Question

On her first three 5 point math quizzes Amy scores were 4,3 and 5. she will take three more quizzes this semester what three scores would give her an average that is a whole number? A repeating decimal?

Answer 1

Score 1 = 4

Score 2 = 3

Score 3 = 5

Score 1 + Score 2 + Score 3 = 12

Amy plans to give three more 5 point quizzes

Part A

Total number of quizzes Amy would have taken by the end = 6

Average score = Sum of all Scores / Total Number of quizzes

Let sum of the last 3 quizzes be x

⇒ Average Score =
`(x+12)/(6)`

Hence, for the average to be a whole number, the sum of all her scores needs to be divisible by 6.

⇒ x+12 should be divisible by 6

Since 12 is divisible by 6, we need to make sure now that x is divisible by 6

The possible values for the scores are 0, 1, 2, 3, 4 and 5. Out of these values we will pick those 3 numbers for which the sum is a multiple of 6:

0+0+0=0

2+2+2=6

3+3+0=6

4+2+0=6

4+4+4=12

5+1+0=6

5+4+3=12

Hence, her average score would be a whole number, if she scores: 0,0,0 OR 2,2,2 OR 3,3,0 OR 4,2,0 OR 4,4,4 OR 5,1,0 OR 5,4,3

Part B

For the average to be a repeating decimal, we will definitely ignore the score combinations determined above as they lead to an average that is a whole number.

Out of 0,1,2,3,4,5, the sum of the three numbers (x) should be such that
`(x)/(6)` is a repeating decimal

So if a person scores 0,0,1 then
`(x)/(6)` would be 0.16666.... i.e. repeating decimal (and average in this case would be
`(13)/(6)` i.e. 2.166666.... , which is also a repeating decimal)

Another such combination can be 0,1,1, then
`(x)/(6)` would be 0.3333..... and average in this case would be
`(14)/(6)` i.e. 2.33333....

There can be many such combinations.