Round up or down to most significant digits, then work backwards from there, if needed.
Then...
7 + 8 = 15, last digit is 5, just as the last digit of the quick answer is 5, so the quick answer is probably correct.
I don't actually think about any of this stuff, I just do it.
Years ago, when every penny counted, I'd keep counts accurate to the cent in my head while grocery shopping. It was partly a game, partly necessity.
I'm probably the last generation of kids to use slide rules and printed paper tables in high school, so that might make a difference.
If I'm doing something beyond in-my-head or back-of-the-envelope stuff, I'll use formal systems that were beaten into me in college physics and chemistry classes, keeping track of all the units, canceling and simplifying where appropriate as I work the problem through, generally to end up with some ugly fraction.
I once worked for a machinist who would frequently do complex set-up problems in his head without reaching for his scientific calculator. Sure, I understood the math, but no way could I do it in my head.
And then of course, there is the Trachtenberg system. It was developed by the Russian Jewish engineer Jakow Trachtenberg in order to keep his mind occupied while being in a Nazi concentration camp.
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