I need some help. Here is the problem: Give an estimate for the following expression. Then state whether you believe that the exact answer is greater than or less than the estimate. Explain how you decided. Here is what I have for an answer so far: 978*66 can be converted to 980*70 that the exact answer is less than the estimate because 978*66=64548 where as 980*70=68600. How do I go about explaining how I decided this?

Answer 1

980*70
=(978+2)*(66+4)
=978*66 + (2*66 + 4*978 + 4*2)
Since the terms in parentheses are all positive, so the estimate must be more than the correct answer.

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# following text for mental calculation
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In fact, you can actually calculate the exact value mentally, if you are prepared to exercise your grey matters a little more than usual. Proceed as follows:
978*66
=(1000-22)*66
=1000*66 - 22*66
=66000 - 2*66*11
=66000 - 132*11
=66000 -1452
=64548

When a number is multiplied by 11, you only have to add the adjacent digits to make a new digit. Assume you have a leading and a trailing zero. For example,
12*11
= 0+1 || 1+2 || 2+0
= 132

22*66 can be calculated as follows:
=(2*6)*11*11
= 132*11
= 0+1 || 1+3 || 3+2 || 2+0
= 1 || 4 || 5 || 2
= 1452