The base exchange rate of a currency $X$ with respect to a currency $Y$ is the number of units of currency $Y$ which is equivalent in value to one unit of currency $X$. Currency exchange outlets buy currency at buying exchange rates that are lower than base exchange rates, and sell currency at selling exchange rates that are higher than base exchange rates.
A currency exchange outlet uses the local currency $L$ to buy and sell three international currencies $A$, $B$, and $C$, but does not exchange one international currency directly with another.
The base exchange rates of $A$, $B$ and $C$ with respect to $L$ are in the ratio $100:120:1$. The buying exchange rates of each of $A$, $B$, and $C$ with respect to $L$ are $5$% below the corresponding base exchange rates, and their selling exchange rates are $10$% above their corresponding base exchange rates.
The following facts are known about the outlet on a particular day:
- The amount of $L$ used by the outlet to buy $C$ equals the amount of $L$ it received by selling $C$.
- The amounts of $L$ used by the outlet to buy $A$ and $B$ are in the ratio $5:3$.
- The amounts of $L$ the outlet received from the sales of $A$ and $B$ are in the ratio $5:9$.
- The outlet received $88000$ units of $L$ by selling $A$ during the day.
- The outlet started the day with some amount of $L$, $2500$ units of $A$, $4800$ units of $B$, and $48000$ units of $C$.
- The outlet ended the day with some amount of $L$, $3300$ units of $A$, $4800$ units of $B$, and $51000$ units of $C$.
What was the base exchange rate of currency $B$ with respect to currency $L$ on that day?