Question

The demand function of a light rail transit system (LRTS) of a city is represented as \(Q = 43000 - 850P\), where \(Q\) is the ridership/day and \(P\) is the fare/ride. If the existing fare of Rs. 30 per ride is reduced to Rs. 25 per ride, then the consequent increased ridership of the LRTS (in percentage) is (rounded off to two decimal places).

Hint:

Find ridership at Rs. 30 and Rs. 25 using \(Q=43000-850P\), then take the percentage change over the original ridership.

Answer 1

Correct Answer: 24.29

Step 1: Read the slope of the demand line.
Write the demand function as $Q = 43000 - 850P$. Here the coefficient $850$ tells us how many riders are gained or lost for every rupee change in fare: $\frac{dQ}{dP} = -850$.

Step 2: Apply this rate to the actual fare cut.
The fare drops from Rs. 30 to Rs. 25, so $\Delta P = 25 - 30 = -5$.
Using the constant slope, the change in ridership is:
\[ \Delta Q = \left(\frac{dQ}{dP}\right) \times \Delta P = (-850)(-5) = 4250 \]
Ridership goes up by 4250 riders/day, since the fare fell.

Step 3: Get the base ridership to compare against.
At the original fare $P = 30$:
\[ Q_1 = 43000 - 850(30) = 17500 \]

Step 4: Express the change as a percentage of the base.
\[ \% change = \frac{4250}{17500} \times 100 = 24.2857\ldots\% \]
Because this is a straight line demand curve, the slope method and the plug-in method must agree, and they do.

Step 5: Round off and state the result.
To two decimal places, the ridership rises by 24.29%, which lies within the accepted range of 24.00% to 25.00%.
\[ \boxed{24.29\%} \]