Review of Arithmetic

Objectives

This chapter introduces foundational arithmetic concepts essential for business mathematics and college algebra. Students will learn to simplify expressions, work with fractions and decimals, calculate averages, and apply arithmetic to payroll and tax scenarios.

• Simplify arithmetic expressions using the basic order of operations.

• Determine equivalent fractions and convert fractions to decimals and vice versa.

• Compute simple and weighted averages through problem solving.

• Calculate gross earnings for employees paid by salary, hourly wages, or commissions.

• Compute GST, HST, PST, sales taxes, and property taxes through problem solving.

Basic Order of Operations (BEDMAS)

Definition and Steps

The order of operations ensures that arithmetic expressions are evaluated consistently. The acronym BEDMAS stands for:

1. Perform all operations inside brackets first.

2. Evaluate exponents.

3. Perform multiplication and division from left to right.

4. Perform addition and subtraction from left to right.

Examples

• Example 1: (work inside the bracket first)

• Example 2: (do multiplication before subtraction)

• Example 3: (do multiplication and division before adding)

• Example 4: (work inside the bracket first, then do division before subtraction)

Common Fractions

Definition and Types

A common fraction represents a part of a whole. The fraction means four parts out of five.

Equivalent Fractions

Changing Terms Without Changing Value

• Higher Terms: Multiply both numerator and denominator by the same number. For any fraction, there are unlimited equivalent fractions in higher terms.

• Lower Terms: Divide both numerator and denominator by the same number(s). This process is called reducing to lower terms.

Example

is equivalent to (multiply numerator and denominator by 2).

Lowest Common Denominator (LCD)

Definition and Calculation

The LCD is the lowest common multiple of the denominators of two or more fractions. It is useful for adding, subtracting, or comparing fractions.

1. Divide denominators by integers (2 or greater) until all are reduced to 1.

2. Multiply all divisors to determine the LCD.

LCD = 2 × 3 × 3 × 5 = 90

Converting Fractions and Mixed Numbers

Fractions to Decimals

• Divide the numerator by the denominator.

• For repeating decimals, use a bar or period above the repeating sequence.

Mixed Numbers to Decimals

• Mixed numbers consist of a whole number and a fraction (e.g., ).

• Convert the fraction to a decimal and add to the whole number.

Rounding

Rules and Examples

• If the first digit to be dropped is 5, 6, 7, 8, or 9, increase the last digit retained by 1.

• If the first digit to be dropped is 0, 1, 2, 3, or 4, leave the last digit retained unchanged.

Examples

• (drop the digit 4)

• (round the digit 8 up to 9)

• (drop the 48)

• (round the second digit 9 up to 0)

Complex Fractions

Definition

Complex fractions are expressions containing one or more fractions in the numerator, denominator, or both.

Using a calculator, the 1/x function can be useful for evaluating complex fractions.

Percents

Meaning and Conversion

Percent means "per hundred". The symbol % means "parts of one hundred".

Changing Percents to Fractions and Decimals

• Replace the % symbol by and reduce to lowest terms.

• To convert percent to decimal, drop the % symbol and move the decimal two places to the left (divide by 100).

Changing Decimals to Percents

• Move the decimal two places to the right and add the % symbol (multiply by 100).

Changing Fractions to Percents

• Convert the fraction to a decimal, then convert the decimal to percent.

Averages

Arithmetic Average (Mean)

The arithmetic mean is the sum of all values divided by the number of values.

• Formula:

• Example: For test scores 82, 68, 83, 72, 76, 96, and 83:

Weighted Average

A weighted average uses a weighting factor to indicate the number of items or their relative importance.

• Multiply each item by its weight, sum the products, and divide by the total weight.

Weighted Average GPA:

Applications: Payroll Salaries, Wages, and Commissions

Payroll Salaries

• Employees may be paid monthly, semi-monthly, biweekly, or weekly.

• If paid weekly or biweekly, the year is assumed to have 52 weeks.

Wages

• Compensation paid to hourly employees.

• Gross earnings formula:

Gross Pay and Overtime

• Most common workweek is 40 hours.

• Overtime is paid at least at time-and-a-half the regular rate.

• Two methods for calculating overtime:

  • Method A: Add overtime pay to regular gross pay.

  • Method B: Calculate overtime premium separately and add to gross earnings.

Example Calculation: Regular pay: Overtime pay: Gross earnings:

Commissions

• Straight commission: Percent of net sales.

• Graduated commission: Increasing percent for higher sales levels.

• Salary plus commission: Guaranteed minimum income plus commission on sales above quota.

Applications: Taxes

Types of Taxes

• Provincial Sales Tax (PST): Applied as a percent of retail price in some provinces.

• Goods and Services Tax (GST): Federal tax, currently 5%.

• Harmonized Sales Tax (HST): Combined PST and GST in some provinces.

Example: Calculating Total Cost with HST and Tip

• Food: $60, Wine: $32, Total: $92

• HST on food:

• HST on wine:

• Total including taxes:

• Tip (15%):

• Total spent:

Property Tax

Definition and Calculation

• Municipal tax charged on the assessed value of real estate.

• Calculated by applying a percent (millage rate) to the assessed value.

• In some municipalities, assessed value is divided by 1000, then percent is applied.

Example

• Property assessed at $400,000, millage rate 15 mills.

• Tax = $400,000 / 1000 × 15 = $6,000

Summary

Common business applications such as payroll, commissions, and taxes require proficiency in arithmetic operations, fractions, percents, and decimals. Understanding averages and weighted averages is essential for data analysis in business contexts.