This blog is the online extension of our intermediate classrooms. Our goal is to enhance and document our learning experience throughout the school year, and share this journey with teachers, parents and students. We welcome your constructive feedback, and we look forward to learning with you!
Congrats to Richie, Leo, Calista, and Stanley for correctly determining that the smaller rectangles in POTW #10 were 4cm by 20cm (remember Area = length x width).
Gauss probably made 51 pairs of 100 and subtracted the extra 50 in the middle. Meaning, he took pairs from the numbers 1-100 That add up to 100 and then multiplied like this: 1. 0+100=100 2.0 1+99=100 3. 2+98=100 4. 3+97=100 ... 50. 49+51=100 51. 50+50 (Extra 50 late subtracted) 51 pairs multiplied by 100 is simply 5100 and since there is only one 50, you subtract it to 5100 and get 5050 which is the answer.
Now, to add the digits of the numbers from 1-100 First I added the numbers from 1-9 which applies to all numbers. 1+2+3+4+5+6+7+8+9=45 We can multiply 45 by 10 as with 0-9 in the tenth digit column it, they all include these numbers. 45*10=450 Next we have the tens column now that all the ones are eliminated. They are still the numbers from 1-9 so we can do the same thing. 45*10(for the 10 numbers per tens column) which is 900. Finally we add them together + 1 for the 100. 450+450+1. Therefore 901 is the sum of all the digits of the number 1 through 100.
the answer is 5050 because there are only 51 pairs that take a number from one half of all the numbers between 1 and 100. For example, 1+100=100, 2+98=100, 3+97=100 and so on. So if you multiply 51*100, you will get 5100, but because there is only one 50, you have to subtract the other from 5100 to get 5050.
Haha I know how he did this because I learned in SMS: 1, 2, 3, 4.......100 Sum = 100(100 + 1)/2 100 x 101/2 10100/2 5050 Now for the actual question: I decided to find the sum from 1- 10, 11-20, and so on and so forth. 1-10: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 1 = 46 11-20: 46 + 7 x 1 = 53 21-30: 2 x 9 + 46 + 3 = 67 31-40: 3 x 9 + 46 + 4 = 77 41-50: 4 x 9 + 46 + 5 = 87 51-60: 5 x 9 + 46 + 6 = 97 61-70: 6 x 9 + 46 + 6 = 107 71-80: 7 x 9 + 46 + 8 = 117 81-90: 8 x 9 + 46 + 9 = 127 91-100: 9 x 9 + 46 + 1 = 128 Next, I added the numbers together: 46 + 53 + 67 + 77 + 87 + 97 + 107 + 117 + 127 + 128 = 906 Therefore, the sum of the digits of the numbers from 1-100 is 906.
Haha I know how he did this because I learned in SMS: 1, 2, 3, 4.......100 Sum = 100(100 + 1)/2 100 x 101/2 10100/2 5050 Now for the actual question: I decided to find the sum from 1- 10, 11-20, and so on and so forth. 1-10: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 1 = 46 11-20: 46 + 7 x 1 = 53 21-30: 2 x 9 + 46 + 3 = 67 31-40: 3 x 9 + 46 + 4 = 77 41-50: 4 x 9 + 46 + 5 = 87 51-60: 5 x 9 + 46 + 6 = 97 61-70: 6 x 9 + 46 + 6 = 107 71-80: 7 x 9 + 46 + 8 = 117 81-90: 8 x 9 + 46 + 9 = 127 91-100: 9 x 9 + 46 + 1 = 128 Next, I added the numbers together: 46 + 53 + 67 + 77 + 87 + 97 + 107 + 117 + 127 + 128 = 906 Therefore, the sum of the digits of the numbers from 1-100 is 906.
Since each of the rows have 9 of each digit(ones), we can add the number from 1-9 together by matching up numbers to get the same number, 9+1, 8+2, 7+3, 6+4 and 5. which is 10, 4 times plus 5 of 45. next we multiply that by 10 to get 450. then we do the same to the tens column, we get 450 but we need to subtract the top row since it doesn't have 10s, this gives us 405. Next we add the last column which had 1-9 but then 100 which only had a 1, so get 45+1 or 46, 405+46= 451+450=901, the answer is 901
1+2+3+4+5+6+7+8+9=45 45x10=450 We can multiply 45x10=450, because with 0-9 in the tenth digit column it, all these numbers are there 450x2=900 100 has 1, so 900+1=901 The answer is 901
to solve this, instead of working in groups, I used the chart and worked in columns (if that makes sense)
I started with all numbers under 1. if you add the digits for the number under 1, you get 2 (1+1 for 11). under that is 21 and 2+1=3. this goes on until u reach 91 which is 9+1=10. I did the same thing with all the numbers for 2 except this time I start with 2 so it would be 2,3 4 etc. instead of 1,2,3,4. this continued for all until 10,20,30, where we were back at 1,2,3. (after finding all the numbers, I added them up). my results were 1,2,3,4,5,6,7,8,9,10=55 2,3,4,5,6,7,8,9,10,11=65 3,4,5,6,7,8,9,10,11,12=75 4,5,6,7,8,9,10,11,12,13=85 5,6,7,8,9,10,11,12,13,14=95 6,7,8,9,10,11,12,13,14,15=105 7,8,9,10,11,12,13,14,15,16=115 8,9,10,11,12,13,14,15,16,17=125 9,10,11,12,13,14,15,16,17,18=135 1,2,3,4,5,6,7,8,9,1=46 55+65+75+85+95+105+115+125+135+46=901 therefore the sum of the digits 1-100 is 901
The answer is 901. First, I added 1-9 and got 45 these are the only 1 digit numbers so they won't be as complicated to add. I then added 1-9 once more except added another 1 to the equation. This represents the 10,20,30,40,50,60,70,80,90 and the 1 being 100. 45+45+1=91.
To add the rest separately, I first added the first row of 2 digit numbers (excluding the 20 of course) adn got, 2+3+4+5+6+7+8+9+10=54. The next row was 3+4+5+6+7+8+9+10+11=63
The sum of the rows have a difference of 9. I then realized that the next rows' sums will keep raising by 9 each time, because there are 9 numbers we are adding in every row row, and each rows' tens column raises by 1 every time. This gives us 54+63+72+81+90+99+108+117+126=801.
Lastly, we must add the 2 numbers we calculated, 810+91=901. The answer is 901.
*Ritchie
ReplyDeleteGauss probably made 51 pairs of 100 and subtracted the extra 50 in the middle. Meaning, he took pairs from the numbers 1-100 That add up to 100 and then multiplied like this:
ReplyDelete1. 0+100=100
2.0 1+99=100
3. 2+98=100
4. 3+97=100
...
50. 49+51=100
51. 50+50 (Extra 50 late subtracted)
51 pairs multiplied by 100 is simply 5100 and since there is only one 50, you subtract it to 5100 and get 5050 which is the answer.
Leo aren't we supposed to find the sum of the digits of the numbers from 1-100.
DeleteNow, to add the digits of the numbers from 1-100
DeleteFirst I added the numbers from 1-9 which applies to all numbers.
1+2+3+4+5+6+7+8+9=45
We can multiply 45 by 10 as with 0-9 in the tenth digit column it, they all include these numbers.
45*10=450
Next we have the tens column now that all the ones are eliminated. They are still the numbers from 1-9 so we can do the same thing. 45*10(for the 10 numbers per tens column) which is 900. Finally we add them together + 1 for the 100.
450+450+1. Therefore 901 is the sum of all the digits of the number 1 through 100.
the answer is 5050 because there are only 51 pairs that take a number from one half of all the numbers between 1 and 100. For example, 1+100=100, 2+98=100, 3+97=100 and so on. So if you multiply 51*100, you will get 5100, but because there is only one 50, you have to subtract the other from 5100 to get 5050.
ReplyDeleteI think you meant 1+99
DeleteHaha I know how he did this because I learned in SMS:
ReplyDelete1, 2, 3, 4.......100
Sum = 100(100 + 1)/2
100 x 101/2
10100/2
5050
Now for the actual question:
I decided to find the sum from 1- 10, 11-20, and so on and so forth.
1-10: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 1 = 46 11-20: 46 + 7 x 1 = 53
21-30: 2 x 9 + 46 + 3 = 67 31-40: 3 x 9 + 46 + 4 = 77 41-50: 4 x 9 + 46 + 5 = 87
51-60: 5 x 9 + 46 + 6 = 97 61-70: 6 x 9 + 46 + 6 = 107
71-80: 7 x 9 + 46 + 8 = 117 81-90: 8 x 9 + 46 + 9 = 127
91-100: 9 x 9 + 46 + 1 = 128
Next, I added the numbers together:
46 + 53 + 67 + 77 + 87 + 97 + 107 + 117 + 127 + 128 = 906
Therefore, the sum of the digits of the numbers from 1-100 is 906.
Haha I know how he did this because I learned in SMS:
ReplyDelete1, 2, 3, 4.......100
Sum = 100(100 + 1)/2
100 x 101/2
10100/2
5050
Now for the actual question:
I decided to find the sum from 1- 10, 11-20, and so on and so forth.
1-10: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 1 = 46 11-20: 46 + 7 x 1 = 53
21-30: 2 x 9 + 46 + 3 = 67 31-40: 3 x 9 + 46 + 4 = 77 41-50: 4 x 9 + 46 + 5 = 87
51-60: 5 x 9 + 46 + 6 = 97 61-70: 6 x 9 + 46 + 6 = 107
71-80: 7 x 9 + 46 + 8 = 117 81-90: 8 x 9 + 46 + 9 = 127
91-100: 9 x 9 + 46 + 1 = 128
Next, I added the numbers together:
46 + 53 + 67 + 77 + 87 + 97 + 107 + 117 + 127 + 128 = 906
Therefore, the sum of the digits of the numbers from 1-100 is 906.
Oops I sent it twice.
ReplyDelete1-9 grid=19x9=171
ReplyDelete10 colum=21
192
Since each of the rows have 9 of each digit(ones), we can add the number from 1-9 together by matching up numbers to get the same number, 9+1, 8+2, 7+3, 6+4 and 5. which is 10, 4 times plus 5 of 45. next we multiply that by 10 to get 450. then we do the same to the tens column, we get 450 but we need to subtract the top row since it doesn't have 10s, this gives us 405. Next we add the last column which had 1-9 but then 100 which only had a 1, so get 45+1 or 46, 405+46= 451+450=901, the answer is 901
ReplyDelete1+2+3+4+5+6+7+8+9=45
ReplyDelete45x10=450
We can multiply 45x10=450, because with 0-9 in the tenth digit column it, all these numbers are there
450x2=900
100 has 1, so
900+1=901
The answer is 901
to solve this, instead of working in groups, I used the chart and worked in columns (if that makes sense)
ReplyDeleteI started with all numbers under 1. if you add the digits for the number under 1, you get 2 (1+1 for 11). under that is 21 and 2+1=3. this goes on until u reach 91 which is 9+1=10.
I did the same thing with all the numbers for 2 except this time I start with 2 so it would be 2,3 4 etc. instead of 1,2,3,4. this continued for all until 10,20,30, where we were back at 1,2,3. (after finding all the numbers, I added them up). my results were
1,2,3,4,5,6,7,8,9,10=55
2,3,4,5,6,7,8,9,10,11=65
3,4,5,6,7,8,9,10,11,12=75
4,5,6,7,8,9,10,11,12,13=85
5,6,7,8,9,10,11,12,13,14=95
6,7,8,9,10,11,12,13,14,15=105
7,8,9,10,11,12,13,14,15,16=115
8,9,10,11,12,13,14,15,16,17=125
9,10,11,12,13,14,15,16,17,18=135
1,2,3,4,5,6,7,8,9,1=46
55+65+75+85+95+105+115+125+135+46=901
therefore the sum of the digits 1-100 is 901
The answer is 901.
ReplyDeleteFirst, I added 1-9 and got 45 these are the only 1 digit numbers so they won't be as complicated to add.
I then added 1-9 once more except added another 1 to the equation. This represents the 10,20,30,40,50,60,70,80,90 and the 1 being 100.
45+45+1=91.
To add the rest separately, I first added the first row of 2 digit numbers (excluding the 20 of course) adn got, 2+3+4+5+6+7+8+9+10=54.
The next row was 3+4+5+6+7+8+9+10+11=63
The sum of the rows have a difference of 9. I then realized that the next rows' sums will keep raising by 9 each time, because there are 9 numbers we are adding in every row row, and each rows' tens column raises by 1 every time. This gives us 54+63+72+81+90+99+108+117+126=801.
Lastly, we must add the 2 numbers we calculated, 810+91=901. The answer is 901.