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13 Dec 2005 - 03:11
What is the digit in the hundreds places of the sum of the following addition problem: 7 + 77 + 777 + 7777 + ... + 77777777777777777777
(The final number has 20 7s)
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This is a place-value problem. (I would also term it an algebra problem -- but possibly not in the way that the general public thinks of algebra.) By what we know about place value, we can solve this problem by disregarding everything to the left of the hundreds place in all of the numbers. Thus, the sum in question will have the same digit in the hundreds place as would: 7 + 77 + (18)(777). Or, we can imagine adding in columns, like the standard addition algorithm. Our right-most colum will yield 20*7. Adding this to the values in the tens place, that result would be added to 19*70. And working with the hundreds, we add on 18*700. If I haven't screwed up, this should be the same number that I defined before. I don't have a calculator handy, so evaluation is left as an exercise to the reader. :) -- RudbeckiaHirta - 13 Dec 2005
THANK YOU -- CatherineJohnson - 13 Dec 2005
I looked up the FERPA stuff today. VERY helpful. Thank you2 -- CatherineJohnson - 13 Dec 2005
Rudbeckia beat me to it. I'll lay out my approach in case it helps too. I thought of it as follows:
7 + 77 + 777 + ... +77777777777777777777 ---------------------When I laid it out mentally like that, and thought about what I do when doing multi-digit multiplication. So, start by summing the 1s column. There are 20 7s there. 20 7s = 140. In the 10s column there are 19 7s. 19 7s = 133. And we should put this over one place. In the 100s column there are 18 7s. We should put this over two places. Then sum the answers.
7 + 77 + 777 + ... +77777777777777777777 --------------------- 140 1330 12600 ------- 070To me it's a critical thinking question and a good one for an accelerated class for gifted kids. -- TracyW - 13 Dec 2005
(20X7) + (19X70) + (18X700) = 140 + 1330 + 12600 1 + 3 + 6 = 10 answer: 0 -- KtmGuest - 13 Dec 2005
Christopher, of course, insisted on writing EVERY 7 out.... -- CatherineJohnson - 13 Dec 2005
now we have sobbing and crying -- CatherineJohnson - 13 Dec 2005
thank god Ed is dealing with it -- CatherineJohnson - 13 Dec 2005
And the sum is: 86419753086419700000 -- SmartestTractor - 13 Dec 2005
86419753086419700000 good lord -- CatherineJohnson - 13 Dec 2005
ok, so Ed and I just went through all his tests, etc..... basically, our conclusion is that he's being de-instructed things he knew going into the course are disappearing fast -- CatherineJohnson - 13 Dec 2005
I realize that means he didn't know these things to mastery so what if he hadn't gone to middle school he would know them to mastery by now, because I would have been teaching them to mastery -- CatherineJohnson - 13 Dec 2005
we were working our way through ALL the fraction units in PRIMARY MATHEMATICS that's done no time -- CatherineJohnson - 13 Dec 2005
my last hope at this point is KUMON, which can be done on the side -- CatherineJohnson - 13 Dec 2005
KUMON, summer re-teaching, and I'm going to figure out SOME way to distribute practice of ROTE knowledge -- CatherineJohnson - 13 Dec 2005
how is this algebra, by the way? (That reminds me, I want to ask everyone exactly how I should be defining algebra....) I didn't do problems like this in algebra. -- CatherineJohnson - 13 Dec 2005
When I laid it out mentally like that, and thought about what I do when doing multi-digit multiplication. So, start by summing the 1s column. There are 20 7s there. 20 7s = 140. oh, that's cool Christopher wrote every single number (thank God we had graph paper), then crossed out all the extra digits, then did this: 18 x 777 + 7 + 77 -- CatherineJohnson - 13 Dec 2005
one mom of an incredibly bright little guy in K-3 says her child is losing the math he knew TRAILBLAZERS is confusing him terribly that's exactly what we see happening here the math he knew to some degree is getting jumbled -- CatherineJohnson - 13 Dec 2005