Which decimal can be multiplied by the result of $(x \times 1.15)$ to return $15$% of $x$?

Question

Which decimal can be multiplied by the result of $(x \times 1.15)$ to return $15$% of $x$?

For example, when I want to add $15$% sales tax to an item before tax, I would multiply by $1.15$.

$100 \times 1.15 = 115$

I was trying to figure out which decimal I can multiply against the result $(115)$ that would essentially give me the same result of $[(x \times 1.15) - x]$.

Through trial and error I've come up with the decimal $0.13044$ which approximately gave me what I want. Example:

$(100 \times 1.15) - 100 = 15$

vs

$4(100\times1.15) \times 0.13044 = 15.0006$

Is there a way to calculate the exact or close to exact decimal that I am looking for, that when multiplied by a number plus $15$% would return the value of the $15$%.

This is purely out of curiosity rather than for any practical purpose.

Answer 1

You want to have your decimal $q$ such that $$q \times (1.15 \times x ) = 1.15 \times x - x = 0.15 \times x$$ so $$q \times 1.15 \times x = 0.15 \times x$$ for all $x$ Finally, you have $$q = \frac{0.15}{1.15} \simeq 0.1304347826$$

In general, if you increase by $p$ (i.e. $100p$ %), $$q = \frac{p}{p+1}$$

Answer 2

Let $y = 0.15 x$, and let $z = 1.15x$.

Then, you want to uncover the relationship between $z$ and $y$.

$$\frac{z}{y} = \frac{1.15 x}{0.15 x} = \frac{1.15}{0.15} = \frac{23}{3}.$$

So, $y = \frac{3}{23}z$, so we see that

$$\frac{3}{23} 1.15 x = 0.15x.$$