Ten students sit a test consisting of 20 questions. Two students get 8 questions correct and one student gets 9 questions correct. The remaining seven students all get at least 10 questions correct and the average number of questions answered correctly by these seven students is an integer. If the average number of questions answered correctly by all ten students is also an integer, then what is that integer?
http://www.mathsisfun.com/whole-numbers.html
This is all the info I was given. It was also a question I was given in a math competition that I participated in and the answer is either
A.10
B.11
C.12
D.13
E.14
sum for the 7 over 10 must be divisible by 7 and from 77 to 140 by sevens
77, 84 , ...... 105 , 112 ..... 140
that sum + 2*8+9 or 25 must be divisible by 10
LOL 105 !!!
105 + 25 = 130
130/10 = 13
even for a contest.
Yes it was a hard question but thank u so much for helping
You are welcome.
To solve this problem, we need to find the average number of questions answered correctly by all ten students.
Let's start by finding the total number of questions answered correctly by the two students who got 8 questions correct and the student who got 9 questions correct. The total for these three students is 2 x 8 + 9 = 25.
Next, let's find the total number of questions answered correctly by the remaining seven students who all got at least 10 questions correct. Since the average number of questions answered correctly by these seven students is an integer, we can infer that each of these seven students got the same number of questions correct.
Let's call the number of questions answered correctly by each of these seven students "x." Since each student answered at least 10 questions correctly, we have the inequality 10 ≤ x.
The total for these seven students is then 7x.
Finally, to find the average for all ten students, we need to find the total number of questions answered correctly by all ten students, which is 25 + 7x, and divide it by 10 (the total number of students).
So, the equation is (25 + 7x) / 10. For the average to be an integer, the numerator (25 + 7x) must be divisible by 10.
Let's find the values of x that make (25 + 7x) divisible by 10, keeping in mind that x must be greater than or equal to 10.
The possible values for x are: 10, 17, 24, 31, 38, ...
Let's test these values:
For x = 10, (25 + 7x) / 10 = (25 + 7(10)) / 10 = 95 / 10 = 9.5 (not an integer)
For x = 17, (25 + 7x) / 10 = (25 + 7(17)) / 10 = 144 / 10 = 14.4 (not an integer)
For x = 24, (25 + 7x) / 10 = (25 + 7(24)) / 10 = 193 / 10 = 19.3 (not an integer)
For x = 31, (25 + 7x) / 10 = (25 + 7(31)) / 10 = 254 / 10 = 25.4 (not an integer)
For x = 38, (25 + 7x) / 10 = (25 + 7(38)) / 10 = 295 / 10 = 29.5 (not an integer)
Since none of these values give an integer average, we need to increase the value of x.
For x = 40, (25 + 7x) / 10 = (25 + 7(40)) / 10 = 305 / 10 = 30.5 (not an integer)
For x = 41, (25 + 7x) / 10 = (25 + 7(41)) / 10 = 312 / 10 = 31.2 (not an integer)
For x = 42, (25 + 7x) / 10 = (25 + 7(42)) / 10 = 319 / 10 = 31.9 (not an integer)
We can see that increasing x further does not give an integer average.
Therefore, none of the possible values for x that are greater than or equal to 10 give an integer average. This means that our assumption that each of the remaining seven students got the same number of questions correct is not possible.
Hence, it is not possible for the average number of questions answered correctly by all ten students to be an integer.