#8358;2,100+ &#8358;3,000= &#8358;5,100$
The cash price for both pairs is:
\begin{align} \text{discounted price} &= \text{normal price} \times \frac{100-\%\text{ discount}}{100} \\ &= ₦5,100 \times \frac{95}{100} \\ &= ₦4,845 \end{align}
The average price per pair is:
$\text{price per pair}=\frac{ &#8358;4,845}{2}= &#8358;2,422.50$
A trader buys an item priced at ₦7,500 at a discount of 15%. He sells the same item for ₦6,885. Calculate the profit/loss he made on the item.

\begin{align} \text{discounted price} &= \text{normal price} \times \frac{100-\%\text{ discount}}{100} \\ &= ₦7,500 \times \frac{85}{100} \\ &= ₦6,375 \end{align} \begin{align}\text{profit/loss} &= \text{selling price} - \text{cost price} \\ &= ₦6,885 - ₦6,375 \\ &= ₦510 \end{align} \begin{align} \% \text{ profit} &= \frac{\text{profit}}{\text{cost price}} \times 100 \\ &= \frac{ ₦510}{ ₦6,375} \times 100 \\ &= 8 \% \end{align}

Commission

Commission is money that an agent is paid to perform a service. Sales people, insurance agents, debt collectors and lawyers are examples of agents that earn commission for services they provide.

Sometimes an agent's payment is a combination of a fixed salary and commission. The commission that an agent receives is normally calculated as a percentage of sales or collections.

The commission is paid only when the agent makes a sale, collection, or other service. It is worked out for each service, or on a certain value of services. It is paid at a regular time, such as once a month.

commission Commission is money paid to an agent such as a sales person each time the agent performs a service.

We calculate commission as follows:

$\text{commission received}=\text{commission percentage} \times \text{value of sales/services}$

Also note:

$\text{commission percentage}=\dfrac{\text{commission received}}{\text{value of sales/services}}\times 100$ $\text{value of sales/services}=\dfrac{\text{commission received}}{\text{commission percentage}}$

Worked example 3.9: Calculating commission received

A sales person receives a basic salary of ₦85,000 and commission of 15% on all sales above ₦25,000. If this sales person sold stock worth ₦60,000, what is her total pay for the month?

Step 1: Calculate the sales that commission is earned on.

Commission is earned on:
\[ ₦60,000- ₦25,000= ₦35,000\]
Step 2: Calculate the amount of commission received.
\begin{align} \text{commission received}&=\text{commission percentage} \times \text{value of sales/services} \\ &= 15\% \times ₦35,000 \\ &= \frac{15}{100}\times ₦35,000 \\ &= ₦5,250 \end{align}
Step 3: Calculate the total pay.
\[ ₦85,000+ ₦5,250= ₦90,250\]

A debt collector receives 12% commission on all money he manages to get in from debtors. Calculate the commission he receives on collections of ₦246,000.

\begin{align} \text{commission received}&=\text{commission percentage} \times \text{value of sales/services} \\ &= \frac{12}{100}\times ₦246,000 \\ &= ₦29,520 \end{align}
A sales person sells ₦125,500 of books to customers. She receives ₦11,295 commission. What is her commission percentage?

\begin{align} \text{commission percentage}&=\frac{\text{commission received}}{\text{value of sales/services}}\times 100 \\ &= \frac{ ₦11,295}{ ₦125,500} \times 100 \\ &=9\% \end{align}
A sales person that gets a basic salary of ₦50,000 takes home ₦110,250 in a specific month. The value of her sales was ₦753,125. Calculate her commission percentage.

$&#8358;110,250- &#8358;50,000= &#8358;60,250$ \begin{align} \text{commission percentage}&=\frac{\text{commission received}}{\text{value of sales/services}}\times 100 \\ &= \frac{ ₦60,250}{ ₦753,125} \times 100 \\ &=8\% \end{align}
The author of a book receives 2% commission on every book sold. If his commission for one month was ₦600, what was the value of the books sold?

\begin{align} \text{value of sales/services}&=\frac{\text{commission received}}{\text{commission percentage}} \\ &= \frac{ ₦600}{2\%}\\ &=\frac{ ₦600}{\frac{2}{100}} \\ &= ₦600\times \frac{100}{2} \\ &= ₦30,000 \end{align}
An agent that sells air conditioning units receives a basic salary of ₦60,000 per month, plus 20% commission on the units she sells. Her total income for one month is ₦102,942. Calculate the value of the units she sold.

$&#8358;102,942- &#8358;60,000= &#8358;42,942$ \begin{align} \text{value of sales/services}&=\frac{\text{commission received}}{\text{commission percentage}} \\ &= \frac{ ₦42,942}{20\%}\\ &=\frac{ ₦42,942}{\frac{20}{100}} \\ &= ₦42,942\times \frac{100}{20} \\ &= ₦214,710 \end{align}

3.2 Household expenditure

Electricity bills

A bill is a statement of the money you owe for goods or services. An electricity bill shows how much you owe for the electrical energy you used during a specific period. If you have a prepaid electricity meter, you pay for electricity before you use it. Otherwise, you get a bill at the end of each month for the electricity you used during that month.

bill A bill is a statement of the money you owe for goods or services.

Electricity is sold in units called kilowatt-hours (kWh). The main components of an electricity bill are the following:

Units used: The reading on your electricity meter is taken at the end of the month. The reading at the end of the previous month is subtracted. This gives the number of kilowatt-hours used for the month.
Tariff rate: This is the cost for 1 kWh that applies to you.
- Nigeria has five major classes of electricity users. If you use a private house or flat only for living in (not to rent out, or run a business), you fall in the residential class.
- Each class has different tariff rates. A tariff rate depends on your location, how much electricity you use and how the electricity is supplied to your house or flat. For example, if you receive electricity from the Ikeja Distribution Company, the tariff rate for 2018 was ₦18.94 per kWh.
VAT: Value Added Tax of 5% is added to the cost of your electricity.

At the end of 2015, the Nigerian Electricity Regulatory Commission removed fixed charges for all classes of electricity users. Before that, an electricity bill included a fixed monthly charge that you had to pay, irrespective of how much electricity you used.

Worked example 3.10: Calculating the balance owed on an electricity bill

The reading on the electricity meter of a house that receives electricity from the Benin Distribution Company is 102,518 at the end of October and 103,479 at the end of November. The tariff rate is ₦30.98 per kWh. Calculate the amount due on the electricity bill for November.

Step 1: Calculate the number of kilowatt-hours used by subtracting the second meter reading from the first.
$\begin{array}{r} 103,479 \\ -102,518 \\ \hline 961 \\ \hline \end{array}$
Step 2: Multiply the kilowatt-hours used by the tariff rate.
\[961\times ₦30.98 = ₦29,771.78\]
Step 3: Calculate 5% VAT and add it to the amount in Step 2.
\[₦29,771.78 \times \frac{5}{100}= ₦1,488.59\] \begin{array}{r} ₦29,771.78 \\ +\; ₦1,488.59 \\ \hline ₦31,260.37 \\ \hline \end{array}

Instead of first calculating the 5% VAT and then adding it, you can do the following:
$\text{amount including VAT}=\text{amount excluding VAT} \times \frac{105}{100}$

Water bills

Water is commonly sold in cubic metres (m $^3$ ). Water and sanitation providers may include the following in a water bill:

Units used: The reading on your water meter is taken at the end of the month. The reading at the end of the previous month is subtracted. This gives the number of cubic metres used for the month.
Tariff: This is the cost for 1 m $^3$ of water used. Domestic tariffs apply to private houses. Non-domestic tariffs apply to places such as schools, hospitals, businesses and factories.
Wastewater and connections surcharges: These are fixed amounts that you pay every month.
VAT: Value Added Tax of 5% is added to the total amount.

Worked example 3.11: Calculating the balance owed on a water bill

A family used 4.5 m $^3$ of water in a specific month. In the area they live, the domestic tariff is ₦420. The water company charges a connection fee of ₦75 per month. Calculate the balance on their water bill at the end of the month.

Step 1: Calculate the cost of the units of water used.
\[4.5\times ₦420 = ₦1,890\]
Step 2: Add any fixed amounts.
\begin{array}{r} ₦1,890.00 \\ +\phantom{00} ₦75.00 \\ \hline ₦1,965.00 \\ \hline \end{array}
Step 3: Calculate 5% VAT and add it to the amount in Step 2.
\[₦1,965.00 \times \frac{5}{100}= ₦98.25\] \begin{array}{r} ₦1,965.00 \\ +\phantom{0} ₦98.25 \\ \hline ₦2,063.25 \\ \hline \end{array}

Family budget

A budget is an estimate of your income and expenditure for a certain period of time. Your estimated income is the amount of money you expect to receive. Your estimated expenditure is the amount of money you plan to spend. A budget helps you to plan the best way to spend your income.

budget A budget is an estimate of your income and expenditure for a certain period of time.

estimated income Your estimated income is the amount of money you expect to receive in a certain period of time.

estimated expenditure Your estimated expenditure is the amount of money you expect to spend in a certain period of time.

Worked example 3.12: Drawing up a family budget

In a family of four, the wife earns a fixed salary. After deductions, ₦238,450 is paid into her bank account. The husband earns a basic salary of ₦80,000 plus commission on the sales he makes every month. His commission ranges between ₦45,000 and ₦60,000 after deductions. The family pays rent of ₦158,300 and school fees of ₦12,500 every month. Their electricity bill and water bill together are usually about ₦30,000. They pay transport costs of about ₦15,000. Their food, cleaning agents and personal care items cost about ₦110,000 and they allow for ₦30,000 if someone needs clothing. Draw up a budget for the family and advise them how much they can put away as savings.

Step 1: Draw up a table with income in one column and expenditure in another column.
\begin{array}{|r|l r|} \hline \text{Income}\phantom{00} & \text{Expenditure} & \\ \hline \phantom{00} & \phantom{00} & \phantom{00} \\ \phantom{00} & \phantom{00} & \phantom{00} \\ \hline \end{array}
Step 2: Insert all the given amounts. For the commission, insert the lowest expected amount.
\begin{array}{|r|l r|} \hline \text{Income}\phantom{00} & \text{Expenditure} & \\ \hline ₦238,450 & \text{Rent} & ₦158,300 \\ ₦80,000 & \text{School fees} & ₦12,500 \\ ₦45,000 & \text{Electricity and water} & ₦30,000 \\ & \text{Transport} & ₦15,000 \\ & \text{Groceries} & ₦110,000 \\ & \text{Clothes} & ₦30,000 \\ \hline \end{array}
Step 3: Calculate the total income and total expenditure and add it at the bottom of each column.
\begin{array}{|r|l r|} \hline \text{Income}\phantom{00} & \text{Expenditure} & \\ \hline ₦238,450 & \text{Rent} & ₦158,300 \\ ₦80,000 & \text{School fees} & ₦12,500 \\ ₦45,000 & \text{Electricity and water} & ₦30,000 \\ & \text{Transport} & ₦15,000 \\ & \text{Groceries} & ₦110,000 \\ & \text{Clothes} & ₦30,000 \\ \hline ₦363,450 & & ₦355,800 \\ \hline \end{array}
Step 4: Find the difference between the family's income and expenditure.
\begin{array}{r} ₦363,450 \\ - ₦355,800 \\ \hline ₦7,650 \\ \hline \end{array}
Step 5: Determine whether there are any other amounts, apart from the one determined in Step 4, that they put away as savings.

The commission included in the budget is the lowest expected amount. This means the family can save ₦7,650 plus any commission over ₦45,000 that the wife earns.

At the end of February, the reading on the electricity meter of a house that receives electricity from the Enugu Distribution Company is 75,892. At the end of March, the reading is 76,745. The tariff rate is ₦25.40 per kWh. Calculate their outstanding balance at the end of March.

$\begin{array}{r} 76,745 \\ -75,892 \\ \hline 853 \\ \hline \end{array}$
Cost without VAT:
$853\times &#8358;25.40 = &#8358;21,666.20$
Cost with VAT:
$&#8358;21,666.20 \times \frac{105}{100} = &#8358;22,749.51$
A household that receives electricity from the Jos Distribution Company, used 702 kWh units during June. The balance on their electricity bill, including VAT, for June is ₦23,955.75. Calculate the tariff rate they were charged at.

Cost without VAT:
$&#8358;23,955.75 \times \frac{100}{105} = &#8358;22,815.00$
Tariff rate:
$\frac{ &#8358;22,815.00}{702} = &#8358;32.50$
At the end of April, the reading on the electricity meter of a house that receives electricity from the Kano Distribution Company is 211,015. The tariff rate is ₦24.82. The residents receive an electricity bill with a balance of ₦12,658.20 before VAT and ₦13,291.11 after VAT. Determine the reading on the electricity meter at the end of March.

$\text{kWh used}=\frac{ &#8358;12,658.20}{ &#8358;24.82}=510$
The reading at the end of March, before the 510 kWh units were used, was:
$\begin{array}{r} 211,015 \\ -510 \\ \hline 210,505 \\ \hline \end{array}$
At the end of August, the reading on the water meter of a house is 7,192,000 litres. At the end of September, the reading is 7,195,750 litres. The domestic tariff in the area they live is ₦375 per 1 m $^3$ . The water company requires a monthly wastewater surcharge of ₦80. Calculate their outstanding balance at the end of September.

Remember: 1,000 L = 1 kl = 1 m $^3$

$\begin{array}{r} 7,195,750 \\ -7,192,000 \\ \hline 3,750 \\ \hline \end{array}$ $3,750 \text{ L} = 3.75 \text{ m}^3$
Cost of water used:
$3.75 \text{ m}^3 \times &#8358;375 = &#8358;1,406.25$
Cost of surcharge and water without VAT:
$&#8358;1,406.25+ &#8358;80= &#8358;1,486.25$
Cost with VAT:
$&#8358;1,486.25 \times \frac{105}{100} = &#8358;1,560.56$
At the end of September, the reading on the water meter of a house is 125,324 litres. The domestic rate for water is ₦375 per 1 m $^3$ and the monthly connection fee is ₦65. The residents receive a water bill with a balance of ₦2,408.75 before VAT and ₦2,529.19 after VAT. Determine the reading on the water meter at the end of August.

Cost of water used:
$&#8358;2,408.75- &#8358;65= &#8358;2,343.75$ $\text{units used}=\frac{ &#8358;2,343.75}{ &#8358;375}=6.25\text{ m}^3$
The reading at the end of August, before the use of $6.25\text{ m}^3\; (6,250\text{ L})$ :
$\begin{array}{r} 125,324 \\ -6,250 \\ \hline 119,074 \\ \hline \end{array}$
The total water bill, including VAT, for a family is ₦2,073.75. A fixed monthly connection fee of ₦50 is included and the family used 5.5 m $^3$ . Calculate the domestic tariff in that area.

Total cost without VAT:
$&#8358;2,073.75 \times \frac{100}{105} = &#8358;1,975.00$
Cost of water used:
$&#8358;1,975.00- &#8358;50= &#8358;1,925.00$
Domestic rate:
$\frac{ &#8358;1,925.00}{5.5} = &#8358;350$
A family live in an area where the domestic water rate is ₦380 per 1 m $^3$ . The family used 6 m $^3$ water in January. The total water bill, including VAT, is ₦2,493.75. Calculate the fixed monthly fee charged by the water company.

Total cost without VAT:
$&#8358;2,493.75 \times \frac{100}{105} = &#8358;2,375.00$
Cost of water used:
$6\text{ m}^3 \times &#8358;380 = &#8358;2,280.00$
Fixed fee:
$&#8358;2,375.00- &#8358;2,280.00= &#8358;95$
A primary school teacher earns ₦50,000. She spends 62% of her salary on rent, 27% on food and personal care and the rest on transport. Calculate how much money she spends on transport.

Percentage spent on transport:
$\text{Transport}=100-(62+27)=11\%$
Amount spent on transport:
$\text{Transport}=11\%\times &#8358;50,000= &#8358;5,500$
A student gets a monthly allowance of ₦120,000 from her parents to cover her living costs while she is studying. She pays ₦80,000 rent per month, which includes water and electricity. She needs about ₦20,500 for food and personal care, ₦7,500 for transport and ₦2,000 for mobile data.
1. How much money does she have left for entertainment?
\begin{array}{|r|l r|} \hline \text{Income}\phantom{00} & \text{Expenditure} & \\ \hline ₦120,000 & \text{Rent} & ₦80,000 \\ & \text{Groceries} & ₦20,500 \\ & \text{Transport} & ₦7,500 \\ & \text{Mobile data} & ₦2,000 \\ \hline ₦120,000 & & ₦110,000 \\ \hline \end{array}
She has ₦10,000 left for entertainment.
1. What percentage of her allowance is spent on rent?
$\text{Rent}=\frac{ &#8358;80,000}{ &#8358;120,000}\times100=66.67\%$
A family usually spends its total income as follows: 45% on rent, 22% on groceries, 5% on school fees, 9% on transport, 5% on clothing, and 12% on electricity and water. They save the rest for unforeseen expenses.
1. If they spent ₦77,000 on groceries, what was the total family income?
They spent ₦22 out of every ₦100 on groceries.
$\therefore \text{ Total income}= &#8358;77,000 \times \frac{100}{22}= &#8358;350,000$
1. Calculate the amount they save.
Percentage saved:
$\text{Saved}=100-(45+22+5+9+5+12)=2\%$
Amount saved:
$\text{Saved}=2\%\times &#8358;350,000= &#8358;7,000$

3.3 Summary

The cost price of an item is the price that a trader pays when she buys the item. It also includes any other money she spends on the item before she sells it.
The selling price of an item is the price at which a trader sells the item to a customer.
When the selling price of an item is more than its cost price, a trader makes a profit on the item.
When the selling price of an item is less than its cost price, a trader makes a loss on the item.
We can express profit or loss as an amount of money or as a percentage:
$\text{profit/loss} = \text{selling price} - \text{cost price}$ $\% \text{ profit/loss} = \dfrac{\text{profit/loss}}{\text{cost price}} \times 100$
Interest is extra money you pay on a loan or money you earn on an investment.
A loan is an amount of money you borrow, that you must normally pay back with interest.
An investment is an amount of money you give to a business or bank in the hope of getting more money back.
The principal amount is the original amount of money borrowed when a loan is taken out, or paid in when an investment is made.
For simple interest, the interest rate on the loan or investment is the percentage of the principal amount that you pay or earn per specified time period.
The number of time periods is the number of months or years over which a loan or an investment runs.
The actual amount is the actual amount of money you pay back on a loan or get out from an investment. It is the principal amount plus the simple interest.
We calculate simple interest as follows: $SI=P \times i \times n$
We calculate the actual amount of a loan or investment as follows:
\begin{align} A &= P+SI \\ \therefore A&=P + P\times i \times n \\ \text{or } A&=P(1+in) \end{align}
A discount is a reduction in the marked price of an item, so that the customer pays less than the normal price of the item.
We can express discount as an amount of money or as a percentage:
$\text{discount} = \text{normal price} - \text{discounted price}$ $\% \text{ discount} = \dfrac{\text{discount}}{\text{normal price}} \times 100$
Commission is money that an agent such as a sales person is paid to perform a service.
We calculate commission as follows: $\text{commission received}=\text{commission percentage} \times \text{value of sales/services}$
A bill is a statement of the money you owe for goods or services.
Electricity is sold in units called kilowatt-hours (kWh).
Water is commonly sold in cubic metres (m $^3$ ).
Value Added Tax of 5% is added to the cost of your electricity or water.
A budget is an estimate of your income and expenditure for a certain period of time.

Chapter 2: Fractions

Table of Contents

Chapter 4: Approximation