Ten students roll three dice respectively and consider a number of sums which come out as the person's score.
The sum total of the ten persons' score was 100.
Moreover, after dividing total score of each student by 3 and omitting below the decimal point, ten students’ sum total was 30.
After dividing total score of each student by 3 and rounding off the number of 1st decimal digit, ten students’ sum total was 34.
In these ten students, find the number of students whose score is the number when it is divided by three, one remains.
The sum total of the ten persons' score was 100.
Moreover, after dividing total score of each student by 3 and omitting below the decimal point, ten students’ sum total was 30.
After dividing total score of each student by 3 and rounding off the number of 1st decimal digit, ten students’ sum total was 34.
In these ten students, find the number of students whose score is the number when it is divided by three, one remains.
Answer
The sum total at the time of omitting is 30 and the sum total in rounding off is 34.
It turns out that there was the four numbers of 2 remainder group.
Moreover, if 30 which is the sum total of the number after omitting is multiplied by 3, it will be set to 30 × 3 = 90.
There is a 100 - 90 = 10 point difference in 100 which is score's sum total of all the students.
This difference indicates that the sum of remainders when each number of 1 remainder group and 2 remainder group was divided by 3 was ten.
The sum of remainders when each number of 2 remainder group was divided by 3, is 2 × 4 = 8 points since there are four numbers of 2 remainder group. 10 - 8 =2 points.
This 2 is the sum of remainders of the number 1 remainder group being divided by 3.
Therefore the number of students to be found is 2 / 1 = 2 persons.
Two persons
Solution
Range of scores are from 3 (1+1+1) to 18 (6+6+6) and these numbers are categorized by the remainder when the number is divided by 3 as following table.
1st decimal digit
| |||||||
0 remainder group
|
0
|
3
|
6
|
9
|
12
|
15
|
18
|
1 remainder group
|
3
|
4
|
7
|
10
|
13
|
16
| |
2 remainder group
|
6
|
5
|
8
|
11
|
14
|
17
|
As for numbers in 2 remainder group, in case they are divided by 3 and omitting below the decimal point or rounding off the number of 1st decimal digit, the number after rounding off is larger than the number after omitting by one.
The sum total at the time of omitting is 30 and the sum total in rounding off is 34.
It turns out that there was the four numbers of 2 remainder group.
Moreover, if 30 which is the sum total of the number after omitting is multiplied by 3, it will be set to 30 × 3 = 90.
There is a 100 - 90 = 10 point difference in 100 which is score's sum total of all the students.
This difference indicates that the sum of remainders when each number of 1 remainder group and 2 remainder group was divided by 3 was ten.
The sum of remainders when each number of 2 remainder group was divided by 3, is 2 × 4 = 8 points since there are four numbers of 2 remainder group. 10 - 8 =2 points.
This 2 is the sum of remainders of the number 1 remainder group being divided by 3.
Therefore the number of students to be found is 2 / 1 = 2 persons.
