Is there an equation/algorithm to so you get the same number back when multipling by decimal

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GetThroughDiffEq

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I know this seems like a stupid question because if you did say 5000*.78 to get 3900, you'd just divide by .78 to get back 5000.

We're not going to use a calculator or software, so let's just assume your mind can't do division with large numbers (that is the point here):

What if I'm not a mathematical genius and wanted to know in a very fast time period (say 6 seconds) what to multiply 3900 by to get 5000?

I can guess and say 25%, sure, but it gets harder and harder if it's 562,705*.67 for instance.

What I'm getting at is can this be broken down to an algorithm where every number has the same proportion lost e.g. if 62000 loses 20% (62k*.8) then must gain x?
 
It's very hard to follow this; each paragraph seems to be asking for something entirely different, and you've never quite stated the goal. But after a lot of pondering, I think I've figured out what you're trying to ask.

My guess is that what you're looking for is an easy way to "back out" a percent decrease, so that if y is p% less than x, then x is q% more than y. What counts as easy enough is unclear.

Taking "p%" to mean the decimal, so that for 20%, I'll say p is 0.20, we have

y = x - px​
y = (1 - p)x​

and can solve for x, to get

x = y/(1 - p)​

which has to equal (1 + q)y .

Solving 1/(1 - p) = 1 + q for q gives us

q = 1/(1 - p) - 1​
q = p/(1 - p)​

That would be your formula, and it's kind of nice. For example, the reverse of a 20% decrease is q = 0.20/0.80 = 0.25 = 25%.

As far as I can tell, you are aware of at least part of that, but want to calculate it without having to do any work. Maybe you didn't get as far as an actual formula, and this is all you need. Or do you also want a way to do the division in your head without effort?

For the 20% case, we can see that the answer is just 20/80 = 1/4, which I can do in my head. If we started with a 22% decrease (your first example), to undo it we need to increase by 22/78 = 28.2%. I used a calculator to do that. Is that allowed?
 
When you multiply any (positive) number by less than 100% you get a smaller number. After all less than 100% of something means part of that thing.

What if I'm not a mathematical genius and wanted to know in a very fast time period (say 6 seconds) what to multiply 3900 by to get 5000? I can guess and say 25%. This is not true at all, sorry. If you compute 25% of 3900 the result will be much less than 3900 so clearly not 5000. What you mean to say is that you can see that 125% of 3900 is around 5000 or that you multiply 3900 by 125% to get around 5000. Do you see why you should multiply 3900 by 125% instead of 25%??
 
What you are asking about seems to be mental arithmetic, which is a very handy thing to know, especially if you are negotiating.

I'd say 3900 is close to 4000 and 5000 divided by 4000 is exactly the same as 5 divided by 4, which is 1.25.

How good is that answer?

3900 * 1.25 = 4875.

You wanted 5000. What is the percentage error? Less than 3%.

If you want exact answers, use your calculator. If you want quick but approximate answers, use your common sense and imagination.

I probably conducted a hundred negotiations without ever pulling out a calculator.
 
That's not quite I was asking. I should of been more clear, but there's no way to edit it now. I wanted an Algorithm that applies to what number + decimal (e.g. 1.3) to multiply by the result of a number multipled by a decimal less than one (e.g. .8).

So, if 5000*.8=4000, we need to multiply it by 1.25 to break even, a difference of 25% (20% loss vs 25% gain or .2 or .25).

On the other hand, this can't be universal because if you multiplied the same number by .75, you can't just tack on 25% to .25 (31.25%) to get back to 5000 (1.3125*3750=4921.875).

Again, to be more clear, I wanted it to apply using decimals (like .5, .6, .8) or whole numbers + decimals (like 1.7, 1.8, 1.92) only (no percentages), which is no easy task.
 
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Expressed in terms of a multiplier rather than a percent change, this is simple: the reciprocal. That is, we undo multiplication by dividing. But I thought you knew that, from the first line of the OP.

Given x*0.8 = y, x = t*1.25, because 1/0.8 = 1.25. In terms of fractions, 0.8 = 4/5, and 1.25 = 5/4, the reciprocal.

Given x*0.75 = y, x = y*1.333..., because 1/0.75 = 1.333... . In terms of fractions, 0.75 = 3/4 and 1.333... = 4/3.

Given x*1.7 = y, x = y*0.5882..., because 1/1.7 = 0.5882... . In terms of fractions, 1.7 = 17/10 and 0.5882... = 10/17.

What you're calling "whole number + decimal" is commonly called a mixed number. But this applies to any number (except zero).
 
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