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?
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
# enthusiasts only
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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
To explain how you decided that the exact answer is less than the estimate, you can break it down into steps. Here's a step-by-step explanation:
1. Start with the given expression, 978*66.
2. To estimate, you rounded the numbers to the nearest multiple of 10, so 978 becomes 980, and 66 becomes 70.
3. Multiply the rounded numbers: 980*70 = 68600. This is your estimate.
4. Now, calculate the exact answer by performing the multiplication: 978*66 = 64548.
5. Compare the estimate and the exact answer. In this case, your estimate is 68600, which is greater than the exact answer of 64548.
6. Therefore, you can conclude that the exact answer is less than the estimate.
When explaining, you can communicate these steps, highlighting the rounding to estimate and the comparison between the estimate and the exact answer.