Percentages of $\$100,000$

Question

$\$6,200$ is $6.2\%$ of $\$100,000$. That leaves $\$93,800$ as $93.8\%$ of that $\$100,000$. But when I take $\$93,800$. and multiply it by $6.2\%$, I get $\$99,615.60$ instead of $\$100,000$. Why is this? And how do I get back to my $\$100,000$. Do I need to multiply the $\$93,800$ by more? Why?

Answer 1

You're thinking about it wrong. Sure, $\$93,800$ is $93.8\%$ of $\$100,000$. However, it doesn't mean that $106.2\%$ of $\$93,800$ is $\$100,000$ and as you saw, that's not true. You have to set it up like so:

$$\$93,800=93.8\%\text{ of }\$100,000 \\ \implies \$93,800=(0.938)(\$100,000)$$

Now the question you're asking is - what times $\$93,800$ will give me $\$100,000$? In other words, what is $c$ in this equation:

$$c \times \$93,800=\$100,000$$

Using basic algebra, if you re-arrange the formula shown at the top, you'll see that:

$$c=\frac1{0.938}$$

Giving you an approximate answer of:

$$c=1\frac{31}{469}\approx 1.066098$$

And this makes sense. To get to $\$100,000$ from $\$93,800$, you'd need to add $100,000-93800=\$6,200$. What percentage of $\$93,800$ is $\$6,200$? Well:

$$\frac{\$6,200}{\$93,800}=\frac{31}{469}\approx 6.6098\%$$

Meaning that you'd need to take $100\%$ of $\$93,800$ in addition to $\approx6.6098\%$ of it to get to $\$100,000$.

Answer 2

Instead of taking $6.2\%$ of $\$93,800$, you should be taking $6.2\%$ of $\$100,000$.

Think of it as taking a percentage of the total amount, where the total amount in this case is $\$100,000$.

Hence, \[ 0.062\times100000=6200 \] \[ 0.938\times100000=93800 \] And \[ 6200+93800=100000 \] No algebra is needed, just an understanding of what your taking a percentage of.

Answer 3

You need to multiply with $\frac1{0.938}\approx 1.0660980810234541577825159914712153518\ldots$, i.e. you have to add $6.6098\ldots \,\%$. For small percentages, you assumed method almost works (i.e is a good approximation), but the larger the percentagem the bigger the problem (i.e. the diffference to the correct method) becomes. For example, if you subtract $50\,\%$ (cut in half), you have to add $100\,\%$ (double) to get back to the original; and if you subtract $100\,\%$, you get zero and can add percentages as high as you want and wonÄt succeed ...