The lowest common denominator
The lowest common denominator
To add or subtract fractions, we need to have fractions with the same denominator. Take one-sixth plus two-ninth. A simple way you may have learned is to expand the fractions by multiplying both denominators and numerators by the other fraction's denominator. The denominators in both fractions will then be the same. But this trick expanding by other fractions denominator, is not always so elegant.
If you want to add or subtract more than one fraction or if the denominators are both large numbers that you can end up with very big denominators, and the rest of the calculation gets harder. This is why it's a good idea to find a smaller common denominator, even the smallest possible. The lowest common denominator. Let's take the same fractions again. Instead of multiplying each fraction by the other fraction's denominator, we factorize both denominators as far as possible.
Six can be factored into three times two. We can't get any further since three and two are prime numbers. Prime numbers are numbers that can only be divided by one and by themselves. Nine can be written as three times three. This leaves us with prime numbers only.
Now we expand both fraction's denominators to get the exact same value. There is a three in both from the start so we don't have to do anything with that. But the two in the first denominator has to be in the second denominator as well. So, we expand the other fraction by two. Then there is only the second fraction's three left.
So, we expand the first fraction by three. And now we have the same number in the denominators of both fractions. We can write them in the same order to make it easier to see. In this way, the fraction becomes a bit simpler and you do not have to deal with large numbers. The answer is the same since 21/54 which we've got before gives the same quotient as 7/18th.
Maybe you know six and nine timetables so well that you can see at once that 18 is the smallest number divisible by both. Then you have the LCD right there. For small denominators, this is the fastest way to find them. Let's take a harder problem where you probably can't use the multiplication table, 26 over 84 plus seven over 15. Start by factoring the denominators, 84 can be factored as two times 42 and 42 as two times 21 and 21 as three times seven.
This leaves us with prime numbers only. The other fractions denominator can be factored as three times five. As we have prime factorization for both denominators, we cannot divide them any further.Now we expand the fractions until the denominators are identical. Start from the left. Two is missing from the second denominator. Expand the second fraction by two.
The next factor is also two and we do the same thing again. Then comes three but it is already in the second denominator so we do not need to expand by three. The last factor in the first fraction is seven. It is not in the second denominator so we expand it by seven. In the second denominator only number five is not in the first denominator already so we expand the first fraction by five.
Now both denominators consist of exactly the same factors and we did not expand the fractions more than necessary. The lowest common denominator is therefore 420 but we'll let the multiples and the denominators remain factorized into their prime numbers for now. Multiply the factors and the numerators. Now you can use a shared fraction line and add the numerators. And 326 can be simplified by two.
This is as far as we can go so we multiply the factors in the denominators to get 210, 163, 210s. That was a lot of steps and it might seem difficult when you see it for the first time. But there are actually only two things to remember when finding the LCD using prime factorization. One, factorize both denominators as far as possible to prime numbers. Two, expand the fractions until both denominators are identical but no further like that.
Now both denominators are identical and the fractions have the lowest common denominator. The rest is familiar fraction subtraction. One, factorize to prime numbers. Two, expand until both denominators are the same