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I would think the values would be the same but the increase is $638.1$ an $138$ dollar increase and the decrease is $386.89$ an $113.10$ decrease.

Can someone explain?

Teddy38
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Warriors Live Forever
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  • Your title is clearly wrong. As $1.05\gt 0.95$ you should expect that $1.05^5 \gt 0.95^5$ – Ross Millikan Nov 08 '17 at 16:08
  • Let's replace $5$ with $1$ to begin. Would you expect $1.05^1$ equal $0.95^1$? And if they are not equal, why should $500\cdot 1.05^1$ equal $500\cdot 0.95^1$...? – CiaPan Nov 08 '17 at 16:13

2 Answers2

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It is a general phenomenon. If you increase a number by some percentage then decrease the result by the same percentage, you get a smaller number. That is because the reduction applies to a larger number than the increase. If $x$ is the increase, you have $(1+x)(1-x)=1-x^2 \lt 1$

Ross Millikan
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  • This answer seems a bit off-topic — OP decreases the same base $500$, not a result of the previous increase... – CiaPan Nov 08 '17 at 16:14
  • Thank you for the clarification Ross. – Warriors Live Forever Nov 08 '17 at 16:28
  • @CiaPan: but after the first reduction, the base is smaller. – Ross Millikan Nov 08 '17 at 17:17
  • @RossMillikan I disagree. The question, specified in the title actually, is: 'why is $500(1.05)^5$ not equal to $500(0.95)^5$?' With clearly the same base of $500$, increased $5$ times by $5%$ in the first case and decreased 5 times by $5%$ in the second case. You just have guessed what OP meant (and you apparently did it right!), but still you answer your guess, not the question actually asked. – CiaPan Nov 08 '17 at 21:18
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An increase is when something gets bigger. A decrease is when something gets smaller. The different names indicate those are different effects, hence they give different results. Why would you expect the results equal?

As a result of an increase of an initial value of $500$ you get more than $500$. On the other hand, a decrease of the same initial value of $500$ results in less than $500$.

Possibly you meant applying some percentage-defined increase to a given initial value $500$ and then applying a decrease to a result of the previous growth, not to the same initial $500$...?
If so, you have two steps:

  1. Initial_Value = $500$ increases, say, by $10\%$, which is $500\cdot\frac{10}{100} = 50$.
    As a result you get Increased_Value = $500 + 50 = 550$.
  2. Then the Increased_Value = $550$ decreases by apparently "the same" $10\%$. But this time the percentage applies to the Increased_Value, which is $550$, so a decrease is $550\cdot \frac{10}{100}=55$.
    As a result we get the Final_Value = $550 - 55 = 495 < 500$.

The crucial part is that we apply "the same" percentage to different values: when increasing, we multiply the initial, not-yet grown value by $0.1$, but to decrease we multiply the grown value by $0.1$. As a result the actual increase is smaller than the actual decrease ($50$ vs. $55$) hence the net result of $+p\%$ followed by $-p\%$ is negative, as Ross Millikan shows in more general way in the answer.

CiaPan
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