Percent Change: Finding and Applying It

A question we got in March asked about working with percent increases. As I replied to this rather common topic, referring to an archived answer, I was reminded that this would is a very common type of question, and would be a good topic to cover in the blog. So that will be the topic of the next couple posts, starting with how to calculate the percent change between two numbers, then how to apply a percent change to an existing number, then (next time) how to reverse a percent change to find the original number, and finally some additional issues involving percent change.

Applying a percent decrease

Let’s start with a basic question from 2002:

Finding Sale Price

We are studying how to find the sale price by subtracting the discount from the regular price. Can you help me?

I answered:

We have a number of answers to questions in this area, which you can find by searching our site. Most of these deal with the reverse problem, finding the original price when you know the sale price. Here's a sample problem in the forward direction:

If the regular price is $8.99, and the item is being sold at 30% off, what is the sale price?

Well, 30% of $8.99 means 30/100, or 0.30, times $8.99:

    0.30 * 8.99 = 2.697

That much is being taken off the price, so the sale price is

    $8.99 - $2.697 = $6.293

which would be rounded down to $6.29.

So, the basic idea of a percent decrease, such as a discount, is to find that percent of the given number, and subtract it. The formula for this would be $$\text{new value} = \text{old} – \frac{\text{percent change}}{100}\times \text{old}.$$ In the example, I directly converted 30% to the decimal 30/100 = 0.30, and multiplied by that.

Another way to do this would be to recognize that if 30% is being taken off, then what's left is 70%; so you can just multiply $8.99 by 70% to get the sale price:

    0.70 * $8.99 = $6.293

This method looks like $$\text{new value} = \left(1 – \frac{\text{percent decrease}}{100}\right) \times \text{old}.$$

Using complement to calculate a discount

That second method is the subject of this 2001 question:

Opposite of Percentage

I am beginning to wonder if there *is* an answer to this.

I am trying to write a document explaining a common procedure, namely, calculating a discounted price. To calculate the discount, you multiply the "base" by the "percent discount" to yield "percentage."  At least, these were the terms I was taught.

So, if the price is $32 and the discount is 15%, then you multiply 32 times .15 to get the amount that is to be subtracted from the initial price ($4.80) to yield the discounted price ($27.20). I have been taught that the "initial price" ($32) is the "base," the .15 is the "percent" or "discount rate," and the resulting discount amount ($4.80), is the "percentage."

Another way to calculate the final price would be to subtract the percentage from 100 (100% - 15% = 85%) and then multiply that result by the original price (.85 times $32 = 27.20).  Or, as it appears in many computer programs, net = price * (1-discount_rate), where it is understood that discount_rate is a decimal fraction.

My question is one of terminology. What is the name given to the term "(1-discount_rate)" in this example? I have looked up reciprocal, complement, inverse, etc., but none of those fits. I need to be able to define this word once and then use it repeatedly in my document, rather that explaining it every time I use it, which will be many times.

One of the terms Marty looked up turns out to be the answer, as I explained, using what is now a dead link:

What you want is "the complement of the discount rate." Here is one site I've found that defines it:

   Trade Discounts - Business Mathematics, Confederation College

   Trade discounts are used in the retail sector. 
   They are used to calculate how much a retailer
   will pay a manufacturer for a given product...

   Complement of a Trade Discount: This is the
   difference between the discount rate and 100%.
   The complement can be any percent that results
   when you subtract the trade discount rate from 100%.

In general mathematicians use the word "complement" to mean "all except ..." or "the whole minus a given part"; here it means 1 minus a fraction - though I haven't found any dictionary that gives this definition.

So we can apply a discount (percent decrease) using the complement: $$\text{new value} = \text{old}\times (1 – \text{percent decrease}).$$

Similarly, we can apply a percent increase (markup – we’ll be discussing this later) this way: $$\text{new value} = \text{old}\times (1 + \text{percent increase}).$$

Combining discount and tax

Here’s an interesting question from 2002:

Order of Discounts and Taxes

I'm wondering if you could tell me why it doesn't matter in determining the cost of an item whether you take off the tax before or after you take off the saving rate. 

For instance, if I buy an item that originally costs 100 dollars during a 20% off sale and the tax is 10% (for sake of simplicity), intuitively I think that I want to take the 20% off before I calculate the tax, so I will be taxed on a lower amount, but it doesn't seem to matter whether I take the tax off the 100 dollars first and then calculate the savings, or I take the savings first and then calculate the tax. Am I right that order doesn't matter, and is there an explanation?

Since the question was about why it works rather than how to do it, Doctor Ian replied with an algebraic view:

As with most things, it's often easier to see what's going on if you use variables instead of actual numbers.  

Suppose the normal price of an item is P, and the tax rate is T. Then the amount you'd normally pay for the item is 

  P(1 + T)

Now, suppose you discount the price by some percentage, D.  If you do it before the tax, you end up paying

  [P(1 - D)](1 + T)

But if you apply the discount to the after-tax amount, you end up paying

  [P(1 + T)](1 - D)

Since you're just multiplying things together, the order doesn't matter; so you end up paying the same thing in both cases.

He is using the complement to find the discount, and its equivalent to apply the tax. As an example, take that 20% discount and 10% tax. The discount results in paying 80% of the full price; the tax leads to paying 110%. So the net price is \($100\times0.80\times1.10 = $88\). Taking them one at a time, the discount on $100 would be $20, leaving $80; the tax on $80 is $8, so you pay $88. In the other order, the tax on $100 is $10, so you owe $110; but the discount on $110 is $22, so you only pay $88, the same as before.

Finding or applying the percent change

Here is a question from 2005 that focuses on finding the percentage, given the two numbers:

Formula to Calculate Percent Increase

I want to compare two figures and show the percent increase that has been made.  For example, if 55 has increased to 75, what percentage increase is that?

Whenever I get a figure, I am never sure if there is a way of checking back that it is correct.

Doctor Ian took this, explaining the appropriate formulas for both the calculation (finding the percent) and the check. First, what does a percent increase mean?

When we say

  Q increased by P percent

what we _mean_ is that the new value of Q is 

  Q' = Q + (P percent of Q)

which is to say, 

  Q' = Q + (P/100 * Q)

So far, so good?

He is defining percent change in terms of how it is applied: If we increase Q by P%, we are adding P% of Q to Q itself. For example, increasing 120 by 5% means finding 5% of 120, which is 5/100 times 120 (that is, 6), and adding that to 120, so that the result is 126.

Now we reverse the calculation to find the percentage by which Q was increased to get Q‘:

If you have Q (the old value) and Q' (the new value), then you want to find P.  If we solve the equation for P, we get

            Q' = Q + (P/100 * Q)

        Q' - Q = P/100 * Q

        Q' - Q
        ------ = P/100
           Q

  Q' - Q
  ------ * 100 = P
     Q

So you subtract the old value from the new one, and divide the difference by the old value.  Then multiply by 100 to get the percentage.

Having seen the algebra, you don’t need to repeat it each time; you can just use that formula: $$\text{percent change} = \frac{\text{new } – \text{ old}}{\text{old}}\times 100\%.$$

Another way to think of that is:

  amount of change
  ---------------- x 100 = percent change
  original amount

I find this easier to remember than an actual formula: We are simply asking, “What percentage of the original amount is the change from that to the new amount?” That, of course, is exactly what the phrase “percent change” means.

I also like to think of the multiplication by 100 as a unit conversion, where we multiply a number (typically a decimal) by 1, in the form of 100%, to get an answer measured in percent.

An example will make it clearer. First we calculate a percent change, then check by applying it:

If you want to check your work, you can go back to the definition, plug the numbers in, and see if what you get is true.  For example, suppose I have an increase from 40 to 50.  The formula gives me

       50 - 40         10
   P = ------- * 100 = -- * 100 = 25
         40            40

So I think it's 25%.  Let's go back to the definition:

  50 = 40 + (25% of 40)

     = 40 + 10

     = 50

This is true, so I have the right percentage increase.

The change divided by the old value was 0.25, which is 25%. And the check is to apply that percent change to the old value and see that we get the new value: \(40 + 0.25\times40 = 50\), the final value we started with. It’s good!

The same sort of work applies to finding a percent decrease; we just get a negative number if we subtract old from new. We still divide by the original amount.

Another version of the formula is obtained by dividing each term of the numerator by the denominator: $$\text{percent change} = \left(\frac{\text{new }}{\text{ old}} – 1\right)\cdot 100\%.$$ For example, in the case above, we have $$\left(\frac{50}{40} – 1\right)\cdot 100\% = (1.2 – 1)\cdot 100\% = 0.2\cdot 100\% = 20\%.$$

Percentage error in an estimate

Here is a similar question from 2002, with a different context:

Percentage Error

Please help me find the percentage error for the following numbers.

   5.7 estimated findings
   5.8 actual findings

Doctor Achilles took the question, starting with the formula:

Percentage error is just how much your guess was off from the actual value. The formula is:

  |estimate - actual|/actual * 100%

[That is: the absolute value of (the estimate minus the actual) all divided by the actual, all multiplied by 100%.]

Note that this is the same formula we saw above, except that, by taking the absolute value, we are ignoring the direction of the error. We are treating the actual value as the “original”, from which the estimate deviates by some percentage, called the percentage error.

But he goes beyond the mere answer, focusing on the reason for the calculation:

Let's think about why we use this formula. If you want to know how close your estimate is, the first thing to do is just to ask how much you missed by, that is the absolute value of the difference between the two numbers (the absolute value is used because you're only concerned with how much you missed by, not whether you were too high or too low). So in this case, you missed by 0.1.

0.1 is a small number, so it sounds like your guess is pretty good.

So why do we bother with this business of dividing by the actual amount?

Let's take a couple of other examples:

Example 1:

  100.0 (estimated)
  105.3 (actual)

Example 2:

  10.0 (estimated)
  15.3 (actual)

Example 3:

  1.0 (estimated)
  6.3 (actual)

Example 4:

  0.1 (estimated)
  5.4 (actual)

In all cases, you missed by the same amount (5.3). But in the first example, it seems as if your guess was a lot better. Even though it's off by just as much as the last example, missing by 5.3 out of 105.3 isn't too bad, but missing by 5.3 out of 5.4 seems pretty darn bad.

So we're concerned here not just with how much you missed by, but with what percent of the actual value you missed by.

So in example 1, you missed by 5.3 out of a total of 105.3:

  5.3/105.3 * 100%

equals

  0.0503 * 100%

equals

  5.03%

And in example 4, you missed by 5.3 out of total of 5.4:

  5.3/5.4 * 100%

equals

  .9815 * 100%

equals

  98.15%

This is a good way to represent the intuition that missing by 5.3 out of 105.3 is a pretty good guess, but missing by 5.3 out of 5.4 is a pretty rotten guess. In one case, your percent error is only about 5% (small error), while in the other case, your percent error is a big 98% (huge error).

So the point of the percent error is to put the amount of error into context; it’s also called relative error.

Percent change between numbers

The discussion above related to a percent increase or decrease from one number to another. The next question, from 1997, gets us into another kind of percent change, through variations in wording.

Percent Change, Increase, Difference

I haven't had math in years and I need a little help here. Say you have two numbers 5 and 7. You want to know what is the difference in percent between the two numbers.

Below is an real example of the problem I'm working on.

1991 = $2346.80

1992 = $3608.29

Percent change 91-92 = ?

The two examples are potentially quite different, because the first just has two numbers with no context, while the second is a practical type of problem with a specific meaning. Doctor Rob answered, starting with the latter, which is the same type we did above:

First you need to compute the difference between the two amounts. Then you make a fraction of that difference over the first of the two amounts. Finally you convert the fraction to a percentage by dividing the denominator into 100 times the numerator.

In your example, dropping the dollar signs, the difference is 3608.29 - 2346.80 = 1261.49. The fraction is 1261.49/2346.80, and the percentage comes from dividing 2346.80 into 126149. The result is 53.75, and since the sign is positive, this is a 53.75 percent increase from 1991 to 1992.

Here we are specifically asked for the change from 1991 to 1992, with a definite “old” and “new”, so we do just what was done above.

But the first example is different, and a little ambiguous. Which number comes first?

In the other example, 5 and 7, the difference is 2, the fraction is 2/5, and the percentage is 5 divided into 200, or 40. Thus 7 is 40 percent larger than 5.

On the other hand, if you take 7 and 5, the difference is -2, the fraction is -2/7, and the percentage is 7 divided into -200, or -28.57. Thus 5 is 28.57 percent smaller than 7.

So the percent change from 5 to 7 is \(\displaystyle\frac{7 – 5}{5}\cdot 100% = \frac{2}{5}\cdot 100% = 40%\), while the percent change from 7 to 5 is \(\displaystyle\frac{5 – 7}{7}\cdot 100% = -\frac{2}{7}\cdot 100% = -28.57%\).

It may seem paradoxical that if we start with 5 and add 40 percent, we get 7, but to get back to 5 we have to subtract only 28.57 percent.  The reason these percentages are different is that they are percentages of different amounts: 40 percent of 5, and 28.57 percent of 7 - both are equal to 2.

We’ll be coming back to this issue (increase and decrease being asymmetrical because of the different bases) next time.

Different kinds of percent change

But there’s another issue, which was brought up by a reader, Jerone, in 2003 (and edited into the same page):

The above explanation defines percent difference as ((q1-q2)/q2)*100, which is used in many calculations for percent error rather than percent difference. Percent difference is defined as (|q1-q2|/((1/2)*(q1+q2)))*100 for comparing values.

I am confused, as the terms percent difference and percent error are not consistent with what is expected in many classes.

There is both a contextual and a semantic issue here; technically, no one above used the exact term “percent difference”, and what Allistair was asking for (using the inaccurate term “difference in percent”) is indeed a percent change, equivalent to what Jerone calls a percent error (as we saw above under that title). The percent difference is a case we haven’t seen yet.

I replied:

The above is really about "percent change" or "percent increase," rather than "percent difference," since there is an "old" and a "new" value. In that case, you take the percentage of the old value.

When there is no directionality to the difference, so you can't distinguish an "old" value from a "new" value, you have to use the average of the two as the standard of comparison, as in your formula.

This is one of many cases where terms are used differently in various contexts, and what is a minor error of terminology in one question can lead to major confusion for people coming at it from a different direction.

“Percent difference” properly refers to a comparison between two numbers, neither of which takes priority as “original” or “correct”.

To sum up, when we are describing a change (increase or decrease) from one value to another, or the difference from a correct value to an estimate (percent error), the formula is $$\text{percent change} = \frac{\text{new } – \text{ old}}{\text{old}}\cdot 100%;$$ but when we just have two numbers, treated as equals, we use $$\text{percent difference} = \frac{a – b}{\frac{a + b}{2}}\cdot 100% = \frac{2(a – b)}{a + b}\cdot 100%.$$

A recent example

This is the context of the recent question, which came from George in March:

If I’m baking a clay brick in an oven and know that it shrinks by 25% as a result of this process, and I need the final size to be one meter, I’m looking for the best way to determine the initial size.

I’m thinking the percentage difference equation probably comes into play:

x% = (a-b) / ((a+b)/2)

So, if x=25 and b=1, what’s the equation we use to solve for a? Let me know.

Here George is using the percent difference formula, but he is really doing a percent decrease. I replied:

Your equation is for the percentage difference between two numbers that are not distinguished (e.g. as original and new, or true and measured, for example). See 

Percent Change, Increase, Difference.

For a percent change from a given value, you just add or subtract the percentage of that value. in your case, you have an (unknown) initial size x that is decreasing by 25%, so 25% of x is subtracted from x. That leaves

x – 0.25x = 1x – 0.25x = 0.75x

To reverse this, you can divide by 0.75. So if the final size is to be 1, we have to solve

0.75x = 1

so the solution is

x = 1/0.75 = 1.33… = 4/3.

In other words, the new size is 100% – 25% = 75% of the original, so the original is the new divided by 0.75. This is a division by 3/4, which is the same as multiplying by 4/3. The item has to be 1/3 larger to offset a decrease of 1/4.

We’ll be getting to this reversal process next time. But the question raises the interesting question: How do you reverse a percent difference, to find the “other” value when you know one value and the percent difference. Give it some thought!