One form of estimation involves rounding numbers to get a "ballpark" answer. There are some situations where this is a valuable technique, and others where it is not. For example, if you were an astronaut on your way to the moon, and mission control said, "It's about 238,000 miles from the earth to the moon, but that's an ugly number, so we rounded it to 200,000 miles, and used that number to calculate how much fuel to put in your rocket..." you might be a little bit doubtful about whether or not you want to get into that rocket!

And sometimes it'll depend on which way we do our rounding. If mission control said, "that's an ugly number, so we rounded it to 250,000 miles," you might be more willing to climb in the rocket, knowing that it's less likely that you'll run out of fuel!

A more "down to earth" example would be a shopping trip, in which you have a certain amount of money to spend, and you don't want to go over budget. You don't have a calculator, so you simplify the process by rounding your purchase amounts. Let's say you have $100 to spend, and your cart contains 5 items that each cost $1.99, 4 items that each cost $4.75, 10 items that each cost $3.19, and 19 items that each cost $0.89.

Here's one way you can estimate the cost: "Well, $1.99 is close to $2.00, $4.75 is close to $5.00, $3.19 is close to $3.00, 19 is close to 20, and $0.89 is close to $1.00, so that's about 5 x 2 + 4 x 5 + 10 x 3 + 20 x 1 = $80.00, which is under $100, so I'm all set!"

One thing to watch out for is whether or not you're rounding up or down; in almost all of these cases I rounded up, but in one case (rounding $3.19 to $3.00) I rounded down. This is like the rocket example; rounding down results in running the risk of running out of "fuel" (in this case, money). The only reason I felt comfortable rounding down was because I knew that I had rounded everything else up, and all those round-ups were going to easily outweigh the round-downs.

If I wasn't so confident of that, I would have rounded $3.19 up to $4.00, just to be on the safe side. If you're concerned about running out of money (or fuel, or anything else!), it's better safe than sorry, so round up!

If you're doing multiplication, and rounding, it's a good idea to - if possible - round one number up, and one down, since errors get compounded when you're multiplying. For example, if you're multiplying 2.5 times 3.5, don't round both of them up; round one up and one down.

If you're dividing, round in such a way that you get numbers that work out without remainders.

For example, could be rounded as = 4 or = 5.  The actual answer is between these two approximations.

For each question below, perform rounding, and show how you rounded, and then use your calculator to find the exact answer, and in parentheses show how much your estimate was off from the actual answer, as a percent error.

Percent error = 100()

For example:

Problem
98 x 102

Solution

Estimate: 100 x 100 = 10,000

Actual: 98 x 102 = 9,996

Percent error = 100() = 0.04%

You would write: 100 x 100 = 10,000 (off by 0.04%)

1.

26 times 30

2.

5(8.9) + 12(9.5) + 31(18)

3.

32/14

4.

99/32 + 108/11

5.

(105 + 85)(225 - 20)