First, let's break down the information given:
1. Purchase by Eddy:
- Eddy bought a bicycle at a discounted price of \$Y from Ranger Bicycles.
2. Sale to Friend:
- Eddy sold the bicycle to his friend at the initial marked price (let's call this \$M).
- By doing this, Eddy made a 20\% profit on his purchase price (\$Y).
3. Quantities to Compare:
- Quantity A: The discount percentage offered by Ranger Bicycles to Eddy when he bought the bicycle.
- Quantity B: 20\%

Our goal is to compare Quantity A and Quantity B to see which is greater, or if they're equal.
Defining the Variables
Let's define the variables clearly:
- Marked Price (Initial Price): \$M (the price before any discount)
- Purchase Price (Discounted Price): \$Y (the price Eddy paid after discount)
- Selling Price: Eddy sold it at the marked price, so \$M
- Profit: $20 \%$ on the purchase price (\$Y)

Step 1: Express Profit in Terms of Y
Eddy made a 20\% profit on his purchase price when he sold the bicycle for \$M.

Profit is calculated as:

$$
\(\text { Profit }=\text { Selling Price }- \text { Purchase Price }\)
$$


Given that the profit is 20\% of the purchase price:

$$
\(0.20 \times Y=M-Y\)
$$


Now, solve for $M$ :

$$
\(\begin{gathered}
0.20 Y=M-Y \\
M=Y+0.20 Y \\
M=1.20 Y
\end{gathered}\)
$$

So, the marked price $M$ is $\mathbf{1 . 2 0}$ times the purchase price $Y$.
Step 2: Determine the Discount Percentage
The discount percentage is the reduction from the marked price $M$ to the purchase price $Y$.
Discount is calculated as:

$$
\(\begin{aligned}
\text { Discount } & =M-Y \\
\text { Discount Percentage } & =\left(\frac{M-Y}{M}\right) \times 100
\end{aligned}\)
$$


We know $M=1.20 Y$, so:

$$
\(\begin{gathered}
\text { Discount }=1.20 Y-Y=0.20 Y \\
\text { Discount Percentage }=\left(\frac{0.20 Y}{1.20 Y}\right) \times 100 \\
\text { Discount Percentage }=\left(\frac{0.20}{1.20}\right) \times 100 \\
\text { Discount Percentage }=\left(\frac{1}{6}\right) \times 100 \\
\text { Discount Percentage } \approx 16 . \overline{6} \%
\end{gathered}\)
$$


Step 3: Compare Quantity A and Quantity B
- Quantity A: Discount percentage $\(\mathbf{= 1 6 . 6 7 \%}\)$
- Quantity B: 20\%

Clearly, $\(\mathbf{2 0 \%} \mathbf{> 1 6 . 6 7 \%}\)$, so Quantity B is greater than Quantity A.