Question 1178328
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Let x be the average number of questions correct by the group of 7 students.
Let y be the average of all the students.<br>
According to the statement of the problem, x and y are both integers.<br>
The total number of questions correct for all 10 students is 2(8)+1(9)+7(x), or 7x+25.<br>
The total number of question correct for all 10 students is also 10 times their average number, or 10y.<br>
So<br>
{{{10y = 7x+25}}}<br>
That equation, with the given constraints, is easily solved using logical reasoning.<br>
10y is a multiple of 5, and 25 is a multiple of 5; that means 7x is a multiple of 5.<br>
And we know x is a number between 10 and 20.  So x has to be 15.<br>
The number the problem asks for is the average of all the students, which is y.<br>
{{{10y = 7x+25 = 7(15)+25 = 130}}}
{{{y = 13}}}<br>
ANSWER: The average number of questions correct for all 10 students is 13.<br>