For how many days, x, will they have eaten the same total pieces of cheese?

Question

Mickey has eaten 5 pieces of cheese each day this year. Minnie has eaten 3 pieces of cheese each day this year, but also ate 12 pieces last year. 

Answer 1

1) It looks like it will only be on one day if we assume the problem is asking for the same day, which seems unlikely.  The table below shows the cumulative amount of cheese eaten by Minnie and Mickey.  Since there is no mention made of how many pieces of cheese Mickey ate last year, the assumption is he did not eat any

 

Day   Minnie   Mickey

0        12          0

1        15          5

2        18         10

3        21         15

4        24         20

5        27         25

6        30         30

7        33         35

 

No need to go any further. 

 

As an equation:   12 + 3x =  5x   where x is the day of the new year

12 = 2x

x = 6

There is only one value for x, and it is x =6 which means they will have eaten the same number on the 6th day.

 

2) Another way of looking at the problem is to say that the days on which they have eaten the same total number of pieces of cheese do not need to be the same day. For example, in the above table, Minnie has eaten 15 pieces of cheese on day 1 whereas Mickey has eaten 15 pieces of cheese on day 3. On day 6, they both have eaten 30 pieces of cheese. On day 9, Mickey will have eaten 45 pieces of cheese (5*9) and on Day 11 Minnie will also have eaten 45 pieces of cheese (12 + 3*11). It turns out, that every 3 days, there will be a number where Minnie and Mickey have eaten the same quantity of cheese. The easiest way to see what is happening is to create an Excel spreadsheet. The first column will be days, numbered from 0 to 365. The second column will be the number of pieces of cheese Minnie eats, the third column the number Mickey eats, and then two more columns that use the Excel "Match" function to find a match whenever the number of pieces Minnie eats on a specific day matches the number Mickey eats on any day within the year, and vice-versa. If you do this, you will find the number of days in a year where there is a match is 73. Why 73? Because there will be a match every 5 days, and 365 days in a year / every 5 days = 73. If you look at it from Mickey's perspective, he will match a Minnie number every 3 days, but this will only continue until day 219. And 219/3 is again equal to 73. What is the significance of day 219? By day 219, Mickey has eaten 5*291 = 1095 pieces of cheese. In the entire year, Minnie will only eat 12+(365*3) = 1107. So it turns out there is no opportunity for Mickey to match Minnie after Mickey gets to Day 219. So the answer is 73.