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I'm a total noob and need help with this, I don't understand it at all. I need to find out the percentage change using the 2 numbers for each question, some have increased some have decreased.

Calculate the percentage (%) change in mass of carbon in the

atmospheric 662 becomes 850

soil and detritus 1700 becomes 1460

plant biomass 600 becomes 530

surface ocean reservoirs 918 becomes 1000

hardmath
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Will
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  • I think those should be arrows instead of minus signs, no? –  Mar 07 '15 at 13:32
  • Do you know how much $10%$ of $600$ is? So what would a $10%$ increase be from $600$? What would a $10%$ decrease be? – AlexR Mar 07 '15 at 13:36
  • sorry my mistake, the lines aren't minuses, i was trying to separate the two numbers – Will Mar 07 '15 at 14:16
  • Er, actually '>' is wrong as well; it means 'greater than'. I know there's no arrow on the keyboard so there's no simple way to do it. I'm just trying to avoid confusion. –  Mar 07 '15 at 14:44

1 Answers1

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A change in percentage depends on the ratio $\frac{\text{after}}{\text{before}}$. The concrete formula would be $$\text{change in percent} = 100 \cdot \Big(\frac{\text{after}-\text{before}}{\text{before}}\Big)$$ Where $\text{after}-\text{before}$ is the change (with sign). This can be rewritten to $$\text{change in percent} = 100 \cdot \Big(\frac{\text{after}}{\text{before}} - 1\Big)$$ For example starting with $100$ and ending with $90$ you get a change percentage of $100\cdot (\frac{90}{100}-1) = 100\cdot (0.9 - 1) = 100\cdot(-0.1) = -10$.

A last example: Start at $123$ (before) and end at $234$ (after): $$100\cdot(\frac{234}{123} - 1) \approx 100\cdot(1.902 - 1) = 100\cdot 0.902 = 90.2 $$

Working out the formula if you know how to compute $x$ percent of a number: We will write the percentage directly as a number here. $x = 0.1 = 10\%$ for example.
To compute $x$ percent of a number $n$, we just compute $n\cdot x$. If we want to compute a new number $m$ wich is the old number increased by $x$ percent, this is then $$m = n + n\cdot x = n\cdot(1+x)\tag1$$ In the same way we could decrease using $m=n-n\cdot x = n\cdot(1-x)$. If we view $x$ as a signed percentage, for example $x = -0.1$ for a $10\%$ decrease, we can use the formula $(1)$ in both cases. Now we want to find $x$ given $n$ and $m$ in the assignment:

$$m = n\cdot (1+x)\\ \Leftrightarrow \frac mn = 1+x\\ \Leftrightarrow x = \frac mn - 1$$

Now since we want $x$ as a percentage number instead, we have to multiply the result by $100$ again: $$x[\%] = 100 \cdot x = 100\cdot \Big( \frac mn - 1 \Big )$$ Above $n$ was labelled $\text{before}$ and $m$ was labelled $\text{after}$.

AlexR
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  • I know this is gonna sound stupid but is there a simpler way to understand this, or could someone figure out the answers cause im going crazy trying to figure this out – Will Mar 07 '15 at 14:22
  • @Will I'm afraid this is no one-stop shop for homework. If you're trying to learn something, show us what you tried so far and what you don't understand. – AlexR Mar 07 '15 at 14:23
  • I know it isn't I just really don't understand it and im stuck on my uni assignment. All I have so far are the numbers written down on the page. I know how to work out 10% of a number, I just don't know how to work out percentage changes in two numbers or numbers higher than 100 – Will Mar 07 '15 at 14:29
  • Can you show me an example with a two high numbers, so I can understand how it works? – Will Mar 07 '15 at 14:30
  • @Will I have added an example with $123$ and $234$ five minutes ago. Do you need even higher numbers to see where to plug in what? – AlexR Mar 07 '15 at 14:32
  • Sorry I didn't see it. Just so i'm clear is the 90.2 at the end what the percentage change would be? – Will Mar 07 '15 at 14:35
  • @Will Right. From $123$ to $234$ is an increase by (approximately) $90.2%$. – AlexR Mar 07 '15 at 14:40