In 2008, a certain factory produced 30% more widgets

I don't think you'll ever need to worry about this on the real GMAT - in real GMAT questions asking for estimates, the answers will usually be far enough apart that you can actually estimate fairly easily. But even in this case, there are a couple of interesting techniques we can use to bypass long division.

Here, if you can tell the right answer choice is either 39% or 41%, you really just need to know if 69/169 > 0.4 to pick the right answer. There are a few ways to make that decision quickly. You might locate a fraction which is exactly equal to 4/10 and that looks similar to 69/169. Notice if we multiply by 17 in the numerator and denominator, we find that 4/10 = 68/170. But notice now that 69/169 > 68/170, since 69/169 has both a larger numerator and a smaller denominator than 68/170. So 69/169 > 0.4, and 0.41 is a better estimate than 0.39.

Or you could notice that (0.4)*169/169 is exactly equal to 0.4 (just cancel the 169), and is exactly equal to 67.6/169. Since this is clearly smaller than 69/169, we can see that 69/169 is larger than 0.4.

If you prefer to avoid decimals, you can instead write down the following inequality (this method is mathematically the same as the one above, though) :

69/169 > 4/10

We want to know if this is true. If it is, then when we rewrite it, what we arrive at must be true. If you multiply by 169 and 10 on both sides, you find

690 > 4*169 = 676

and since this is true, the original inequality was true as well.

If you know that rounding 69/169 to 70/170 = 7/17 has only a negligible influence on the overall value of the fraction, then the above techniques are even easier to apply, but I've not done that here since it may not be clear how much that approximation will affect the fraction's value.