This guide provides useful resources for a wide variety of math topics. It is targeted at students enrolled in a math course or any other Centennial course that requires math knowledge and skills.

- Welcome
- Learning Math Strategies (Online)Toggle Dropdown
- Study Skills for MathToggle Dropdown
- Simply Math
- Business MathToggle Dropdown
- How to use a scientific calculator
- Place Value in Decimal Number Systems
- Arithmetic Operations
- Basic Laws
- Operations on Signed numbers
- Order of Operations
- Fractions
- Decimals
- Percents
- Ratios and Proportions
- Exponents
- Statistics
- Factoring
- Rearranging Formulas
- Solving Linear Equations
- Solving Systems of Linear Equations
- Trade and Cash Discounts
- Multiple Rates of Discount
- Payment Terms and Cash Discounts
- Markup
- Markdown
- Simple Interest
- Equivalent Values
- Compound Interest
- Equivalent Values in Compound Interest
- Nominal and Effective Interest Rates
- Ordinary Simple Annuities
- Ordinary General Annuities

- Hospitality Math
- Place Value in Decimal Number Systems
- Arithmetic Operations
- Order of Operations
- Basic Laws
- Prime Factorisation and Least Common Multiple
- Fractions
- Decimals
- Percents
- Exponents
- Units of Measures
- Fluid Ounces and Ounces
- Metric Measures
- Yield Percent
- Recipe Size Conversion
- Ingredient Ratios
- Food-Service Industry Costs

- Engineering MathToggle Dropdown
- Basic Laws
- Order of Operations
- Prime Factorisation and Least Common Multiple
- Fractions
- Exponents
- Radicals
- Reducing Radicals
- Factoring
- Rearranging Formulas
- Solving Linear Equations
- Areas and Volumes of Figures
- Congruence and Similarity
- Functions
- Domain and Range of Functions
- Basics of Graphing
- Transformations
- Graphing Linear Functions
- Graphing Quadratic Functions
- Solving Systems of Linear Equations
- Solving Quadratic Equations
- Solving Higher Degree Equations
- Trigonometry
- Graphing Trigonometric Functions
- Graphing Circles and Ellipses
- Exponential and Logarithmic Functions
- Complex Numbers
- Number Bases in Computer Arithmetic
- Linear Algebra
- Calculus
- Set Theory
- Modular Numbers and Cryptography
- Statistics
- Problem Solving Strategies

- Upgrading / Pre-HealthToggle Dropdown
- Basic Laws
- Place Value in Decimal Number Systems
- Decimals
- Significant Digits
- Prime Factorisation and Least Common Multiple
- Fractions
- Percents
- Ratios and Proportions
- Exponents
- Radicals
- Reducing Radicals
- Metric Conversions
- Factoring
- Solving Linear Equations
- Solving Quadratic Equations
- Functions
- Domain and Range of Functions
- Polynomial Long Division
- Exponential and Logarithmic Functions
- Statistics

- Nursing MathToggle Dropdown
- Arithmetic Operations
- Order of Operations
- Place Value in Decimal Number Systems
- Decimals
- Fractions
- Percents
- Ratios and Proportions
- Nutrition Labels
- Interpreting Drug Orders
- Oral Dosages
- Dosage Based on Size of the Patient
- Parenteral Dosages
- Intravenous (IV) Administration
- Infusion Rates for Intravenous Piggyback (IVPB) Bag
- General Dosage Rounding Rules

- Transportation MathToggle Dropdown
- PhysicsToggle Dropdown

A **proper fraction** is a fraction whose numerator is smaller than its denominator.

For example: \( \dfrac{1}{2} , \dfrac{4}{5} , \dfrac{3}{7} \)

Consider the proper fraction \( \dfrac{4}{5} \).

It can be visualised as:

An **improper fraction** is a fraction whose numerator is greater than its denominator.

For example: \( \dfrac{3}{2} , \dfrac{7}{4} , \dfrac{9}{5} \)

Consider the improper fraction \( \dfrac{7}{4} \).

It can be visualised as:

A **mixed ****fraction **or **mixed number** is a combination of a whole number and a proper fraction.

For example: \( 3\dfrac{1}{2} , 5\dfrac{2}{5} , 1\dfrac{3}{4} \)

The mixed fraction \(1\dfrac{3}{4}\) means there is one whole part plus \(\dfrac{3}{4}\) of a whole part.

In fact, \(1\dfrac{3}{4} = 1 + \dfrac{3}{4}\) and can be visualised as:

Notice that the shaded region in the circles representing \(\dfrac{7}{4}\) and \(1\dfrac{3}{4}\) are the same!

In fact, \(1\dfrac{3}{4} = \dfrac{7}{4}\).

They are different ways of representing the same thing!

**To switch from mixed to improper:**

- Multiply the whole number by the denominator of the proper fraction
- Add to the numerator of the proper fraction.

`Examples`

- \( 4\frac{2}{3} = \dfrac{4\times3+2}{3} = \dfrac{12+2}{3} = \dfrac{14}{2}\)

- \( 5\frac{1}{4} = \dfrac{5\times4+1}{4} = \dfrac{20+1}{4} = \dfrac{21}{4}\)

- \( 2\frac{3}{7} = \dfrac{2\times7+3}{7} = \dfrac{14+3}{7} = \dfrac{17}{7}\)

**To switch from improper to mixed:**

- Use long division to divide the denominator into the numerator.
- We will use the Quotient and Remainder to write our Mixed Fraction.

The improper fraction in mixed form is: \(\text{Quotient}\dfrac{\text{Remainder}}{\text{Denominator}}\)

`Example`

Suppose we want to change \( \dfrac{21}{4} \) to a mixed fraction.

\(21\) divided by \(4\) is \(5\) with remainder \(1\).

So:

\( \dfrac{21}{4} = 5\frac{1}{4} \)

**To add/subtract fractions:**

- Make sure the denominators are the same.**

- Add/Subtract the numerators and put that answer over the denominator.

- Simplify the fraction if necessary.

(Done by dividing numerator and denominator by their Greatest Common Factor (GCF)).

**If they are not, this can be done by rewriting one/both of the fractions by multiplying both numerator and denominator by the same value.

`Example`

\( \dfrac{1}{2} \) is the same as \( \dfrac{2}{4} \) because \( \dfrac{1}{2} = \dfrac{1\times 2}{2\times 2} = \dfrac{2}{4} \)

For simplicity, we aim to multiply both fractions separately to ensure each denominator becomes the **least common multiple (LCM)** of the two original denominators. See Example 2.

Suppose we want to add: \( \dfrac{3}{12} + \dfrac{5}{12} \).

The denominators are the same, so we add the numerators and simplify:

\begin{align} \dfrac{3}{12}+\dfrac{5}{12} &= \dfrac{8}{12} \\ &= \dfrac{8 \div 4}{12 \div 4} \\ &= \dfrac{2}{3} \end{align}

The above calculation can be visualised as:

Suppose we want subtract: \( \dfrac{2}{5} - \dfrac{1}{4} \).

The denominators are not the same so we must rewrite one/both of the fractions.

The least common multiple of \(5\) and \(4\) is \(20\), so we need to multiply each fraction above and below so that their denominator becomes \(20\).

\begin{align} \dfrac{2}{5} &= \dfrac{2\times4}{5\times4} \\ &= \dfrac{8}{20} \end{align}

and

\begin{align} \dfrac{1}{4} &= \dfrac{1\times5}{4\times5} \\ &= \dfrac{5}{20} \end{align}

Now that the denominators are the same, the fractions can be subtracted as usual:

\begin{align} \dfrac{2}{5}-\dfrac{1}{4} &= \dfrac{8}{20}-\dfrac{5}{20} \\ &= \dfrac{3}{20} \end{align}

The above calculation can be visualised as:

**To multiply fractions:**

- Multiply the numerators together.
- Multiply the denominators together.
- As always, simplify if necessary.

In other words, remember the rule "**Top by top, Bottom by bottom**".

`Example`

\begin{align} \dfrac{3}{5} \times \dfrac{2}{7} &= \dfrac{3\times2}{5\times7} \\ &= \dfrac{6}{35} \end{align}

**To divide fractions:**

- Change the division symbol to a multiplication symbol.
- Flip the divisor.

The divisor is the dividing fraction in this case, like \(\frac{1}{2}\) in \( \frac{3}{4} \div \frac{1}{2}\) - Perform the multiplication, and as always, simplify if necessary.

In other words, remember the rule "**Keep, Change, Flip**".

`Example`

\begin{align} \dfrac{2}{9} \div \dfrac{3}{4} &= \dfrac{2}{9} \times \dfrac{4}{3} \qquad \qquad \qquad \text{(1)} \\ &= \dfrac{2\times4}{9\times3} \\ &= \dfrac{8}{27} \end{align}

(1)* Keep the* \( \dfrac{2}{9} \) , *Change the division to multiplication, Flip the* \( \dfrac{3}{4} \)

- Last Updated: Jun 21, 2024 3:54 PM
- URL: https://libraryguides.centennialcollege.ca/mathhelp
- Print Page

chat loading...