There are any number of practical examples. You're about to split a tip among three people for a $57.98 restaurant meal. (The service was such that you've all decided to leave about a 20% tip.) One member of the group says "OK, each of us should leave a $10 bill." You can immediately know that he's wrong by quickly estimating that 20% of your approximate $20 share of the bill is about $4. The exact answer doesn't need to be calculated to know that your friend is off by more than a factor of two.
I’m still guessing the only practical application is to give a passing grade to students who can’t successfully calculate enough math problems correctly to earn a passing grade.
Any case where an exact answer isn't required is a candidate for estimating the result. You're about to pick up paint for a wall in your house. The paint covers 950 square feet per gallon. The wall is about 8 feet, 4 and 1/2 inches high by 19 feet wide. Do you need one quart can of paint? Two? Or should you buy a gallon can? The exact answer is 0.3536111... gallons but that doesn't really matter. A quick estimate is (8 x 20) / 500 = 0.32 gallons. You can easily determine that two quarts will be sufficient without doing the exact calculation.
Most real world calculations are checked with something called doubling checking, or methods built into computer programs or spreadsheets.
I'm a programmer. Believe me -- Bugs, like math errors, do happen. I've found that it can be very difficult to locate bugs in my own code since I put them there in the first place. The same thing happens with math errors.
NASA lost a Mars mission because someone sent them data in kilometers and NASA treated as if it were in miles (possibly vice versa). I don't know that it's possible to estimate what the calculation results should have been. If it is, though, you don't need to calculate the exact answers to realize that the computer-generated results are about 60% of the expected values or more than 1 and 1/2 times the estimated values.
Your examples in #139 agree with my contention. The examples make use of correct calculations using rounded or estimated numbers. The numbers are estimated; the calculations are exact.
This started out with 9 x 9 as an example to be estimated. What the sense of estimating that with 10 x 10, or 9 x 10. It’s nonsensical and serves no purpose. And if a kid hasn’t learned the multiplication tables, how can they calculate 10 x 10, or 9 x 10 to make the nonsensical estimate rather than the exact 9 x 9 calculation.
The 9 x 9 example asked for an estimated calculation of exact numbers. Utter nonsense. We don’t estimate the calculation, but we do estimate numbers to be used in exact calculations, as your examples show.
The 9 x 9 test question was an totally asinine exercise for a kid learning math to be asked to answer as some sort of estimate, and it’s not really an estimation. It’s nonsense. Test questions involving real estimates and exact calculations could have been easily developed.