1. Amy wrote down a 4-digit number. Then Roxanne wrote down the sum of the digits of Amy's number. Mary wrote down the sum of the digits of Roxanne's number. What is the greates possible number Mary could have written down? Justify your answer.

11

Step-by-step explanation:

We may think that in order to have the biggest sum of digits for Mary, we need the biggest number for Amy, so when we sum the digits, we have the biggest number, but that is incorrect:

The maximum number we can have with 4 digits is 9999.

So, The number that Roxanne would write is 9 + 9 + 9 + 9 = 36, and then Mary would write 3 + 6 = 9

But, if the number written by Roxanne was 29, the number of Mary would be 2 + 9 = 11, which is bigger than 9.

So, if the maximum number Roxanne can have is 36, the biggest sum of digits for the numbers from 1 to 36 is achieved with the number 29, that gives us 2 + 9 = 11.

So the greatest number possible Mary could have written down is 11.

x=6 and x=3

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