# Thread: How do I work out a percentage in reverse?

1. ## How do I work out a percentage in reverse?

Hi,

Sorry if this is in the incorrect section, I really couldn't see anywhere else more relevent.

Problem:
In my line of work, prices of my contracts vary annually via a retail price index. Between 2010 and 2011 this rose by 4.64%.

For example: Lets say I have a contract in 2010 priced at £10,000. I add 4.64% and we get £10,464 for 2011 that's no problem. However, lets say 6 months later I lose the 2010 paperwork and have completely forgotten the details for 2010 which I need to replace. I know the rise was 4.64% and I know it rose to £10,464. What calculation would I use to work out the previous figure of £10,000?

I also asked this on Yahoo and was told to divide 10,464 by 1.0464 to get 10,000. This works out but how do you get to 1.0464 without knowing the 10'000 figure? (10,464 / 10,000?? But I lost the paperwork for 2010, so I have no idea about the figure '10,000' remember)

Hopefully an easy one for yourselves

2. ## Re: How do I work out a percentage in reverse?

Originally Posted by Bikkembergs
Hi,

Sorry if this is in the incorrect section, I really couldn't see anywhere else more relevent.

Problem:
In my line of work, prices of my contracts vary annually via a retail price index. Between 2010 and 2011 this rose by 4.64%.

For example: Lets say I have a contract in 2010 priced at £10,000. I add 4.64% and we get £10,464 for 2011 that's no problem. However, lets say 6 months later I lose the 2010 paperwork and have completely forgotten the details for 2010 which I need to replace. I know the rise was 4.64% and I know it rose to £10,464. What calculation would I use to work out the previous figure of £10,000?

I also asked this on Yahoo and was told to divide 10,464 by 1.0464 to get 10,000. This works out but how do you get to 1.0464 without knowing the 10'000 figure? (10,464 / 10,000?? But I lost the paperwork for 2010, so I have no idea about the figure '10,000' remember)

Hopefully an easy one for yourselves
x + 4.6% of x = 10,464

=> x + (4.6/100)x = 10,464

=> x + 0.046x = 10,464

=> x(1.046) = 10,464

=> x = 10,464/1.046 = 10,000.

3. ## Re: How do I work out a percentage in reverse?

I'm still not quite sure though

4.6% / 100 for 0.046, thats fine. But then you appear to multiply this by 10,000 to reach 1.046, before you've calculated the end figure of 10,000..? I dont see how you can x by 10,000 unless it's an unrelated figure which is coincedentally the same figure as the desired result? =/

Sorry to be a pain.

4. ## Re: How do I work out a percentage in reverse?

Originally Posted by Bikkembergs

I'm still not quite sure though

4.6% / 100 for 0.046, thats fine. But then you appear to multiply this by 10,000 to reach 1.046, before you've calculated the end figure of 10,000..? I dont see how you can x by 10,000 unless it's an unrelated figure which is coincedentally the same figure as the desired result? =/

Sorry to be a pain.
No worries! I'm not sure I see x being multiplied by 10,000 in mr fanstastic's answer, though...
I do see it being divided by 100, and that's happens with percentages.
Let's go back to mr fantastic's method. I'll repeat it, maybe tweak it a bit, and annotate each step:

We know some amount of pounds, x, increased by 4.64% and and thus stands at a total of £10,464:
x + 4.64% of x = 10,464

Now, let's convert the percentage to a number. I find the easiest way to deal with percent is to think of it as "divide by one hundred". So let's do that:
x + 4.64*(1/100)*x = 10,464

Then we work the math on that second term:
x + 0.0464x = 10,464

Now, we collect like terms:
1.0464x = 10,464

Then, we isolate x by dividing both sides by 1.0464:
x = 10,464/1.0464

Thus
x=10,000

5. ## Re: How do I work out a percentage in reverse?

Originally Posted by Bikkembergs

I'm still not quite sure though

4.6% / 100 for 0.046, thats fine. But then you appear to multiply this by 10,000 to reach 1.046, before you've calculated the end figure of 10,000..? I dont see how you can x by 10,000 unless it's an unrelated figure which is coincedentally the same figure as the desired result? =/

Sorry to be a pain.
mr fantastic factored out "x".

$x+0.0464x=x(1)+x(0.0464)=x(1+0.0464)=x(1.0464)$

6. ## Re: How do I work out a percentage in reverse?

Hi again,

That's great stuff . Sorry, just couldn't see you'd simplified it to get 1.0464x rather than doing some fancy equation

I'm going to explain to colleagues how to retrieve old figures should they need to tomorrow. I'm simply going to tell them to divide 4.64 by 100, then add 1 and divide the new price by this to reveal the old price. Incorrect, but the result will always be the same as the RPI is fixed. What do you think?

Lastly, just to clarify I'm on this. If the RPI for 2011 hiked up to a massive 904.64% from £10,000 in 2010, we'd do:

x + 904.64% of x = 100,464
x + (904.64/100)x = 100,464
x + 9.0464x = 100,464
x(10.0464) = 100,464
x = 100,464/10.0464 = 10,000

..right?

Thanks again to the 3 of you. I can go in with a bit more confidence now tomorrow

7. ## Re: How do I work out a percentage in reverse?

Originally Posted by Bikkembergs
Hi again,

That's great stuff . Sorry, just couldn't see you'd simplified it to get 1.0464x rather than doing some fancy equation

I'm going to explain to colleagues how to retrieve old figures should they need to tomorrow. I'm simply going to tell them to divide 4.64 by 100, then add 1 and divide the new price by this to reveal the old price. Incorrect, but the result will always be the same as the RPI is fixed. What do you think?

Lastly, just to clarify I'm on this. If the RPI for 2011 hiked up to a massive 904.64% from £10,000 in 2010, we'd do:

x + 904.64% of x = 100,464
x + (904.64/100)x = 100,464
x + 9.0464x = 100,464
x(10.0464) = 100,464
x = 100,464/10.0464 = 10,000

..right?

Thanks again to the 3 of you. I can go in with a bit more confidence now tomorrow
Correct on all three counts.