Why Do We Need a Common Denominator When Adding and Subtracting Fractions? | Math with Mr. J

welcome to math with Mr J [Music] in this video I'm going to cover why we need common denominators when we add and subtract fractions or even when we compare and order fractions now fractions show up all throughout math and some of us may fully understand how to add and subtract fractions but we may not understand why we need a common denominator when doing so no matter what level of math you're in or what your situation may be understanding the why of something is very important so I hope this helps you understand the why of common denominators now simply put we need common denominators in certain situations like again adding subtracting comparing and ordering fractions that way all of the parts or pieces are the same size of those fractions common denominators make everything fairer so to speak let's jump into our example where we have one half plus one-fourth and see exactly what this all means we're actually going to work through this problem twice we have part A and Part B we are going to do the problem incorrectly and then correctly in order to see why a common denominator is needed we will start with part A where we will do this incorrectly without a common denominator so let's add these fractions as they currently are and go straight across we will start with the numerators so the top numbers 1 plus 1 that equals two now for the denominators the bottom numbers two plus four that equals six so we get 2 6 which is incorrect and if we think about it this doesn't make any sense we started with one half and added one-fourth to it but somehow ended up with something less than a half two-sixths is less than a half so again this is incorrect and doesn't make sense let's take a look at the rectangles below to help us visualize what's going on here let's fill in one half so one half here one half plus one-fourth so 1 4 here we can see that these pieces are not the same size even though we have one piece of each fraction filled in the pieces are not of equal size halves are not equal to fourths therefore we cannot add these yet as is this would give us one piece plus one piece which is two pieces out of six total pieces if we are adding the denominators as well which again is incorrect and doesn't make sense we need a common denominator which will make all of the pieces the same size that way we can add the fractions and when we add fractions we actually keep the common denominator through the entire problem even in our answer we will talk about why that is as well in Part B when we do this correctly let's move on to Part B here we have one half plus one-fourth again let's do this correctly by finding a common denominator and renaming the fractions with that common denominator the common denominator specifically the least common denominator between one half and one-fourth is four so let's rename and add and we will rename underneath here let's rename one half first so how do we get that denominator of 2 to equal four well two times two equals four whatever we do to the denominator the bottom we must do to the numerator the top in order to keep this equivalent so one times two equals two -fourths is equivalent to one-half we're not changing the value of this problem at all these are equivalent fractions we just have that common denominator now now we can move to 1 4 which already has a denominator of four so we do not need to rename so we end up with two-fourths plus one-fourth now that we have that common denominator of four we are able to add so let's start with the numerators the top numbers two plus one equals three and then we keep our denominator of four the same we are working with fourths so again we keep that denominator our answer is three-fourths now let's take a look at the rectangles below to make better sense of this we will start with one half but we renamed this fraction with that common denominator of four so we now have two fourths we have fourths so four equal pieces and we have two fourths so we need to fill in two out of those four pieces and we can see that this is equivalent to one-half then we add one-fourth so 1 out of four all of the pieces are now the same size what we have here is two one fourths or two pieces of one-fourth which is two-fourths so one-fourth and then 1 4. so again two pieces of one-fourth which is just two-fourths two out of four plus one piece of one-fourth which is just one-fourth one out of four so that gives us three total one-fourths three of those one fourth pieces one two three one fourths is just three fourths our final answer so again our final answer three fourths and that's why we keep that denominator the same we are working with fourths so we keep that denominator of four throughout the entire problem again the final answer three fourths so there you have it there's why we need a common denominator in certain situations like adding subtracting comparing and ordering fractions that was a very simple example but this applies to any time we need a common denominator common denominators make all of the parts or pieces of a fraction the same size they make everything Fair once we have a common denominator we have Apples to Apples so to speak another way to think of this is to relate it to units of measure for example 12 inches plus two feet so inches and feet would not equal 12 plus 2 which is 14. so 14 inches or 14 feet we can't just add as is we would need to have them both in inches or both in feet the same unit the answer would be either 36 inches or 3 feet depending on which unit you used so we can relate this needing a common unit to needing a common denominator so something to think about as well I hope that helped thanks so much for watching until next time peace