Transcript Counting Pills

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.
28 pills
21 pills
15 pills
How could the tray be used to measure out
a) 22 pills b) 34 pills
Number written on
c) 60 pills d) 200 pills
the tray
Imagine if the tray was larger.
1st
The 5th number on the try is 15.
2nd
The 6th number is 21.
3rd
The 7th is 28.
4th
Copy and complete
5th
the table up to the
fifteenth number
6th
28 pills
21 pills
15 pills
7th
8th
9th
etc
Pills in the tray
1
3
6
10
By forming a difference table, or otherwise, show that the nth
number on the tray is given by
1
n(n  1)
2
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
1
n(n  1)
2
Using this formula then the 40th number to be written
on the tray will be
1
40(41)  820
2
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
1
n(n  1)  7021
2
Cross multiplying and simplifying
gives n2 + n – 14042 = 0 and this factorizes
as ( n – 118)(n+ 119) = 0, giving n = 118.