Find the result of one fifteenth plus one half plus four fifth giving the answer in its simplest form.
We need to add these three fractions together, but we notice that they all three have different denominators. And we know that in order to add fractions together, they must have a common denominator. So before we move any further, we have to find a common denominator for all three of these fractions. To find the common denominator, we’ll first need to find the least common multiple between 15, two, and five. Before we can find the least common multiple, we’ll just need to list out multiples of 15, two, and five.
Starting with 15, it’s of course 15 as a multiple. The next multiple for 15 would be 15 times two or a 30. After that comes 45 and then 60. Next, we’ll find the multiples of two, but remember we’re trying to find a common multiple between 15, two, and five. Two of course has a multiple of two, then four, six, eight, 10, 12, 14, 16, 18, 20. The thing is we still haven’t found a common multiple between two and 15.
When I look at the list of multiples of 15 though, I see 30 and I know that 30 is an even number, which means 30 is divisible by two. 30 is in fact a multiple of two. So we can go ahead and put 30 in the list of multiples of two. Now, we need to make sure that 30 is also a common multiple of five. The first multiple of five is five, then 10, 15, 20, 25, and finally 30. So we found a common denominator. What’s next?
We need to convert each of these fractions into a fraction with a denominator of 30. To move from 15 in the denominator to 30 in the denominator, we multiply by two. But if we multiply by two on the bottom, we have to multiply by two on the top. We need to multiply our numerator of one by two as well. One fifteenths then becomes two thirtieths.
We need to do the same thing with one-half. How would we get from two to 30? We multiplied by 15. And if we multiply our denominator by 15, we must multiply our numerator by 15. One times 15 is 15. We do this process one more time with the four-fifths. How do we get from five to 30? Multiply by six. Then we need to multiply our numerator by six; four times six is 24. So now, we have two thirtieths plus fifteen thirtieths plus twenty-four thirtieths.
I’ll just bring that down, so it’s a little bit easier to see. And we found our common denominator. Now that we have our common denominator, our next step is just to add: two plus 15 plus 24 equals 41. The denominator doesn’t change, so we have forty-one thirtieths. And now we’re finished adding.
We need to do one more thing though; we need to write this in its simplest form. 41 over 30 is an improper fraction; its numerator is larger than its denominator. To simplify here, we’ll want to take out any whole parts. I know that 30 out of 30 equals one whole. If I take one whole or 30 over 30 away from this forty-one thirtieths, in other words if I subtract 30 from 41, I have two pieces: 30 over 30 plus eleven thirtieths. What I wanna do now is rewrite 30 over 30 as one. We can then write our improper fraction 41 over 30 as a mixed number of one and eleven thirtieths.