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May 25th, 2017, 05:17 PM   #1
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Why don't individual percentages add up?

My problem is simple.

I have the following data
2016 2017 Change in %
Cars 300 400 [(400-300)/400]*100 =0.333
Bikes 400 500 [(500-400)/400]*100 =0.25

Total 700 900 [(900-700)/700]*100 =0.2857143

Why didn't the Cars + Bikes Change in % = Total Change in %?

Last edited by skipjack; June 10th, 2017 at 07:03 PM.
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May 26th, 2017, 02:39 AM   #2
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Because a percentage by itself is meaningless! A percentage only has meaning when it is associated with a "base". 10% of 100 is 10 but 10% of 1000 is 100. Saying "10%" only doesn't tell you anything.

Here, a change from 300 to 400 is (400- 300)/300 is 100/300= 1/3 or 33.3%.
(The "base" is 300, not 400, because it is 300 that has "changed". It changed to 400. When you are calculating "percentage change" you always take the original amount as "base".)

A change from 400 to 500 is (500- 400)/400= 100/400= 1/4 or 25%.

Counting both cars and bicycles, we had, initially 300+ 400= 700 which changed to 400+ 500= 900. That is a percentage change of (900- 700)/700= 200/700= 2/7= 28.5%. Notice that this is between 33.3% and 25%. It is not a sum, it is a "weighted average".
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June 6th, 2017, 03:51 AM   #3
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Quote:
Originally Posted by idlejam View Post
My problem is simple.

I have the following data
2016 2017 Change in %
Cars 300 400 [(400-300)/400]*100 =0.333
Bikes 400 500 [(500-400)/400]*100 =0.25

Total 700 900 [(900-700)/700]*100 =0.2857143

Why didn't the Cars + Bikes Change in % = Total Change in %?
Yep, Country Boy is right. It matters what the percentage applies to.


What you have found is that

percentage change in cars + percentage change in bikes is not equal to the percentage change in cars and bikes

This is generally true... you can't just add percentages together based on different quantities. However, you can add percentages together if they describe changes to the same thing at the same time.

The bits in bold above are super important to make sure you don't make any mistakes.

Let's illustrate this in your problem, but we need to be careful...

-----

Let's consider a dealership that has cars and bikes. We start with 300 cars and 400 bikes for a total of 700 vehicles.

Let's define two events:

Event 1: 100 cars are added to the total pool of vehicles at a particular time (say, 3pm today)
Event 2. 100 bikes are added to the total pool of vehicles at the same time as event 1.

Okay... so what is the percentage increase in cars and bikes for events 1 and 2?

Event 1: % increase is $\displaystyle (100/700) \times 100$ = 14.286%

Event 2: % increase is $\displaystyle (100/700) \times 100$ = 14.286%

So the total percentage increase for 14.286% + 14.286% = 28.571%. This is the same figure you derived above.

However... notice that in order for me to get the correct answer, I had to make sure that for event 2, the total pool of vehicles was 700, not 800. The events have to occur at the same time to add up, so you shouldn't include the 100 cars that were added to the pool in event 1 when working out your answer...

If event 2 happened at some later time, then the total pool of vehicles becomes 800 and you'll calculate a different percentage. Consequently, you can't add the percentages together anymore.

I hope this helps

Last edited by skipjack; June 10th, 2017 at 07:04 PM.
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June 6th, 2017, 07:38 AM   #4
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See also this article.
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June 10th, 2017, 02:07 PM   #5
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Quote:
Originally Posted by idlejam View Post
My problem is simple.

I have the following data
2016 2017 Change in %
Cars 300 400 [(400-300)/400]*100 =0.333
Bikes 400 500 [(500-400)/400]*100 =0.25

Total 700 900 [(900-700)/700]*100 =0.2857143

Why didn't the Cars + Bikes Change in % = Total Change in %?

% change in cars = $\displaystyle \frac{400-300}{300}*100$.................1

% change in bikes =$\displaystyle \frac{500-400}{400}*100$.................2

Total % change in cars and bikes add (1) and (2) and we have:

$\displaystyle 100*[\frac{400-300}{300}+\frac{500-400}{400}]$

Now if you add the numerators and the denominators of the above fractions,we get:

$\displaystyle 100*\frac{(400-300)+(500-400)}{300+400}$

Which is equal to: $\displaystyle 100*\frac {900-700}{700}$

Which is exactly your last calculation in finding total % in cars and bikes.

But when you add two different fractions YOU CANNOT ADD NUMERATORS AND DENOMINATORS

Here is your basic mistake.

When we add: $\displaystyle \frac{a}{b}+ \frac{c}{d}$ this is not equal to:

$\displaystyle \frac {a+c}{b+d}$

but to $\displaystyle \frac{ad+bc}{bd}$................................3

What you actually did by adding fractions the way you did is to violate the whole axiomatic system of the real numbers.

Because addition of fractions is a theorem resulting from the axioms of the real numbers and the definition:

$\displaystyle \frac{A}{B}= A*\frac{1}{B}$.

Try to prove (3) and perhaps you will see what I mean.

Last edited by skipjack; June 10th, 2017 at 07:04 PM.
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