Luke does 10 spelling test, each out of 10 marks. luke scored his worse result on the first test and then always either improved or scored the same result. when the results were ranked and the middle number was taken, it was 7. the average was 7.7.
what is the highest number of 10s that luke could have scored?

Because the average is 7.7, his total points are equal to 7.7*10=77.

Because 7*10<77<8*10, Luke cannot have scored 8 10s.  We now seek to prove that he can score 7 tens.

This leaves him with three tests left, that sum to 7 marks.  There are clearly many ways to arrange this - for example, 2, 3, and 3 on the first three tests, then ten on all seven other tests.  As this shows that 7 10s works and we have proven that 8 10s does not work, 7 is the maximum amount of 10s Luke could have scored.