Counting Pills An Investigation

Counting pills Pharmacists sometimes use a triangular tray to quickly count pills. The pills are poured into the tray and fill up the tray level by level. The numbers written on the tray will be used to help quickly count the number of pills in the tray. The diagram shows the case when 15 pills are resting on the tray. 15 pills 21 pills 28 pills

How could the tray be used to measure out a) 22 pills b) 34 pills c) 60 pills d) 200 pills Imagine if the tray was larger. The 5th number on the try is 15. The 6th number is 21. The 7th is 28. Copy and complete the table up to the fifteenth number Number written on the tray Pills in the tray 1st1 2nd3 3rd6 4th10 5th 6th 7th 8th 9th etc 15 pills 21 pills 28 pills

By forming a difference table, or otherwise, show that the nth number on the tray is given by Use this formula to find the 40th number written on the tray. How many numbers will have to be written on the tray if you want to use the tray to count out 7021 pills?

Solution The difference table has second differences constant so a quadratic formula will fit the data. This quadratic turns out to be Using this formula then the 40th number to be written on the tray will be This means that 820 pills will be able to be counted up to the 40th number.

If you wish to count up to 7021 pills then you require n such that Cross multiplying and simplifying gives n 2 + n – = 0 and this factorizes as ( n – 118)(n+ 119) = 0, giving n = 118.