Arithmetic, fractions, ratios and proportions for business

Business mathematics is not advanced mathematics. Almost everything a manager, bookkeeper, buyer or small-business owner needs can be done with the four basic operations, fractions, percentages and a handful of formulas. What makes it hard is not the arithmetic itself but knowing which calculation answers the business question, setting it up correctly and checking that the answer makes sense. This course is built around that skill. Each lesson takes one family of business calculations, shows where it comes from, works through realistic examples and points out the mistakes people actually make.

This first lesson lays the foundation: the order of operations, rounding and precision, fractions and decimals, ratios, rates and proportions. These ideas reappear in every later lesson, from sharing profit among partners to converting currencies and allocating overhead costs.

Examples in this course use Canadian dollars, Canadian tax rules and Canadian conventions unless a comparison is useful. Figures that change, such as tax rates and payroll limits, are stamped with the year they apply to.

Order of operations

A business formula is only useful if everyone evaluates it the same way. The standard order is: brackets first, then exponents (powers), then multiplication and division from left to right, then addition and subtraction from left to right. Spreadsheets and calculators follow the same rule, which is why a misplaced bracket is one of the most common sources of spreadsheet errors.

Rounding and precision

Money is usually rounded to the nearest cent, but rounding too early causes errors that grow through a calculation. The safest habit is to keep full precision in intermediate steps and round only the final answer. When a value lands exactly halfway, the usual rule taught in business courses is to round up (so $2.345 becomes $2.35), although some software uses banker's rounding, which rounds halves to the nearest even digit. Interest rates and exchange rates are normally kept to at least four decimal places during a calculation.

Precision also means reporting sensible digits. A forecast that says sales will be $1,284,617.23 next year suggests a precision nobody has. It is usually better to say about $1.28 million. On the other hand, an invoice must be exact to the cent, because a customer is entitled to check it.

Fractions, decimals and percentages

A fraction, a decimal and a percentage are three ways of writing the same number. One quarter can be written as \( \frac{1}{4} \), as 0.25 or as 25 %. To convert a fraction to a decimal, divide the top number (numerator) by the bottom number (denominator). To convert a decimal to a percentage, multiply by 100. To go back, divide by 100.

Repeating decimals such as one third (0.3333…) are a trap. If three partners each receive 33.33 % of a $10,000 profit, the shares add up to $9,999, not $10,000. The usual fix is to calculate two shares and give the remainder to the third, so the total always balances.

Mixed numbers appear in trades and purchasing: a board may be 2\( \frac{3}{8} \) inches thick, or an order may be for 4\( \frac{1}{2} \) cases. To multiply or divide mixed numbers, convert them to improper fractions or decimals first. To add fractions, put them over a common denominator.

Ratios

A ratio compares two or more quantities of the same kind. Ratios are written with a colon, such as 3 : 2, or as a fraction. Businesses use ratios to divide profit among partners, to allocate costs among departments, to mix ingredients and to compare financial results. A ratio can be scaled up or down by multiplying or dividing every term by the same number, so 6 : 4 and 3 : 2 are equivalent. A ratio is in lowest terms when the terms share no common factor.

To allocate an amount in a given ratio, add the terms of the ratio to find the total number of parts, divide the amount by the total parts to find the value of one part, then multiply by each term.

Ratios are also used to allocate shared costs. A company that rents one building for three departments might split the rent by floor area. A factory might allocate the cost of its maintenance crew by machine hours. The choice of basis matters: a basis that does not reflect what really drives the cost will make some products look more profitable than they are. Managerial accounting courses return to this idea under the name activity-based costing.

Rates and unit rates

A rate compares quantities of different kinds, such as dollars per hour, kilometres per litre or units per shift. A unit rate expresses the comparison per one unit of the second quantity, which makes options easy to compare. Unit pricing on store shelves, wage rates, productivity measures and exchange rates are all unit rates.

Proportions

A proportion is a statement that two ratios are equal, such as \( \frac{a}{b} = \frac{c}{d} \). If three of the four values are known, the fourth can be found by cross-multiplying: \( a \times d = b \times c \). Proportions solve many everyday business problems: scaling a recipe, estimating materials for a larger job, converting currencies or projecting costs from a sample.

Exchange rates are quoted both ways, so always check which currency is the base. If 1 US dollar buys 1.3850 Canadian dollars, then 1 Canadian dollar buys 1 ÷ 1.3850 = 0.7220 US dollars. The rates in this lesson are illustrative; real rates change every second the markets are open.

Checking your answers

Good business mathematicians check every answer in three ways. First, estimate before calculating: if 2,400 units cost about $3.50 each, the total should be about $8,400, so an answer of $84,000 or $840 is clearly wrong. Second, check that allocations add back to the total. Third, ask whether the answer makes business sense: a profit margin of 140 % or a negative wage is a sign that something was set up incorrectly.

# Allocate an amount in a ratio, giving any rounding remainder to the last share
def allocate(amount, ratio):
    total = sum(ratio)
    shares = [round(amount * r / total, 2) for r in ratio[:-1]]
    shares.append(round(amount - sum(shares), 2))
    return shares

print(allocate(90000, [4, 3, 2]))
print(allocate(10000, [1, 1, 1]))

[40000.0, 30000.0, 20000.0]
[3333.33, 3333.33, 3333.34]

The short program above allocates an amount in any ratio and makes sure the shares add exactly to the total. Spreadsheet users can do the same by calculating all but the last share with a formula and setting the last share equal to the total minus the others.