Estimating when Multiplying

Date: 08/19/99 at 20:29:56
From: Renae
Subject: Patterns in estimating (multiplication)

How do you estimate in multiplying?

I have already estimated the top number but I am not sure how to 
estimate the second number if it is a 1-digit number.

Date: 08/20/99 at 16:21:28
From: Doctor Rick
Subject: Re: Patterns in estimating (multiplication)

Hi, Renae.

Do you mean that one of the numbers you are multiplying is only one  
digit? If so, then you don't need to round it at all. For example, 
let's estimate:

     427 x 7

I would round 427 to 400. Then 4 x 7 = 28, so 400 x 7 = 2800. That's 
an estimate on the low side, since I rounded down; the exact answer is 
greater than my estimate. In fact, the exact answer is 2989.

If both numbers have more than one digit, then you can round both 
numbers. Multiply the non-zero digits, and add the total number of 
zeros in the two rounded numbers. For example:

     427 x 7360

Round to:

     400 x 7000

Multiply non-zero digits:

     4 x 7 = 28

Add 2 + 3 = 5 zeros:

     400 x 7000 = 2,800,000

That's my estimate. It's on the low side again, since I rounded _both_ 
numbers down. The exact answer is:

     427 x 7360 = 3,142,720

That's pretty different from my estimate, but at least it isn't twice 
as big or half as big as my estimate. The most important thing is to 
get the number of digits right, and we did.

There are ways that you could make the estimate better; they depend on 
developing a "feel" for the way multiplication works. For one thing, I 
could have chosen to round one number up and the other down. Even 
though one of the rounded numbers will be farther from the true value 
this way, the estimated product will be closer to the exact product. 
Watch:

    427 x 7360

    400 x 8000   [round 427 down and 7360 up]

    3,200,000

See how much closer that is to the exact product?

I don't know just what rules for estimation you are being taught. If 
you have any questions about what I said, or about how to do 
particular problems, please write back.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/