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I have a very simple yet somehow confusing question regarding percentage increases/decrease. I am a bit confused about my initial approach for determining percentage difference between two values versus just being just given the easy formula. Without actually knowing the equation by heart this was my initial approach.

If I have an old value A = 350 and a new value of B = 371. What is the percent increase? So I approach this problem in this way

A/B = 0.943397 so it take 4 decimal places. I then take 1-0.943397 = 0.56603 x 100 = 5.6603 percent increase. Therefore (5.6603 * 1/100) * 371 = 21 and this checks out.

But if I use the New Value - Old Value / Old Value I get 6 percent. Where is this error coming from?

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2 Answers 2

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You fell in the trap. A relative change can be computed wrt the old value or the new value, you need to be specific about that.

$$\frac{371-350}{350}=\frac{371}{350}-1=6.00\%$$

$$\frac{371-350}{371}=1-\frac{350}{371}=5.66\%$$

For a similar reason,

$$\begin{align}100+10\%&=110,\\110-10\%&=99\ !?!\end{align}$$

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  • $\begingroup$ So what is the correct Increase. So if I saw increase It would have to be with respect to 350. then the true answer is 6 percent. $\endgroup$
    – Julien Popa-Liesz
    Jan 22, 2020 at 9:32
  • $\begingroup$ @JulienPopa-Liesz: I can't guess what the teacher told you. $\endgroup$
    – user65203
    Jan 22, 2020 at 9:33
  • $\begingroup$ So this is a wording comprehension issue then. I still feel like this doesn't clarify why I fell into the trap in the first place.Simply just determining what percent increase is 350 to 371. I derived the approach as such must imply missing knowledge. Could there be something I could read up on or understand or is this something that simple that it should be obvious and just walk away with the relative change equation in my head as memory. $\endgroup$
    – Julien Popa-Liesz
    Jan 22, 2020 at 9:37
  • $\begingroup$ Thanks by the way. $\endgroup$
    – Julien Popa-Liesz
    Jan 22, 2020 at 9:41
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    $\begingroup$ I see what you mean now. Thanks very much sir for the insanely quick response. $\endgroup$
    – Julien Popa-Liesz
    Jan 22, 2020 at 9:48
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Your second computation is correct (6%).

A “percent increase” compares the change in value (from 350 to 371) to the starting value (350).

Your first computation tells what percent decrease would then have to occur to recover the original value from the increased value. It would be a second change process (from 371 to 350), so its “starting” value would be the increased value (371), so would appear in the denominator.

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  • $\begingroup$ That makes sense, because if you have 100 squares and then you say have 109 percent. Then it means by convention you start with 100 squares as the whole number for the denominator. Then the numerator is now increased which is a rate of change so 9 squares therefore 9 percent increase. I can't believe I didn't see that before. $\endgroup$
    – Julien Popa-Liesz
    Jan 22, 2020 at 9:56

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