Cannot reconcile simple algebra problem

Question

I'm having trouble with a reconciling two statements.

Suppose a merchant has a price he normally charges for goods ($P_{original}$).

Recently he's decided to transition onto Amazon which charges 11.2% of sale, so said merchant must raise his price to compensate exactly for these new fees. We will call his new price $P_{new}$.

Intuitively, we can look at this as: $$\begin{align} P_{new}=1.112 \times P_{original} \tag{1} \end{align} $$ Conversely, we cant think of this as $P_{original}$ being $11.2\%$ less than $P_{new}$: $$\begin{align} P_{original} = 0.888\times P_{new} \tag{2} \end{align} $$

Substituting (1) into (2) we get:

$$\begin{align} P_{original} & =0.888 \times (1.112 \cdot P_{original}) \\ &=0.987456 \cdot P_{original}\\ \tag{3} \end{align}$$

Since I am using exact values and no rounding, (3) cannot be true. $P_{original} \ne 0.987456 \cdot P_{original}$. So I have a contradiction, despite (1) and (2) being intuitive, straightforward assertions. Can somebody help me to reconcile this contradiction?

Answer 1

The problem is that you consider $P_{original}=0.888\times P_{new}$. This $0.888$ comes from making $1-0.112$ which is wrong.The $11,2$% must be calculated over the original price not over the new price.

Since $P_{new}=1.112\times P_{original}$ we have that $P_{original}=\frac{1}{0.112}\times P_{new}$

$P_{original}=\frac{1}{0.112}\times P_{new}=\frac{1}{0.112}\times 1.112\times P_{original}=P_{original}$

Answer 2

The value of $P_{\text{new}}$ will have to be such that $(1-11.2\%)P_{\text{new}}=P_{\text{original}}$. In other words,$$P_{\text{new}}=\frac{P_{\text{original}}}{0.888}\approx1.126P_{\text{original}}.$$