The Fraction Sum You Can Solve Without a Common Denominator

If we're looking at this particular sum and we want to add up these fractions, but we don't want to have to find a common denominator for 12, 20, 30, 42, and 56, which I can tell you is not a particularly small number. What would we do? What could we possibly do? I'll tell you what. I'm going to give you a hint. I'm going to tell you that 112th is equal to 1 over something minus 1 over something else. But let's set that aside for a moment. If we were going to have to do the common denominator type thing, you might pick just a couple at a time to look at because if you look at just a couple at a time, it's not that bad. 112th is the same thing as 1 over 3 * 4. And 120th is the same as 1 over 4 * 5. So, I can tell that they already have that factor of four in common. All I would have to do is bring the 3 over to the second fraction and the 5 over to the first fraction, and I would end up with a common denominator of 60. And I might think to myself, well, why don't I just keep going with this 1 over 4* 5 kind of pattern? Like, isn't 30 just 1 over 5* 6 and then they have the five in common? And isn't 42 just 6 * 7 and then they have the six in common? And it's totally true that as we go down the line, each of the consecutive fractions has some factor in common. But that's different from saying that they all have those factors in common. We'll think specifically about 112th for a moment. 1 12th happens to be the same as 1/3 minus 1/4. If we do our common denominator thing here, you can see why that is. We need a 4 over4 to turn 1/3 into 4 12ths and a 3 over3 to turn 1/4 into 3 12ths. And then of course when we subtract those 4 12ths - 32ths is 112th. But this isn't just happen stance. This isn't just coincidence. This always happens when our denominator is the product of two consecutive integers, which is exactly what was going on for every single denominator here. 120th is the same thing as 1/4 - 1/5. 1/30th, which remember is the same as 1 over 5 * 6 is the same thing as 1/5 - 16. 1 over 42 is the same as 16 - 17th. And 1 over 56 is the same as 17th - 1/8. When we write every single one of these out, we can see that what we are actually looking at is called a telescoping sequence. The minus 1/4 of our 1112th actually cancels out with the positive 1/4 of our 120th. The minus 1/5th of the 120th cancels out with the plus 1/5 of the 130th. and so on until the only thing we have left from this entire process is that original 1/3 which never subtracted away and the minus 1/8 of 1 over 56 which doesn't have a next fraction to cancel it out with. And when we subtract those we get a sum across all five of these fractions of 5 over 24. Never underestimate the power of being a little clever.