Financial Mathematics and Statistics 1: Business Computations

Solving Simple Linear Equations

To solve a linear equation, do the same operation to both sides until the unknown stands alone: first remove added or subtracted numbers, then divide or multiply to remove the number in front of the unknown. Always check the answer by putting it back into the original equation.

Here you are finding the value of the unknown letter that makes both sides equal. Whatever you do to one side, do the same to the other: first clear brackets, then move all the letter terms to one side and the numbers to the other, and finally divide by the number in front of the letter. Check by putting your answer back into the equation. Below, we show you how it is done, step by step.

Solving Business Problems with Linear Equations

The process is to read the problem carefully, let a letter stand for the unknown, write an equation from the facts given, solve it, and answer the question in words, with the correct units.

Here you are turning a business story into an equation. Choose a letter for the amount you do not know, then write each fact in the story using that letter, and join them with an equals sign. Solve the equation, and finish by answering the question in words. Below, we show you how it is done, step by step.

Rates

A rate compares two quantities measured in different units, such as $ per hour, km per litre or units per day. Rates are used to calculate wages, fuel costs and productivity, by multiplying the rate by the quantity.

Here you are using a rate, an amount per one unit of something else, to find a total. Write the rate clearly with its units, such as $ per hour or litres per 100 km. Multiply the rate by the quantity to find a total, or divide the total by the rate to find the quantity. Below, we show you how it is done, step by step.

Comparing Quantities Using Ratios

A ratio compares two or more quantities of the same kind, written as a:b, and is simplified by dividing all parts by the same number, so 40:60 becomes 2:3. To share an amount in a given ratio, add the parts, find the value of one part, and multiply.

Here you are sharing an amount in a ratio or simplifying a ratio. To share, add the parts of the ratio, divide the amount by the total parts to get one part, then multiply by each person's number of parts. To simplify, divide every part by the largest number that goes into all of them. Below, we show you how it is done, step by step.

Fractions, Decimals and Percentages

A percentage is a fraction out of 100. To change a fraction or decimal to a percentage, multiply by 100: ¾ = 0.75 = 75%. To find a percentage of an amount, multiply the amount by the percentage over 100. Percentage change = change ÷ original × 100, and to increase an amount by r%, multiply by (1 + r/100).

Here you are working with percentages of amounts and percentage changes. To find a percentage of an amount, multiply by the percentage over 100. For a percentage change, divide the change by the original figure and multiply by 100; to undo an increase, divide the new figure by (1 + the rate). Below, we show you how it is done, step by step.

Mixtures

A mixture problem combines items with different prices, such as blends of tea or mixed fruit packs, and asks for the price of the mixture or the quantities to mix. The price of a mixture is its total cost divided by its total quantity.

Here you are working out the price of a mixture, or how much of each item to mix. Find the total cost of everything mixed and divide by the total quantity to get the cost per unit. When the quantities are unknown, let one quantity be x and write an equation for the total cost. Below, we show you how it is done, step by step.

Direct and Inverse Proportion

In direct proportion, two quantities increase or decrease together at the same rate, so doubling one doubles the other; for example, the cost of goods rises with the quantity bought. In inverse proportion, one quantity increases as the other decreases, so doubling one halves the other; for example, the more workers on a job, the fewer days it takes.

Here you are deciding whether two quantities move together or in opposite directions. In direct proportion, find the value for one unit and multiply; in inverse proportion, multiply the two given figures to get the total work, then divide by the new figure. Always ask: should my answer be bigger or smaller? Below, we show you how it is done, step by step.

Types of Discounts

A trade discount is a reduction given by a manufacturer or wholesaler to other traders, off the list price, and is not recorded in the ledger. A cash discount (discount allowed or received) is given for paying promptly, such as 5% if paid within 14 days, and is recorded in the books. A quantity (bulk) discount is given for buying large quantities, and a seasonal discount for buying out of season. When there is both a trade and a cash discount, the trade discount is deducted first.

Here you are finding what a customer pays after discounts. Always take off the trade discount first, from the list price, to get the invoice amount. Then take the cash discount off the invoice amount, but only if the customer pays within the time allowed. Below, we show you how it is done, step by step.

Types of Commission

A sales commission is paid to salespeople as a percentage of the sales they make, sometimes on top of a basic salary. An agent's commission is paid to an agent, such as an estate agent or insurance agent, for arranging a sale. A graduated (sliding scale) commission pays a higher rate on sales above a target. Del credere commission is an extra commission paid to an agent who guarantees that customers will pay.

Here you are working out commission as a percentage of sales. Multiply the sales by the commission rate; on a sliding scale, split the sales into bands and use each band's rate. To find the sales from the commission, divide the commission by the rate. Below, we show you how it is done, step by step.

Meaning of Hire Purchase

Hire purchase (HP) is an agreement in which the buyer pays a deposit and then a number of instalments, and uses the goods from the start, but ownership passes to the buyer only when the last instalment is paid. If the buyer fails to pay, the seller may repossess the goods, subject to the law and the terms of the agreement.

Hire Purchase Price and Cash Price

The cash price is the price paid if the goods are bought outright today. The hire purchase price is the total of the deposit and all the instalments. The difference between them is the HP charge, or interest, which is the cost of paying over time. Buyers should compare the two and work out the interest rate before signing.

Solving Hire Purchase Problems

The procedure is: find the deposit, find the total of the instalments (number × amount), add them to get the HP price, then subtract the cash price to get the HP charge, and express it as a percentage of the cash price if asked.

Here you are comparing the hire purchase price with the cash price. Work out the deposit, then the total of all the instalments, and add them to get the HP price. The extra cost is the HP price minus the cash price, and you can show it as a percentage of the cash price. Below, we show you how it is done, step by step.

Meaning of Currency

A currency is the money accepted as a medium of exchange in a country, such as the US dollar (USD), the Zimbabwe Gold (ZiG), the South African rand (ZAR) and the pound sterling (GBP).

An exchange rate is the price of one currency in terms of another, for example US$1 = R18.50, so it tells us how many units of one currency are needed to buy one unit of another.

Different Currency Rates

A bank's buying rate is the rate at which the bank buys foreign currency from customers, and its selling rate is the rate at which it sells foreign currency to customers; the selling rate is less favourable to the customer, and the difference is the bank's profit. The official or interbank rate is set or published by the central bank, while a cross rate is an exchange rate between two currencies worked out through a third currency. Rates change often, so the rate for the actual date must always be used. The rates in the examples are for illustration only.

Currency Conversion Methods

When the rate is given as 1 unit of A = r units of B, multiply to change A into B and divide to change B into A. A cross rate is found by combining two rates through a common currency.

Here you are changing money from one currency to another. Write the rate as 1 unit of one currency = so many units of the other, then multiply to go from the first currency to the second and divide to go back. When buying and selling rates are given, remember the bank always uses the rate that is better for the bank. Below, we show you how it is done, step by step.