I have been looking at the problems from the UK Junior Mathematical Challenge April 2012. My aim with this blog is to keep a record of my maths work and to see if I can develop a richer understanding of my own mathematical way of working.

Onto the first question I looked at:

11. In the following expression 'box' is to be replaced with either + or - in such a way that the result of the calculation is 100.

123      'box'      45      'box'      67      'box'      89

 

 

The number of + signs used is p and the number of - signs is m. What is the value of  p - m ?

 

 

A) -3         B) -1         C) 0         D) 1         E) 3

 

 

The solution given involved a nice bit of algebra using simultaneous equations but I wanted to develop a different method of act. So I looked at the possible solutions of  p - m.

 

So if

• p = 3 and m = 0 then  p - m = 3
• p = 2 and m = 1 then p - m = 1
• p = 1 and m = 2 then p - m = -1
• p = 0 and m = 3 then p - m = -3

So only these values need to be considered.

 

1. If  p - m = 3 . This is impossible since adding all the numbers won't result in the required number of 100.
2. If  p - m = -3. This is also impossible since subtracting the other three numbers from 123 won't result in 100.
3. If  p - m = 1. For this to occur we would have to add two of the numbers to 123 and subtract the remaining one. This is clearly impossible.
4. This only leaves the case  p - m = -1. Where we add one of the numbers and subtract the remaining two others.

Note  To get this solution to the question I haven't had to do any hard calculations since I have just compared the size of the numbers present. I also haven't actually found out which numbers need to be added or subtracted. However I have answered the question.

 

Question  Can I alter this question to make my method too hard to do so that the question ensures that the numbers needed to be added or subtracted have to be identified?