Percent Change: Working Backward

Having discussed how to calculate the percent change between two numbers, and how to apply such a change to one number to get a new number, we need to look at what may be one of the most common types of questions we get: reversing a percent change (increase or decrease) to find the original value.

Price before discount and tax

We can start with this question from 1996:

Price before Discount

A calculator is discounted 20 percent. After adding 7 percent tax, the total cost is $20.07.  What was the list price of the calculator, without tax, before discount?

Here we have to back off both a tax (which amounts to a percent increase) and a discount (a percent decrease). So this question covers both.

Doctor Brian answered:

When 20 percent is discounted from a price, you still have to pay 80 percent of the original price.  When 7 percent tax is added to a price, you have to pay that plus 100 percent of what you normally would, for a total of 107 percent of the actual price.

We'll use these two ideas in the problem.

If x is the original price, then we are paying 107 percent of (the sale price), which is 107 percent of (80 percent of the original price).  And since the word "of" means "times" in percent problems, we get

   1.07 * .80 * x = 20.07

         .856 * x = 20.07

                x = 23.45

Recall from last time that 80% is called the complement of the 20% decrease, and represents the remaining percentage. The 107% is similar (but unnamed, to my knowledge): It represents what percentage the new value is of the old value.

What he has done here is first to combine the two changes (7% increase and 20% decrease) by multiplying 107% and 80% to get a net multiplier of 85.6%. To undo this, we just divide by the multiplier. We could instead have merely divided separately by each multiplier (1.07 and 0.80) to get the answer.

Turning these into formulas, we can reverse a P% decrease using $$\text{original} = \frac{100}{100-P}\cdot \text{new},$$ and we can reverse a P% increase using $$\text{original} = \frac{100}{100+P}\cdot \text{new}.$$

We can also check the answer. From a list price of $23.45, a 20% discount would be $4.69, reducing the price to $18.76. A 7% tax on this would be $1.31 (rounded from 1.3132); adding this, the total price is $20.07, which is just what we want. (I recommend always doing this check in real-life problems, because rounding can influence the results.)

Reversing a percent decrease

The next question, from two months later, picks up where that left off:

Restoring an Original Price

I know there are formulas for restoring a price to its original value after taking a discount on it. How do you determine what the multiplier is for getting back to your original number?

Example: I start with a cost of $48,695. I discount it by 30 percent, bringing it to $34,087. What do I multiply $34,087 by to get back to $48,695?

Your help in this matter would be very much appreciated. Please explain the method of getting the formula.

Doctor Mike answered, first answering the specific question before broadening it:

If you discount by a certain percentage, like 30 percent, the discounted price is 70/100 (or 70 percent) of the original.  You then multiply the discounted price by 100/70 (about 1.4286) to get the original price back.  This multiplier is equivalent to 142.86 percent of the discounted price.

All we’ve done is to take the 70% (complement) and stand it on its head, so to speak: Rather than divide by it, as we did in the first answer, we can multiply by its reciprocal.

We can also turn this into a percent increase:

The 142.86 breaks down to 100 + 42.86, which means that to get back up to the original price you have to take 100 percent of the discounted price and add to that 42.86 percent of the discounted price.  The 42.86 percent part of it is the "percent of increase" over and above the discounted price.  This can be surprising at first; that you decrease by 30 percent but then must increase by about 43 percent to get back up to where you started. This is just another way of saying that "30 percent of the original price" is about the same as "43 percent of the discounted price."

Here we are reversing the process we used to change a 7% increase to a 107% multiple, by subtracting 100% from 142.86%.

And it turns out that the discount (30% of $48,695 = $14,608.50) is about 43% of the original price (42.86% of $34,087 = $14,609.69, a little off due to rounding).

This can be turned into a formula:

What we did for 30 percent we can do for any percentage "P".  The discounted price is (100-P)/100 of the original.  You multiply the discounted price by 100/(100-P) to get back up to the original price.  Multiply this by 100 to get the equivalent percentage (like 142.86 above), and then subtract 100 to get the percent of increase (like 42.86 above).
  
    /   100         \            100*P
    | ------- * 100 | - 100  =  -------  = Grossing Percentage
    \  100-P        /            100-P 
  
Checking for P = 30% , (100*30)/(100-30) = 3000/70 = 42.85714.... 
  
If you want to work with multipliers rather than percentages of increase, just use "100/(100-P)" for the "grossing up" multiplier.

“Grossing up” means determining the “gross” (before discount or tax) amount required to obtain a given “net” (after discount) amount. The formula to reverse a P% decrease is $$\text{% increase} = \frac{100P}{100-P}.$$

Reversing a percent increase

The next question, from 2002, is similar, but about a percent increase:

Undoing Percentage Changes

If 10000 x 102% = 10200, then how do I figure the percentage to get back to my base number (10000), e.g., 10200 * 1.96% - 10200 = 10000.08?

Here we have a 2% increase, and want to find out that it will be undone by a 1.96% decrease. (Elias had his work backward, and meant 10200 – 10200 * 1.96% = 10000.08. And, of course, this is not exactly 10000 because of rounding.)

Doctor Ian started with the formula rather than the specific example:

It's probably easier to see how this works if you use variables instead of numbers. 

If your original amount is A, and the percent increase is p, then the new amount is 

  A' = A(1+p)

You want to decrease it by some percentage q, to get back to A.  That is, you want to find q such that

        A = A(1+p)(1-q)

        1 = (1+p)(1-q)

  1/(1+p) = 1 - q

        q = 1 - 1/(1+p)

          = (1 + p - 1)/(1+p)

          = p/(1+p)

This is much like the formula in the previous answer, except that (a) the sign is changed because it was a percent increase, and (b) Doctor Ian, like most mathematicians, is working with the decimal forms rather than the actual percent number, so that his p is not 2, but 2% = 0.02. Using the same notations as we saw above for grossing up, the formula to reverse a P% increase is $$\text{% decrease} = \frac{100P}{100+P}.$$

Let's check this with a simple example.  If we increase something by 100%, we should have to decrease it by 50% to get back to where we started:

        q = 1.0 / 2.0

          = 0.5
          
If we increase something by 1/3, we should have to decrease it by 1/4:

        q = (1/3) / (4/3)

          = 1/4

So this seems to work okay.  So if p is 2 percent, q would be 

        q = 0.02 / 1.02

Does this help?

And this q turns out to be 0.0196078…, which rounds to Elias’ 1.96%. It could also have been calculated as 2/102.

Percent increase … or not?

Here’s a 2003 question that focused on distinguishing what we’ve been doing here from a straight percentage calculation:

Percentage Increase vs. Percentage

If there were 10,000 claims in 2001, and that is a 300 percent increase since 1999, how many claims were there in 1999? 

My colleagues are giving me all different answers. I think the answer is 2,500. My colleagues say 3,333.

Very likely the large percentage was part of the reason for the dispute, as it can confuse people. (We’ll be looking at that soon!) Doctor Ian replied:

A 300% increase means that 

   (claims in 2001) - (claims in 1999)   300
   ----------------------------------- = ---
              (claims in 1999)           100

That is, we're saying that the _increase_ is 300% of the original value.  

Since the numbers are so nice and round, we can do this:

   (claims in 2001)    (claims in 1999)   300
   ----------------- - ---------------- = ---
   (claims in 1999)    (claims in 1999)   100

                  (claims in 2001)        
                  ----------------- - 1 = 3
                  (claims in 1999)        

                      (claims in 2001)        
                      ----------------- = 4
                      (claims in 1999)        

So the number of claims in 2001 is 4 times whatever it was in 1999, which means there were 2,500 claims in 1999.

Another way to say this would be that a 300% increase means that the new value is 100% + 300% = 400% of the original (that is, 4 times the original); so to get back to the original, we divide by 4.

So how did the others get their answer? They were missing the word “increase”:

Note that if we change the wording slightly, we can come up with the other answer. That is, if we say that the number of claims in 2001 is 300% of the number of claims in 1999, then we're saying

   (claims in 2001)   300
   ---------------- = ---
   (claims in 1999)   100

which we can rearrange to get

                      100 * (claims in 2001)
   (claims in 1999) = ----------------------
                              300

                      100 * 10,000
                    = ------------
                          300

                      10,000
                    = ------
                         3

                    = 3333 1/3

That is, if the new is just 300% of the old (3 times as much), we just divide by 3. Doctor Ian didn’t mention it, but their misreading of the problem may very well have been due to the thinking discussed in this post: Three Times Larger: Idiom or Error?

Let's put the two cases together, so you can compare them more easily:

1) Claims in 2001 are an increase of 300% over claims in 1999.
   (The increase is 300% of the old value.)

   (claims in 2001) - (claims in 1999)   300
   ----------------------------------- = ---
              (claims in 1999)           100

2) Claims in 2001 are 300% of the claims in 1999. 
   (The new value is 300% of the old value.) 

   (claims in 2001)   300
   ---------------- = ---
   (claims in 1999)   100

Does this make sense? 

I find that it's useful to keep a few simple examples in my head. One of my favorites is this: If I start with $1, an increase of 100% is an increase of $1, which gives me $2, but $2 is twice as much as $1, which means it's 200% of $1.

Up and down by different percentages

The fact that a percent increase is reversed by a different percent decrease leads us to a common confusion that is, as we’ll see, complicated by calculators. This question is from 2002:

Dueling Percentages

Can you tell me why, for instance, when you add 15% to 100, you get 115, but if you subtract 15%, you get 86.95...instead of 85?

Pete’s calculation is wrong, so it needs to be diagnosed: What is he really doing? Doctor Ian answered:

Are you sure about that?  

  100 + (15% of 100) = 100 + (0.15 * 100) 
                     = 100 + 15
                     = 115

  100 - (15% of 100) = 100 - (0.15 * 100) 
                     = 100 - 15
                     = 85

Increasing or decreasing by 15% mean adding or subtracting 15% of the same number.

I think what you're asking is why, when you reduce something by 15%, increasing the reduced figure by the same percentage doesn't get you back to where you started, e.g., 

        85 + (15% of 85) < 100

  86.95 + (15% of 86.95) = 100

We get 86.95 by dividing 100 by 1.15, so that increasing this, not the 85, by 15% gets us to 100.

So what Pete called decreasing by 15% was really undoing an increase by 15%. We’ve already seen that these are different things. Why?

The clearest way to show why it works this way is to use symbols instead of concrete values.  Suppose we start with some amount, A, and we reduce it by some percentage, P.  Then the new value is

  A' = A(1 - P)

Now if we increase the reduced amount, by the same percentage, we get 

  A'' = A'(1 + P)

      = A(1 - P)(1 + P)

      = A(1 - P^2)

which is smaller than what we started with!

In our example, decreasing by 15% and then increasing by 15% results in a number that is only \(1 – 0.15^2 = 97.75\%\) of what we started with. And this is because what we added is 15% of a smaller number than the 100.

It can also be easier to grasp if you use extreme examples.  If you start with $100, and I take 99% of it, that leaves you with $1.  If I increase your dollar by 99 percent, will that get you anywhere near $100?

Subtracting 99% of $100 ($99), then adding back on 99% of $1 ($0.99) gets you only to $1.99, nothing near $100. (We’ve seen this idea of going to extremes several times recently.)

And here is a third way to look at it.  Suppose we start with some amount, 

  +---+---+---+---+---+
  |   |   |   |   |   |
  +---+---+---+---+---+
  |   |   |   |   |   |
  +---+---+---+---+---+
  |   |   |   |   |   |
  +---+---+---+---+---+
  |   |   |   |   |   |
  +---+---+---+---+---+
  |   |   |   |   |   |
  +---+---+---+---+---+

and reduce it by some percentage (say, removing one item out of every 5):

  +---+---+---+---+---+
  |   |   |   |   | - |
  +---+---+---+---+---+
  |   |   |   |   | - |
  +---+---+---+---+---+
  |   |   |   |   | - |
  +---+---+---+---+---+
  |   |   |   |   | - |
  +---+---+---+---+---+
  |   |   |   |   | - |
  +---+---+---+---+---+

Now we want to increase it by the same percentage (i.e., adding one item for every 5):

  +---+---+---+---+---+
  |   |   |   |   | - |
  +---+---+---+---+---+
  |   |   |   |   | - |
  +---+---+---+---+---+
  |   |   |   |   | - |
  +---+---+---+---+---+
  |   |   |   |   | - |
  +---+---+---+---+---+
  |   |   |   |   | - |
  +---+---+---+---+---+
  | + | + | + | + | 
  +---+---+---+---+

The increase, being the SAME fraction of a SMALLER number, will be smaller than the reduction, so we end up replacing only SOME of the amount that was removed:

  +---+---+---+---+---+
  |   |   |   |   |-/+|
  +---+---+---+---+---+
  |   |   |   |   |-/+|
  +---+---+---+---+---+
  |   |   |   |   |-/+|
  +---+---+---+---+---+
  |   |   |   |   |-/+|
  +---+---+---+---+---+
  |   |   |   |   | - |
  +---+---+---+---+---+

The difference between the increase and the reduction is the square of the percentage.

That little square at the lower right is the difference.

Calculators with a -TAX button

Pete responded, explaining what his question really meant:

Thanks very much for replying yesterday to my question.

The problem lay in the fact that I was using a desktop calculator, and the -TAX button for some unknown reason does not give minus 10% of 100 as 90.

Any ideas why this is so, as it happens on all models?

Not all calculators have +TAX and -TAX buttons; in fact, those used by mathematicians or in math classes don’t:

I've never seen a -TAX button on a calculator, but it sounds to me as though it might be assuming that you know the price WITH tax, and the tax rate, and you want to figure out the price WITHOUT tax. 

For example, if the price with tax is $100, and the tax is 10%, and the price without tax is P, then

  P(1 + 0.10) = $100

            P = $100 / 1.10

              = 90.91

In other words, a retail price of $90.91, with 10% tax added to it, comes to $100.

So the button is for reversing a percent increase (removing a known tax), not for making a percent decrease (as if there were such a thing as a negative tax). Pete was misunderstanding the label on the button.

Calculators with a % button

The same issue comes up with % buttons that many (non-scientific) calculators have, as this 2006 question shows:

Finding Percentages on a Calculator

How do I use a calculator to check percentages?  For example, how would I check 6% of 180,000 using a calculator?  Would I multiply 180,000 x 6.00%?

It’s encouraging that Donna doesn’t use the calculator to do her work, but to check; that’s what I recommend to students. (Or did she just say that to avoid offending us?)

I took this one, having a lot of experience with it:

Probably!

Unfortunately, I've seen some variation in how calculators handle percents, so you'll want to check the instructions that came with yours to be sure.  But try it out and see!  What I'd do, if I had an unknown calculator with a % key, is to enter something like

  200 x 5 % =

and see if it gave the right answer, namely 10.  (Also note whether it gives the answer even before you press "="; some work that way, and then you should learn NOT to use "=" here.  That is true, for example, of the calculator that comes in Windows.)

Some calculators will immediately replace 5% with 0.05 in their display, so the multiplication above amounts to \(200\times 0.05 = 10\), which is correct. This mirrors how I think about such calculations. On this type, the “=” is needed. (The current Windows calculator, in Standard mode, works this way, not as I described.)

Then I might also try

  5 % x 200 =

and see if that worked too.  (On many calculators, it won't, so I'd have learned not to do it in that order!)

The Windows calculator, when I tried this, replaced 5% with 5% of the number that I’d left in the display from a previous calculation, which would be a disaster! Two calculator apps on my phone both handled this correctly.

There's much more variation in how they handle percentage increases, like

  200 + 5 %

which many calculators, but not all, would take as "increase 200 by 5%", giving 210.  If you need to do things like that, you'll want to be sure to check the manual.

The Windows calculator, and my two calculator apps, all replace 5% here with 10, implicitly calculating 5% of the number being added to, so that pressing “=” results in 210, the correct answer.

On the other hand, two scientific calculators with a % button (a TI and a Casio) both do what I would expect of such a calculator, converting 5% to 0.05, then adding to get 200.05, which is what the expression literally means.

In other words, calculators meant to be used in financial calculations take “+ 5%” to mean “add 5% of the preceding number”, twisting the notation to fit the application. Calculators meant to be used in contexts where math notation is interpreted carefully don’t do that, in part because it would make it impossible to be sure how to interpret more complicated expressions.

Summary: up and down, forward and reverse

Let’s close with a summary of how to apply and reverse percent increases and decreases, in response to this 2002 question:

Markups and Discounts

I have been selling products over the past 30 years, and I still run 
into this question.

What is the correct way to markup or discount a product?

$100 markup (x) 1.1 = $110 = 10%
$100 markup (/) .9  = $111.11 = 10%

$100 discount (/) 1.1 = $90.90 = 10%
$100 discount  (.9)   = $90    = 10%

I find it easier to mark up (/) .9 .8, which equal 10% 20% etc., and 
then mark down (x) .9, .8, .7, which equal 10 to 30%. It seems faster 
if there is no other explanation.  

I have yet to find anyone who can explain it to me.
Thanks,
Gary B.

His methods are a mix of the forward and backward methods. I replied, explaining what each calculation really does:

If you mark up by 10%, you are adding 0.10 to the price of an item:

    new price = old price + 0.10 * old price
              = 1 * old price + 0.10 * old price
              = old price * (1.10)

(I am using "*" for multiplication.)

To determine the original price given the marked up price, you have to 
undo this by dividing:

    old price = new price / 1.10

If you discount by 10%, you are subtracting 0.10 of the price of the 
item:

    new price = old price - 0.10 * old price
              = old price * (0.90)

To determine the original price given the discounted price, you have 
to undo this by dividing:

    old price = new price / 0.90

To emphasize that these are different, I added:

Note that if you mark up by 10% and then discount by 10%, you don't 
get back the original value, because you are taking off 10% of a 
larger amount:

    $100 * 1.10 * 0.9 = $110 * 0.9 = $99

Here we took off $11 for the discount, not the $10 we added in the 
markup.

I think this differs from what you wrote, if I understand it 
correctly. You always use 1.10 (1 plus the percentage) for markups, 
and 0.90 (1 minus the percentage) for discounts. You multiply to apply 
the markup or discount, and divide to undo it.

Next time, we’ll look at another perspective on this: Gary may in part have been confusing markup and margin.

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