- Add, subtract, multiple and divide fractions
Fractions
A fraction is written [latex]\dfrac{a}{b}[/latex], where [latex]a[/latex] and [latex]b[/latex] are integers and [latex]b \ne 0[/latex].
In a fraction, [latex]a[/latex] is called the numerator and [latex]b[/latex] is called the denominator.
Adding Fractions
The first step in adding fractions is to check if they have the same bottom number, also known as a ‘common denominator.’ When the two fractions have a common denominator, then adding the two numbers is straightforward – add the numerators, and then place that value in the numerator and the common denominator in the denominator. When the fraction does not have common denominators, then we have to transform the fractions so that they do have common denominators. This is technically called finding the least common multiple (LCM).
We can find the least common multiple using the prime factorization method.
- Find the prime factors of each denominator. You can use a factor tree or division method to break down each number into its prime factors.
- List down all the unique prime factors that appear in the prime factorization of each number.
- For each unique prime factor, identify the highest power to which it is raised in any of the given numbers.
- Multiply together the highest powers of all the unique prime factors. The result is the least common multiple (LCM) of the given numbers.
Now that we know how to find the least common multiple, adding fractions with unlike denominators becomes easier.
- Find a common denominator.
- Rewrite each fraction using the common denominator.
- Now that the fractions have a common denominator, you can add the numerators.
- Simplify by canceling out all common factors in the numerator and denominator.
Subtracting Fractions
When you subtract fractions, you must think about whether they have a common denominator, just like with adding fractions. When the two fractions have a common denominator, then subtracting the two numbers is straightforward – subtract the numerators, and then place that value in the numerator and the common denominator in the denominator.
Just like when adding fractions, when subtracting fractions that do not have common denominators, we have to transform the fractions so that they do have common denominators. This can be done the same way we did when adding fractions.
- Find a common denominator.
- Rewrite each fraction using the common denominator.
- Now that the fractions have a common denominator, you can subtract the numerators.
- Simplify by canceling out all common factors in the numerator and denominator.
Multiplying Fractions
Just as you add, subtract, multiply, and divide when working with whole numbers, you also use these operations when working with fractions. Multiplying fractions is less complicated than adding or subtracting fractions, as there is no need to find common denominators. To multiply fractions, multiply the numerators, then multiply the denominators, and write the numerator product divided by the denominator product.
Dividing Fractions
Before discussing division of fractions, we should look at the reciprocal of a number. The reciprocal of a number is [latex]1[/latex] divided by the number. For a fraction, the reciprocal is the fraction formed by switching the numerator and denominator. An important feature for a number and its reciprocal is that their product is [latex]1[/latex]. Sometimes we call the reciprocal the “flip” of the other number: flip [latex]\frac{2}{5}[/latex] to get the reciprocal [latex]\frac{5}{2}[/latex].
When dividing two fractions, find the reciprocal of the divisor (the number that is being divided into the other number). Next, replace the divisor by its reciprocal and change the division into multiplication. Then, perform the multiplication.
Caution! Division by zero is undefined and so is the reciprocal of any fraction that has a zero in the numerator. For any real number [latex]a[/latex], [latex]\frac{a}{0}[/latex] is undefined. Additionally, the reciprocal of [latex]\frac{0}{a}[/latex] will always be undefined.