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.

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- Place Value in Decimal Number Systems
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- Basic Laws
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- Basic Laws
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- Decimals
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- Polynomial Long Division
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- Arithmetic Operations
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A **proper fraction** is a fraction whose numerator is smaller than its denominator. For example,

\[ \frac12, \frac45, \frac37 \]

Consider the proper fraction \(\frac45\) can be visualised as:

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

\[ \frac32, \frac74,\frac95 \]

The improper fraction \(\frac74\) can be visualised as:

A **mixed fraction** is a combination of a whole number and a proper fraction. For example,

\[ 3\frac12, 5\frac25, 1\frac34 \]

The mixed fraction \(1\frac34\) means there is one whole part plus \(\frac34\) of a whole part. In fact, \(1\frac34 = 1 + \frac34\) and can be visualised as follows:

Notice that the shaded region in the circles representing \(\frac74\) and \(1\frac34\) are the same! In fact, \(\frac74 = 1\frac34\). They are different ways of representing the same thing!

**Switching from mixed to improper:**

To switch from mixed to improper, multiply the whole number by the denominator of the proper fraction and add to the numerator of the proper fraction:

For example:

- \(\displaystyle 4\tfrac23 = \frac{4\times3 + 2}{3} = \frac{12+2}{3} = \frac{14}{2}\)
- \(\displaystyle 5\tfrac14 = \frac{5\times4 + 1}{4} = \frac{20+1}{4} = \frac{21}{4}\)
- \(\displaystyle 2\tfrac37 = \frac{2\times7 + 3}{7} = \frac{14+3}{7} = \frac{17}{7}\)

**Switching from improper to mixed:**

To switch from improper to mixed, use long division to divide the denominator into the numerator, finding the quotient and the remainder. The improper fraction is mixed form is

\[ \text{Quotient}\frac{\text{Remainder}}{\text{Denominator}} \]

For example, suppose we want to change \(\frac{21}{4}\) to a mixed fraction. \(21\) divided by \(4\) is \(5\) with remainder \(1\). so

\[ \frac{21}{4} = 5\frac14 \]

To add fractions, take the following steps:

- Make sure the denominators are the same.
- Add the numerators and put that answer over the denominator.
- Simplify the fraction if necessary.

For example, suppose we want to add

\[ \frac{3}{12} + \frac{5}{12} \]

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

\[ \frac{3}{12} + \frac{5}{12} = \frac{8}{12} = \frac23 \]

The calculation above can be visualised as follows:

If the denominators are different, you must take the following steps to make them the same:

- Find the least common multiple of the denominators (see Prime Factorisation and Least Common Multiple)
- Multiply each fraction above and below so that the each denominator becomes the least common multiple.
- Add the fractions as shown above.

For example, suppose we want to add

\[\frac25 - \frac14\]

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 \(20\).

\[\frac25 = \frac{2\times 4}{5\times 4} = \frac{8}{20} \]

\[\frac14 = \frac{1\times 5}{4\times5} = \frac{5}{20} \]

Now that the denominators are the same, the fractions can be added as before:

\[\frac25 - \frac14 = \frac{8}{20} - \frac{5}{20} = \frac{3}{20} \]

See the picture below for a visual explanation of what was done.

See the video below for another example of adding and subtracting fractions.

To multiply fractions, just remember the following rule:

**Top by top, bottom by bottom**

For example,

\[ \frac35 \times \frac27 = \frac{3 \times 2}{5\times 7} = \frac{6}{35} \]

To divide fractions, you just need to flip the dividing fraction and then multiply:

Flip and multiply

For example,

\[ \frac29 \div \frac34 = \frac29 \times \frac43 = \frac{2 \times 4}{9 \times 3} = \frac{8}{27} \]

Notice that the \(\dfrac34\) gets flipped to \(\dfrac43\) and multiplied instead.

- Last Updated: Mar 25, 2023 5:34 PM
- URL: https://libraryguides.centennialcollege.ca/mathhelp
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