When you have fractions having different denominators, and you're adding or subtracting, you first need to find a common denominator. This way, you're dealing with apples-to-apples rather than apples-to-oranges. And yes, most textbooks and teachers will have you find the lowest common denominator. And then convert each fraction into it, then add or subtract to get your final answer.

However, I've taught students to find ANY common denominator they happen to see. Or if you cannot find one, just multiply the denominators together to get a common one. This technique sometimes allows you to at least get an answer. And then you can reduce it down to its lowest terms.

With 1/2 and 3/4, most can easily see that a common denominator is 4. So we'd need to convert halves to fourths. 1/2 would become 2/4 by multiplying both numerator and denominator by 2. Once this is done, we'd add 2/4 and 3/4 to get our final answer of 5/4. And if your answer needs to be a mixed fraction, it would be 1 1/4.

But what if you didn't know 4 was the lowest common denominator? You could have just multiplied the two denominators together to get a common one. IN this case that would give you 8. And then convert each fraction to eights. 1/2 would become 4/8 because we'd multiply top and bottom by 4. And 3/4 would become 6/8 because we'd multiply top and bottom by 2. Adding 4/8 and 6/8 together would give us 10/8.

Both numerator and denominator are divisible by 2. So we can reduce this down to 5/4. That matches the answer we got earlier on. So either way, you can arrive at the final answer. It's just a matter of whether you first find the lowest common denominator or ANY, then reduce afterwards. Depending upon the numbers, I find this alternate strategy sometimes is helpful.

Source(s):
I've taught math on the college-level and when subbing in the public schools.