Question

I cannot figure this problem out. I have did it several ways and get conflicting answers. Give an estimate for the following expression: 12*26. Then state whether you believe that the exact answer is greater than or less than the estimate. Explain how you decided. Here is how I started the problem 12*26=312(this is the original problem) 10*30=300 The exact answer is greater than the estimate. I rounded 12 to 10 and 26 to 30. Can someone explain this a little better to me? Thanks.

Answer 1

12*26
= (10+2)(30-4)
= 300 -40 + 60 - 8
= 300 + 12

notice that 300 was our estimation answer and the 12 was the error in the estimation

so the estimation was lower than the actual answer.

Geesh, more time was spent on the arithmetic explaining the error than the time it would take to actually do the multiplication, lol

Answer 2

You have to round to below or above, but you cannot round one number to below and the other to above.

Put 12*26 = N

Then you can say that

10*26 < 12*26 = N ---->

N > 260

Or

12*30 > 10*26 = N -------->

N < 360

Answer 3

I agree that if you round one up and one down, it is not obvious if the estimate is too high or too low.

Here are some schemes from which you can choose, and possibly apply to other situations.

A. Round down:
12*26
≈ 10*20 = 200 (under-estimate)

B. Round up:
12*26
≈ 20*30 = 600 (over-estimate)

C. Round-up and -down:
12*26
≈ 10*30 = 300
It is not easy to be sure in all cases whether we are over- or under-estimating. See Reiny's response for an estimate, or even an exact solution. As he said, it is easier to multiply than to calculate the error.

D. 12*26 (estimate by double/half)
First examine the multiplication by the technique called double and half.
12*25
=6*50 (double 25, half 12)
=300
So
12*26
≈12*25 (round 26 down to 25)
= 300 (under-estimate)

E. 12*26 (exact answer by double/half)
12*26
=12*25 + 12*1
=300 + 12
=312

F. 12*26 Another exact answer by double/half
12*26
=6*52
=3*104
=312

G. 12*26 (estimate by square of the mean)
The mean of 12 and 26 is (12+26)/2=19
so
12*26
≈ 19² = 361 (always over-estimate)

H. 12*26 (exact calculation by square of the mean)
Simple squaring the mean is always an over-estimate. The bigger the difference between the numbers and the mean, the bigger the error, which is equal to the difference between the mean and one of the two numbers. The error is to be subtracted from the square.

12*26
=19² - (26-19)²
= 361 - 49
= (361 - 50) + 1
= 311 + 1
= 312

Answer 4

Why not (10+2)(26) ?
260 + 52
312

Answer 5

I suppose you could estimate low by saying
(10+2)(25)
250 + 50
300
but it is just about as easy to do it exactly.

Answer 6

I agree.
I finally realize that the exercise is basically on rounding, and not the mental calculation part.
The curriculum stresses the importance of knowing whether the estimation is over or under, which is very commendable.