Fractions

A fraction has two numbers separated by a fraction bar: the numerator (on top) and the denominator (on the bottom).

To represent the fraction \(\dfrac{\textcolor{colordef}{2}}{\textcolor{colorprop}{3}}\) on a number line:

1. Draw a straight line and mark the points \(0\) and \(1\).
2. Divide the segment from \(0\) to \(1\) into \(\textcolor{colorprop}{3}\) equal parts (the denominator).
3. Count \(\textcolor{colordef}{2}\) parts from \(0\) (the numerator) and mark the point at \(\dfrac{\textcolor{colordef}{2}}{\textcolor{colorprop}{3}}\).

Two fractions are equivalent if they represent the same amount. You can make an equivalent fraction by multiplying or dividing the numerator and the denominator by the same nonzero number.

To simplify a fraction is to find an equivalent fraction that uses the smallest possible whole numbers for its numerator and denominator.
The procedure is to divide both the numerator and the denominator by the same number (a common factor). This process may need to be repeated until there are no more common factors (other than 1) that can divide both the numerator and the denominator.

Simplify \(\dfrac{4}{6}\).

Solve \(x\) for \(\dfrac{10}{5}=\dfrac{x}{8}\).

For two fractions with the same denominator, the fraction with the larger numerator is larger.

Compare \(\dfrac{6}{8}\) and \(\dfrac{5}{8}\).

The standard procedure for comparing two fractions with different denominators is as follows:

1. Find a common denominator: Identify a common multiple of both denominators. A simple method is to multiply the denominators together.
2. Create equivalent fractions: Convert each fraction into an equivalent fraction with the chosen common denominator.
3. Compare the numerators: Once the denominators are the same, the fraction with the larger numerator is the greater fraction.

Compare \(\dfrac{3}{4}\) and \(\dfrac{5}{8}\).

To add fractions with the same denominator, add the numerators and keep the denominator the same:

To subtract fractions with the same denominator, subtract the numerators and keep the denominator the same:

To add or subtract fractions with unlike denominators, follow this three-step procedure:

1. Find a Common Denominator: Identify a common multiple of the denominators.
2. Create Equivalent Fractions: Convert each fraction to an equivalent fraction with the common denominator.
3. Add or Subtract the Numerators: With the denominators now the same, perform the operation on the numerators and keep the common denominator.

Calculate \(\dfrac{3}{4} + \dfrac{5}{6}\).

For any integers \(a\) and \(b\) (where \(b \neq 0\)), the division of \(a\) by \(b\) is represented by the fraction \(\dfrac{a}{b}\).$$a \div b = \frac{a}{b}$$In this context:

• The numerator (\(a\)) corresponds to the dividend.
• The denominator (\(b\)) corresponds to the divisor.

Consequently, the fraction \(\dfrac{a}{b}\) is the number which, when multiplied by the divisor \(b\), yields the dividend \(a\).$$ \frac{a}{b} \times b = a $$

The fraction \(\dfrac{2}{3}\) is the same as saying "2 divided by 3". The fraction \(\dfrac{2}{3}\) is the number which, when multiplied by \(3\), gives \(2\):$$\dfrac{2}{3}\times 3 = 2 $$

To find a fraction of a quantity, multiply that quantity by the fraction:$$\textcolor{colorprop}{\frac{a}{b}\,\text{of }N=\frac{a}{b}\times N}$$

In a class of 30 students, the ratio of girls to the total number of students is \(\dfrac{2}{3}\) (i.e., two thirds of the class are girls). How many girls are there?

There are two primary methods for converting a fraction to its decimal equivalent.

• Method 1: Direct Division
  Since a fraction \(\frac{a}{b}\) is equivalent to the division \(a \div b\), perform the division of the numerator by the denominator.
• Method 2: Denominator as a Power of 10
  Find an equivalent fraction where the denominator is a power of 10 (e.g., 10, 100, 1000). The numerator of this new fraction can then be written as a decimal.

Convert \(\dfrac{3}{4}\) to a decimal number.

The procedure for converting a terminating decimal to a fraction is as follows:

1. Write the decimal as the numerator of a fraction without the decimal point.
2. The denominator is 1 followed by as many zeros as there are decimal places in the original number.
3. Simplify the fraction to its lowest terms, if necessary.

Convert \(1.3\) to a fraction.

Fractions are classified based on the relationship between the numerator and the denominator.

• A proper fraction is a fraction where the numerator is less than the denominator. Its value is always less than 1. Example: \(\dfrac{2}{3}\).
• An improper fraction is a fraction where the numerator is greater than or equal to the denominator. Its value is always greater than or equal to 1. Example: \(\dfrac{5}{3}\).

A mixed number is an alternative way to represent an improper fraction. It consists of an integer part (the number of whole units) and a proper fraction part.
The improper fraction \(\dfrac{5}{2}\) can be visualized as two whole units and one half.This is written as the mixed number \(2\dfrac{1}{2}\). By convention, the addition sign is omitted:$$ 2\frac{1}{2} \quad \text{is equivalent to} \quad 2 + \frac{1}{2} $$Caution: A mixed number like \(2\frac{1}{2}\) always means \(2+\frac{1}{2}\), not \(2\times\frac{1}{2}\). If you mean multiplication, write \(2\times\frac{1}{2}\), \(2\cdot\frac{1}{2}\), or \((2)\!\left(\frac{1}{2}\right)\).