Question

In the Regional Science Centre, Lucknow the ra increased by \(50 \%\) to increase the revenue, but simultaneously \(20 \%\) of the visitors decreased. What is percentage change in the revenue of Regional Science Centre. If it is known that the centre collects the revenue only from the visitors and it has no other financial supports : (a) \(+20 \%\) (b) \(-25 \%\) (c) \(+30 \%\) (d) can't be determined

Short answer

Answer: The percentage change in the revenue of the Regional Science Centre is +10%.

Step-by-step solution

Understand the given information

We are given that the ticket prices (ra) increase by \(50 \%\) and the number of visitors decreases by \(20 \%\). Let's denote the initial ticket price as T and the initial number of visitors as V. Therefore, the new ticket price will be \(T(1+50/100)= 1.5T\) and the new number of visitors will be \(V(1-20/100) = 0.8V\). The initial revenue can be represented as \(R_i = T \times V\).

Calculate the revenue after the changes

Now, we need to determine the new revenue after the ticket price increase and visitor decrease. To do this, we will multiply the new ticket price and the new number of visitors, i.e., \(R_f = 1.5T \times 0.8V\).

Calculate the percentage change in revenue

We need to find the percentage change in the revenue. To do this, we must calculate the difference in the initial and final revenue and divide it by the initial revenue. The difference in revenue is \(R_f - R_i = (1.5T \times 0.8V) - (T \times V)\). Then, the percentage change in revenue can be calculated as follows: $$ \text{Percentage Change} = \frac{(1.5T \times 0.8V) - (T \times V)}{T \times V} \times 100 $$

Simplify the expression and find the answer

Now let's plug in the values of new ticket price and new number of visitors and simplify the expression: $$ \text{Percentage Change} = \frac{(1.5T \times 0.8V) - (T \times V)}{T \times V} \times 100 $$ $$ = \frac{(0.2T \times 1.5V) - (0.2T \times V)}{T \times V} \times 100 $$ $$ = \frac{(0.3T \times V) - (0.2T \times V)}{T \times V} \times 100 $$ $$ = \frac{0.1T \times V}{T \times V} \times 100 $$ $$ = \frac{0.1}{1} \times 100 = 10\% $$ So, the percentage change in the revenue of the Regional Science Centre is \(+10\%\), which is not present in the given options. Based on the given information, there may be a mistake in the options provided.

Key concepts

Revenue Calculation

Revenue calculation is a critical fiscal concept for any business or service provider, understanding of which can help manage the financial health. In the broadest sense, revenue is the total income generated from normal business operations and other activities. When it comes to visitor-driven organizations like the Regional Science Centre in Lucknow, the equation is quite simple: the revenue (\(R\)) is a product of the number of visitors (\(V\)) and the rate per visitor or ticket price (\(T\)). So, the formula for revenue is given as:
\[R = T \times V\]
Change in either factor leads to a change in the revenue. In our example, we analyzed two variations: the increase in ticket price and decrease in the number of visitors. Multiplying the new ticket price by the new visitor count gave us the revised revenue. This practical example shows that even when a price increase seems advantageous, a corresponding decrease in customers might negate the expected financial boost.

Percent Increase and Decrease

Understanding percent increase and decrease is vital for numerical literacy and everyday decision-making. This concept deals with how much a numerical value has increased or decreased relative to its original amount, expressed as a percentage. For example, a \(\text{50\text{%}}\) increase on the original ticket price means the new ticket price is \(\text{50\text{%}}\) more than the original, which can be calculated as:
\[T_{\text{new}} = T_{\text{original}} \times (1 + \frac{50}{100}) = 1.5 \times T_{\text{original}}\]
Conversely, a reduction in visitors by \(\text{20\text{%}}\) means the number of visitors is now \(\text{20\text{%}}\) less than the original:
\[V_{\text{new}} = V_{\text{original}} \times (1 - \frac{20}{100}) = 0.8 \times V_{\text{original}}\]
To find the overall percentage change in revenue, we calculate the difference between the new and original revenues and then divide by the original revenue, multiplying by \(\text{100\text{%}}\) to express it as a percentage. This process exemplifies how we perceive changes in rates or counts not only in absolute numbers but in terms of percentage points, facilitating a more intuitive understanding of their impact on financial outcomes.

Quantitative Aptitude

Quantitative aptitude encompasses the ability to handle numbers and perform various arithmetic operations efficiently. It is a skill highly valued in academic environments and competitive scenarios where numerical reasoning is required. The exercise from the Regional Science Centre exhibits this aptitude by requiring multiple mathematical operations—percentage calculation, multiplication, and algebraic simplification—to derive a solution.
As part of developing quantitative aptitude, it's important to break down complex problems into smaller, manageable steps, as was done in the step-by-step solution provided. This approach not only simplifies the problem-solving process but also aids in error identification, ensuring a higher degree of accuracy in the answer. Moreover, it's crucial to check whether the final answer makes sense within the context of the problem, and if not, re-assess the steps taken or the information given, as with the unexpected result that conflicted with the given options in the exercise.