Richard took glass, metal and plastic items to the recycling plant and received 1 "green point" for every 4 items in each category that he brought in. In any category, items in excess of a multiple of four were not counted, so, for example, if he brought in 21 glass items, 30 metal items and 43 plastic items, he'd earn [21/4] + [30/4] + [43/4] = 5 + 7 + 10 = 22 green points. (The [] square brackets indicate the floor function--the greatest integer not exceeding the value within.)

One week he brought in a 2-digit number of glass items, a larger 2-digit number of metal items and a still larger 2-digit number of plastic items. For each class of item he received a 1-digit number of green points.

Among the three 2-digit numbers and three 1-digit numbers involved, all the non-zero digits, 1 through 9, appeared exactly once.

If I told you the total number of items brought, you'd be able to deduce how many were glass, how many were metal and how many were plastic.

How many of each category were there?

Given that Glass < Metal < Plastic; and the number of items of each category are 2-digit numbers and the green points of each category are 1-digit numbers; and, each digit 1 to 9 appears in these quantities, the possible set of quantities are one of the following:

Items, green points:
Glass     Metal     Plastic
1. 16, 4     29, 7     35, 8
2. 17, 4     25, 6     38, 9
3. 18, 4     25, 6     37, 9
4. 19, 4     27, 6     35, 8

The first three sets of item totals equal 80 items, the last set of item totals equal 81 items. As the total is given as distinct, the quantity of items must be:

19 Glass items, 27 Metal items, and 35 Plastic items

Posted by Dej Mar on 2009-05-31 18:44:40