Same Difference?

A friend asked me, are there any number pairs which share the same percentage difference in increasing to, and decreasing back to each other? eg. 100 to 150 is a 50% increase, but 150 to 100 is a 33.33% decrease (one third). Are there any which are equal in increasing and decreasing change, so X to Y is an A% increase and Y to X is a B% decrease, where A=B?

If X and Y are equal, yes, it’s always 0%, but if we ignore those, any others? No, it doesn’t seem so, at least not in natural numbers. Maybe something like 10 to 20, no, that’s 100% and then 50%.

But what about unnatural numbers, positive and negative pairs? How about -1 to 1? This is a bit mind-bending because there’s a negative involved. We got into quite the argument about it.

Let’s use the BBC Bitesize given method for working this out:

Calculating percentage increase

1. Work out the difference between the two numbers being compared.  
2. Divide the increase by the original number and multiply the answer by 100.  
3. In summary: percentage increase = increase ÷ original number × 100.

And the other way..

Calculating percentage decrease

1. Work out the difference between the two numbers being compared.  
2. Divide the decrease by the original number and multiply the answer by 100.  
3. In summary: percentage decrease = decrease ÷ original number × 100.

So by that basis, lets give that a crack with -1 and 1, first the increase between -1 and 1:

-1 – 1 = -2
-2 ÷ -1 = 2
2 x 100 = 200%

Now lets try the decrease from 1 to -1:

1 – -1 = 2
2 ÷ 1 = 2
2 x 100 = 200%

They’re the same percentage difference! All pairs of negative and positive numbers are always like this, because x – -x = 2x. Though it’s a bit horrible to think about.

This seems counterintuitive, how could the negative value increased by a magnitude be the same difference from its positive twin?

Ismael on Math Stack Exchange provides an excellent answer to a question about un-twinned negative and positive values, which helps answer the same question and I’ll try to summarise:

It helps to split the problem into two parts, 1) Reaching zero from the first number, and 2) Reaching the the second number from zero.

-1 to 0 using the provided equation is ((-1 – 0) / -1) = 100% but we can’t do the same for the second part because we can’t divide by zero (for reference it would be; ((0 – 1) / 0) ). So instead we peer at result of the first part, it took us 100% to step from -1 to 0, so 0 to 1, a change of also of 1 would be the same, 100%. Completed, these two parts of the journey sum to 200%.

The other way, 1 to -1 works the same way and produces the same result; 1 to 0 is ((1 – 0) / 1) = 100% for our first part, then 0 to -1 is ((0 – -1) / 0), but again we cannot divide by zero, so we glimpse at our first part and see it must be another 1, or 100%, making 200%, a decrease of the same magnitude.

The difference, both ways between -1 and 1 is 200%, it is the same for all pairs of natural numbers and their negative counterparts.

You might be despairing thinking, “no, negative one plus 200% of negative one is negative three! Not positive one!” (-1 + -1 + -1 = -3) and you’re right, sort of. What you’ve observed there is a decrease, while we’re looking for an increase between -1 and 1. The percentage change is the same between -1 and -3 as it is -1 and 1, but it’s going down instead of up. This curious happenstance is the same for all negative and positive pairs; -25 plus 200% of -25 (-50) is -75, but that’s decreasing, while increasing the same 200% grants us 25. Ultimately, the equation of ((start – final) / start) helps us side-step this awkwardness.