Think at basic level #2

Number Theory Level 2

Mrs. Sharma gave an exam in mathematics class of 5 students . She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered . Mrs. Sharma noticed that after each score was entered, the average was always an integer. The scores were 71 , 76 , 80 , 82 , and 91 . What was the last score Mrs. Sharma entered.

The answer is 80.

1 solution

Pranjal Jain
Nov 17, 2014

Sum of all scores = $400$ which is divisible by 4. Now lets say last score is $a$ , then $(400-a)$ must be divisible by 4 (Average of 4 subjects being integer). So a must be divisible by 4.

3 chances, 2 options (76 and 80), anyone can do it right!!

A variation of this question would be to ask for the third score entered, (answer: 91).

Brian Charlesworth - 6 years, 6 months ago

Nope! Third must be 71!! I know the correct solution as well... Ill add it soon! I am currently out so just mentioned that approach!

The possible ways are (76,82,71,91,80) and (82,76,71,91,80)

Pranjal Jain - 6 years, 6 months ago

Yes, the first two scores entered must be $76$ and $82$ in any order, but I still think that $91$ must be third, since then the 'running' average would be

$\dfrac{76 + 82 + 91}{3} = 83$ .

If $71$ were third then the running average would be $76.3333.....$ , i.e., not an integer.

Brian Charlesworth - 6 years, 6 months ago

@Brian Charlesworth – Oops!! A mistake! Sorry... You are correct!

Pranjal Jain - 6 years, 6 months ago

@Pranjal Jain – No problem. :)

Brian Charlesworth - 6 years, 6 months ago

Think at basic level #2

The answer is 80.

1 solution

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