A Problem of Inflation

A Problem of InflationAuthor(s): David NealSource: Mathematics in School, Vol. 16, No. 5 (Nov., 1987), pp. 28-29Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214395 .

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A

problem of

INFtATION

The Holiday Postcard Stamp Book

P18TAI

E1.43

Royal Mail Stamps 6 at 15fp First Class

4 at 12fp Second class

o70 Royal Mail

,Stamps

"Ten at 17p

'Special :Offer See inside back ccver for details

by David Neal Faculty of Education, Birmingham Polytechnic

Books of postage stamps similar to those shown above have become increasingly popular. Some books contain stamps of a single denomination for first or second class letter post whereas others include a useful range of stamps reflecting current postage rates and the use of cash dispensers. A recent edition of Puzzles, Pastimes, Problems (Vol. 14, No. 5) raised the question of a useful design in the following way.

In 1975 postage rates were 7p for first class and 5-1p for second class letters. A book containing both 7p and 5 1p stamps was issued for +1. How many of each type do you think it contained?

In 1985 the corresponding rates were 17p and 13p, and this makes the design of the book of stamps less obvious. What do you consider to be the most appropriate design?

The purpose of this article is to illustrate some of the solutions developed by pupils and to indicate the degree of ingenuity used to develop an appropriate solution.

Daniel Harding (age 10) of Bishop Kirk School, Oxford, presented the following solution to his teacher, Mr D. Hughes, showing a direct approach to a reasonable answer.

1. "I started off working it out by decid- ing that I would make it the same amount of each stamps. I eventually worked out by adding each stamp up that it had to be 8 of each stamp and I arranged it into the square on the right.

ist 1st iSL Isl

1 -& -2ri

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&A 2f 2fa 2 r

2. Here I have made a stamp book worth +1.50 with 5 of each 1985 stamps. I worked it out because I worked out that 13p + 17p = 30p and then worked out a reasonable price +1.50 and a reasonable amount of stamps, 5 of each, and set it out as in the picture."

lst 1AiIs 1 1st Is kst

2CZnd 122d f2hd 2m d

Zachary Beattie, also of Bishop Kirk School, followed a similar method but arrived at a different format for the book of stamps. This introduced the idea of special offers, now popular with the Post Office, in order to create a handy symmetric format.

1. "I worked out that 7p + 51p= 12+p and added 121p to

12+p to 122p and so on until I got the answer of 8 of each stamp.

2. I worked out that 17p + 13p = 30p so ten of each stamp would cost +3."

1 2st 2ist

la l 1 lA a

2,a 2nd ist ish isk 2*Z 2nd

Leslie Davidge, Cofton School, London, investigated a range of combinations of prices before selecting that which

rf or class

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28 Mathematics in School, November 1987



would allow her to develop an attractive format for the booklet. Her comments suggest that the idea of a "reasonable price" has been a matter of thought and discussion.

"The advantage of my design is the price which is quite reasonable."

Nicola Twizell, a first year pupil at Dayncourt Comprehensive School, Nottingham, produced the following table of price combinations

No. of stamps 17p 13p

1 17p 13p 2 34p 26p 3 51p 39p 4 68p 52p 5 85p 65p 6 102p 78p 7 119p 91p 8 136p 104p 9 153p 117p

10 170p 130p

and then chose a design which contained 3 first class and 3 second class stamps, with a book value of 90p.

Tom Ashton, also of Dayncourt School, gave further thought to the problem of designing a stamp book suitable for use in modern vending machines and arrived at the following conclusion.

"The Post Office issues books of stamps which are sold through machines. To avoid complication the machines can't take 2p or 1p coins. In a +1.70 book you get ten first class stamps; 10 x 17p = +1.70. We decided that a +1.20 book containing 4 first class and 4 second class stamps would be a sensible design; 4 x 13p + 4 x 17p = +1.20. The main reason for this is you only need two coins to pay for it."

Roy 40a Il 5amps

It 7p

This choice of a +1.20 book containing 4 first class and 4 second class stamps was the most popular of the solutions received. Most pupils showed concern to preserve exact prices and ignored the current trend of the Post Office in giving price reductions on books of stamps for ease of purchase through machines. A further concern appeared to be the need to preserve symmetry in the format so that most solutions contained an equal num- ber of first and second class stamps.

The problem was extended into an open ended investi- gation for fifth form pupils by Mr N. J. Lord of Tonbridge School. The pupils were provided with a table of current postal charges and invited to design a book of stamps to the value of +1. The following solutions indicate the degree of discussion and thought developed from this simple mathematical exercise.

T. C. F. Barlows, of Tonbridge School, produced the following argument with his solution, which indicated the non trivial nature of the real problem.

"Since the most frequently posted letters are usually those under 60 g, I have taken care in choosing the stamps I did; so that I may have the greatest possible amount of stamps for letters priced either 13p or 17p. With the selection of stamps I have, one can simply send 6 second class letters and one first class, or 5 second class and 2 first class. Also with a certain combination of the correct stamps, one can send any parcel, first or second class, with weight not exceeding 400 g. For the booklet itself, I include 10 stamps ..."

17p 135 13P lo, 2,

13p 13 5, l1

J. S. R. Barrett, also of Tonbridge School, developed a carefully argued criteria for his alternative design, demon- strating the open endedness of the solution.

Letter Post Weight ist 2nd no oe - Ia l7as 17p 2F 60g 17p0 13 150q 31p 2p 30 6 42 13F j j k

,Op 400g 69p51

1. " The total of the stamps is a convenientfigure, represent- ing a single coin or note.

2. The values are so chosen as to make it possible to make any value used in the current 1st and 2nd class postal rates.

3. Ordinary letter postal rates are provided in an easily accessible place and also with two of both 1st and 2nd classes.

4. A table of current postal rates is included in a prominent position in the booklet.

5. The booklet is conveniently small, measuring approxi- mately 8 by 51 cms.

6. The cover is sufficiently hard-wearing to protect the stamps inside.

7. The booklet is relatively simple to mass-produce and at a cheap rate.

8. Care has been taken to ensure that not too many stamps of low denominations have been included.

9. All the denominations are of stamps already in issue."

The imagination of the pupils was not matched by the Post Office who produced a standard book of 10 first class stamps for +1.70 and one of 10 second class stamps for +1.30. Special books of 6 x 17p stamps were obtainable from machines for +1 as were 3 x 17p for 50p, however special discounts are not common for standard books. The mixed book of stamps followed the normal pattern of 10 stamps with 6 x 17p and 4 x 13p being sold for +1.54.

Current prices of 18p for first class and 13p for second class letters raise further problems of design for the books to be sold through vending machines and the mixed book of stamps. At the time of writing it was possible to purchase two different books for 50p and a mixed book of first and second class stamps for +1. Perhaps your pupils can discover their contents and hence produce a better design.

For extended investigational activity readers are referred to Discovering Mathematics by A. Gardiner. Oxford, 1987.

Mathematics in School, November 1987 29



Documents

A Problem of Inflation