“In grade-school math classes, though, an estimate is a quick approximation of the answer to a complicated calculation. If that approximation says that the result should be around 100 and your calculated answer is over 300, you immediately know that your result is suspect.”
But what is the practical application of this exercise, other than as a test question on grade school math exams?
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.
Most real world calculations are checked with something called doubling checking, or methods built into computer programs or spreadsheets.
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.