Calculating Theatre Earnings Change
The total earnings of a theatre depend on the entry fee (price) and the number of people (quantity). The formula is:
$Total Earnings = Entry Fee × Number of People$
We need to find the percentage change in total earnings based on the given changes in entry fee and the number of people.
Analyzing Percentage Changes
Let the initial entry fee be $P$ and the initial number of people be $N$.
- Initial Earnings = $N \times P$
The entry fee is reduced by 25%. The new entry fee is:
- New Entry Fee = $P - (25\% \text{ of } P) = P - 0.25P = 0.75P$
The number of people increases by 30%. The new number of people is:
- New Number of People = $N + (30\% \text{ of } N) = N + 0.30N = 1.30N$
The new total earnings are:
- New Earnings = $(New Entry Fee) × (New Number of People)$
- New Earnings = $(0.75P) × (1.30N)$
- New Earnings = $(0.75 \times 1.30) \times (P \times N)$
- New Earnings = $0.975 \times (P \times N)$
Determining the Change in Earnings
To find the percentage change in earnings, we compare the new earnings to the initial earnings:
Percentage Change = $\frac{\text{New Earnings} - \text{Initial Earnings}}{\text{Initial Earnings}} \times 100\%$
Percentage Change = $\frac{0.975(P \times N) - (P \times N)}{P \times N} \times 100\%$
Percentage Change = $\frac{(0.975 - 1) \times (P \times N)}{P \times N} \times 100\%$
Percentage Change = $(0.975 - 1) \times 100\%$
Percentage Change = $(-0.025) \times 100\%$
Percentage Change = $-2.5\%$
A negative percentage change indicates a decrease.
Alternative Calculation: Net Percentage Change Formula
This problem can also be solved using the net percentage change formula for two consecutive changes:
$Net Change = $x + y + \frac{xy}{100}$
Where:
- $x$ = Percentage change in entry fee = $-25\%$
- $y$ = Percentage change in the number of people = $+30\%$
Plugging in the values:
Net Change = $-25 + 30 + \frac{(-25)(30)}{100}$
Net Change = $5 + \frac{-750}{100}$
Net Change = $5 - 7.5$
Net Change = $-2.5\%$
Therefore, there is a 2.5% decrease in the total earnings.
Important Questions from Percentage (Notes)
- 20% of a number is more than 20% of 620 by 210. The number is:
- Girish scored 572 marks in an examination and was 30 marks short of 28% of maximum marks. In the same examination, his friend scored 473 marks. What is the percentage of marks scored by his friend?
- Anubhav spent 14% of his income on electricity bills, 28% on rent and 18% on shopping. If $\frac{1}{4}$ th of the remaining amount is ₹5120, how much did he spend on electricity bills ?
- The total population of a town is 50,000. The number of males and females increases by 10% and 15% respectively and consequently the population of the town becomes 56,000. What was the number of males in the town?
- Gokul donates $17\%$ of his savings to old age home, $17\%$ of the savings to orphanage and $17\%$ of his savings to foundations for medical help. The remaining amount of Rs.$9800$ of savings is been deposited in bank. Find the salary of Gokul, if $50\%$ of the salary is his savings amount. (In Rs.)
