Academic journal article Australian Mathematics Teacher

Discovery

Academic journal article Australian Mathematics Teacher

Discovery

Article excerpt

Additive combinations

Suppose that there is an inexhaustible supply of $3 and $5 vouchers from the local supermarket. They may only be exchanged for items that cost an exact number of dollars made up from any combination of the vouchers. What is the highest amount not able to be obtained?

This is an interesting problem in mathematical thinking and logic requiring only the ability to add. It is, therefore, suitable for junior high school students.

Using a systematic approach

One way to solve this problem logically is to develop a systematic approach.

* Clearly a $1 item cannot be obtained.

* Consider the smallest voucher value. Using this 0, 1, 2, 3 ... times yields values 0, 3, 6, 9, 12, 15, 18 ... which covers all items that cost 3N dollars.

* If one $5 voucher is now added to each of these, the amounts are now 5, 8, 11, 14, 17, 20 ... These cover all items that cost (31V + 2) dollars except $2.

* If another $5 voucher is added to the last sequence, the amounts are now 10, 13, 16, 19, 22, 25 ... These cover all items that cost (31V + 1) dollars, except $1, $4 and $7.

Hence, the maximum that cannot be obtained is $7.

What happens when the vouchers are changed to $5 and $13?

The systematic approach is still useful in solving this problem, but it is longer to apply.

* Firstly, using only $5 vouchers yields all $(5N) items for N = 0, 1, 2 . …

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