why do fractions need a common denominator before you can add or subtact them?
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So that the fractions can work on the same terms.
To understand why fractions need a common denominator before you can add or subtract them, let's start by reviewing what a denominator represents in a fraction.
In a fraction, the numerator represents the number of parts being considered, while the denominator represents the total number of equal parts into which the whole is divided. For example, in the fraction 3/4, the numerator is 3 (three parts), and the denominator is 4 (total number of equal parts).
When you add or subtract fractions, you need to ensure that the fractions have the same denominator. This is because the denominator determines the size of each part or unit in the fraction. If the denominators are different, the fractions are representing different sized parts or units, making them incompatible for direct addition or subtraction.
To add or subtract fractions with different denominators, you need to make them equivalent fractions with a common denominator. You can achieve this by finding the least common multiple (LCM) of the denominators and then converting each fraction into an equivalent fraction with that common denominator.
Here's a step-by-step process to add or subtract fractions with different denominators:
1. Identify the denominators of the fractions you want to add or subtract.
2. Find the least common multiple (LCM) of the denominators. The LCM is the smallest number that is divisible by both denominators.
3. Convert each fraction into an equivalent fraction with the common denominator. To do this, multiply the numerator and denominator of each fraction by the same factor that will make the denominator equal to the LCM found in step 2.
4. Once the fractions have the same denominator, you can add or subtract the numerators directly while keeping the denominator the same.
5. Simplify the resulting fraction, if necessary, by reducing it to its lowest terms. To simplify, find the greatest common divisor (GCD) of the numerator and denominator, and divide both by it.
By finding a common denominator and performing the addition or subtraction, you ensure that the fractions are expressing the same-sized parts or units, allowing you to accurately combine them into a single fraction.