- Add and subtract fractions
Add Fractions with a Common Denominator
Adding fractions with the same denominator is a straightforward process. The numerators of the fractions are simply added together to get the numerator of the answer. The denominator will stay the same, since the denominators are the same for all the fractions being added. In this section, we will discuss how to add fractions with the same denominator, and also look at a few examples to illustrate the process.
Fraction Addition
If [latex]a,b,\text{ and }c[/latex] are numbers where [latex]c\ne 0[/latex], then
To add fractions with a common denominators, add the numerators and place the sum over the common denominator.
Find the sum:
[latex]\Large\frac{3}{5}\normalsize+\Large\frac{1}{5}[/latex]
Subtract Fractions with a Common Denominator
Subtracting fractions with the same denominator is a simple process. When the denominators are the same, all you need to do is subtract the numerators to get the difference. In this section, we‘ll discuss the steps for subtracting fractions with the same denominator and provide examples to illustrate the process.
Fraction Subtraction
If [latex]a,b,\text{ and }c[/latex] are numbers where [latex]c\ne 0[/latex], then
[latex]{\Large\frac{a}{c}}-{\Large\frac{b}{c}}={\Large\frac{a-b}{c}}[/latex]
To subtract fractions with a common denominators, we subtract the numerators and place the difference over the common denominator.
Find the difference:
[latex]{\Large\frac{23}{24}}-{\Large\frac{14}{24}}[/latex]
Add or Subtract Fractions with Different Denominators
To add or subtract fractions with different denominators, we first must write them as equivalent fractions having the same denominator. We’ll use the techniques from the previous section to find the LCM of the denominators of the fractions. Recall that we call this the LCD (the least common denominator). We only use the denominators of the fractions, not the numerators, when finding the LCD.
Then we can use the Equivalent Fractions Property to algebraically change a fraction to an equivalent one. Remember, two fractions are equivalent if they have the same value. The steps for finding the LCD and the Equivalent Fractions Property are repeated below for reference.
Least Common Denominator
The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.
Equivalent Fractions Property: If [latex]a,b,c[/latex] are whole numbers where [latex]b\ne 0,c\ne 0,\text{then}[/latex]
[latex]\Large\frac{a}{b}=\Large\frac{a\cdot c}{b\cdot c}\normalsize\text{ and }\Large\frac{a\cdot c}{b\cdot c}=\Large\frac{a}{b}[/latex]
Once we have converted two fractions to equivalent forms with common denominators, we can add or subtract them by adding or subtracting the numerators.
How to: Add or Subtract Fractions with Different Denominators
- Find the Least Common Denominator (LCD): Identify the smallest number that both denominators can divide into evenly. This is the least common denominator.
- Convert Each Fraction to an Equivalent Form with the LCD as the Denominator: For each fraction, determine what number you must multiply the denominator by to reach the LCD. Multiply both the numerator and denominator of the fraction by this number to create an equivalent fraction with the LCD.
- Add or Subtract the Fractions: Now that both fractions have the same denominator, you can add or subtract the numerators directly. Keep the common denominator the same.
- Write the Result in Simplified Form: If the numerator is larger than the denominator, you may need to convert it to a mixed number. Reduce the fraction to its simplest form by dividing both the numerator and denominator by their greatest common divisor (GCD).