In a city, 12% households earn less than ₹ 30, 000 per year, 6% households earn more than ₹ 2, 00, 000 per year, 22% households earn more than ₹ 1, 00, 000 per year and 990 house holds earn between ₹ 30, 000 and ₹ 1, 00, 000 per year. How many households earn between ₹ 1, 00, 000 and ₹ 2, 00, 000 per year?
UPSC-2017
The problem involves calculating the number of households within a specific income bracket based on given percentages and the number of households in another income bracket. We will use percentages to determine the proportion of households in various income groups and then use the absolute number of households in one group to find the total number of households, and subsequently, the number in the desired group.
Let the total number of households in the city be \(H\).
We are given the following percentages:
From the last two points, we can deduce the percentage of households earning between ₹ 1,00,000 and ₹ 2,00,000 per year.
Percentage of households earning more than ₹ 1,00,000 = Percentage of households earning between ₹ 1,00,000 and ₹ 2,00,000 + Percentage of households earning more than ₹ 2,00,000.
Let \(P_{100-200}\) be the percentage of households earning between ₹ 1,00,000 and ₹ 2,00,000.
\[ 22\% = P_{100-200} + 6\% \]\[ P_{100-200} = 22\% - 6\% = 16\% \]Now, let's find the percentage of households earning between ₹ 30,000 and ₹ 1,00,000 per year.
Total percentage of households is 100%.
Percentage of households earning less than ₹ 30,000 = 12%.
Percentage of households earning more than ₹ 1,00,000 = 22%.
The percentage of households earning between ₹ 30,000 and ₹ 1,00,000 can be found by subtracting the percentages of the other two groups from the total percentage (100%) or by considering the complement.
Let \(P_{30-100}\) be the percentage of households earning between ₹ 30,000 and ₹ 1,00,000.
The sum of all disjoint income brackets must be 100%:
Percentage (less than ₹ 30,000) + Percentage (₹ 30,000 to ₹ 1,00,000) + Percentage (₹ 1,00,000 to ₹ 2,00,000) + Percentage (more than ₹ 2,00,000) = 100%
\[ 12\% + P_{30-100} + 16\% + 6\% = 100\% \]\[ 34\% + P_{30-100} = 100\% \]\[ P_{30-100} = 100\% - 34\% = 66\% \]We are given that 990 households earn between ₹ 30,000 and ₹ 1,00,000 per year. This corresponds to \(P_{30-100} = 66\%\) of the total households.
So, \(66\%\) of \(H = 990\).
\[ 0.66 \times H = 990 \]\[ H = \frac{990}{0.66} = \frac{990}{\frac{66}{100}} = \frac{990 \times 100}{66} \]\[ H = \frac{15 \times 66 \times 100}{66} = 15 \times 100 = 1500 \]The total number of households is 1500.
We need to find the number of households earning between ₹ 1,00,000 and ₹ 2,00,000 per year, which we found to be 16% of the total households.
Number of households = \(16\%\) of \(H\)
\[ \text{Number of households} = 0.16 \times 1500 \]\[ \text{Number of households} = 16 \times 15 \]\[ \text{Number of households} = 240 \]B) 240
A student has to get 40% marks to pass in an examination. Suppose he gets 30 marks and fails by 30 marks, then what are the maximum marks in the examination?
UPSC-2018
The problem provides information about a student's score, the marks by which they failed, and the passing percentage. We need to find the total maximum marks for the examination. We can determine the passing marks first, and then use the passing percentage to calculate the maximum marks.
The student scored 30 marks and failed by 30 marks. This means that to pass, the student needed 30 more marks than what they obtained.
\[ \text{Passing Marks} = \text{Marks Obtained} + \text{Marks to Pass} \\ \text{Passing Marks} = 30 + 30 \\ \text{Passing Marks} = 60 \]We are given that 40% marks are required to pass the examination. So, 60 marks represent 40% of the total maximum marks.
Let the maximum marks be \(X\).
\[ 40\% \text{ of } X = 60 \]Convert the percentage to a decimal or fraction:
\[ \frac{40}{100} \times X = 60 \]Simplify the equation and solve for \(X\).
\[ \frac{2}{5} \times X = 60 \\ X = 60 \times \frac{5}{2} \\ X = 30 \times 5 \\ X = 150 \]C) The maximum marks in the examination are 150.
A shopkeeper sells an article at ₹ 40 and gets X% profit. However, when he sells it at ₹ 20, he faces same percentage of loss. What is the original cost of the article?
UPSC-2018
This problem involves profit and loss percentages. We are given two selling prices and the condition that the percentage of profit in one case is equal to the percentage of loss in the other. We need to find the original cost price of the article. We can set up equations based on the definitions of profit and loss percentage and solve for the cost price.
Let the original cost of the article be \(C\).
When the article is sold at ₹ 40, the shopkeeper gets \(X\%\) profit. The selling price (\(SP_1\)) is ₹ 40.
The profit percentage is given by:
\[ X = \frac{SP_1 - C}{C} \times 100 \] \[ X = \frac{40 - C}{C} \times 100 \quad \text{(Equation 1)} \]When the article is sold at ₹ 20, the shopkeeper faces the same percentage of loss (\(X\%\)). The selling price (\(SP_2\)) is ₹ 20.
The loss percentage is given by:
\[ X = \frac{C - SP_2}{C} \times 100 \] \[ X = \frac{C - 20}{C} \times 100 \quad \text{(Equation 2)} \]Since the percentage of profit and loss is the same (\(X\%\)), we can equate Equation 1 and Equation 2:
\[ \frac{40 - C}{C} \times 100 = \frac{C - 20}{C} \times 100 \]We can cancel out \(100\) and \(C\) from both sides (assuming \(C \neq 0\), which it must be for a cost price):
\[ 40 - C = C - 20 \]Now, solve for \(C\):
\[ 40 + 20 = C + C \] \[ 60 = 2C \] \[ C = \frac{60}{2} \] \[ C = 30 \]So, the original cost of the article is ₹ 30.
Correct Option: C) ₹ 30
If the price of an article is decreased by 20% and then the new price is increased by 25%, then what is the net change in the price?
UPSC-2021
0%
5% increase
5% decrease
Cannot be determined insufficient data
To find the net change in price, we can assume an initial price for the article. Then, we apply the percentage decrease and subsequent percentage increase sequentially to find the final price. Finally, we compare the final price with the initial price to determine the net percentage change.
Let the original price of the article be \(P = 100\).
Decrease amount \( = 20\% \text{ of } 100 = \frac{20}{100} \times 100 = 20 \).
New price after decrease \( = P - 20 = 100 - 20 = 80 \).
The new price is \(80\).
Increase amount \[= 25\% \text{ of } 80 = \frac{25}{100} \times 80 = \frac{1}{4} \times 80 = 20\].
Final price after increase \( = 80 + 20 = 100 \).
Initial price \( = 100 \).
Final price \( = 100 \).
Since the final price is the same as the initial price, there is no net change.
Net change \[= \frac{\text{Final Price} - \text{Initial Price}}{\text{Initial Price}} \times 100\% = \frac{100 - 100}{100} \times 100\% = 0\%\].
A 20% decrease means the price becomes \(100\% - 20\% = 80\%\) of the original. The factor is \(1 - 0.20 = 0.80\).
A 25% increase means the price becomes \(100\% + 25\% = 125\%\) of the previous price. The factor is \(1 + 0.25 = 1.25\).
Net factor \( = (\text{Factor for decrease}) \times (\text{Factor for increase}) = 0.80 \times 1.25 \).
\[ 0.80 \times 1.25 = \frac{80}{100} \times \frac{125}{100} = \frac{4}{5} \times \frac{5}{4} = 1 \]A net factor of \(1\) means the final price is \(1\) times the original price, indicating no change.
Percentage change \( = (\text{Net factor} - 1) \times 100\% = (1 - 1) \times 100\% = 0\% \).
Correct Option: A) 0%
When a runner was crossing the 12 km mark, she was informed that she had completed only 80% of the race. How many kilometres was the runner supposed to run in this event?
UPSC-2019
This problem involves understanding percentages. We are given the distance covered and the percentage of the race completed. We need to find the total length of the race. We can set up a simple equation where the given distance represents 80% of the total distance.
B) 15
As calculated, if 12 km represents 80% of the race, then the full race distance is 15 km.
A person allows a 10% discount for cash payment from the market price of a toy and still he makes a 10% gain. What is the cost price of the toy which is marked Rs. 770?
UPSC-2016
This problem involves calculating the cost price of a toy given its marked price, a discount percentage, and a profit percentage. We will first calculate the selling price after the discount, and then use the profit percentage to find the cost price.
The discount is 10% of the Marked Price.
\[ \text{Discount Amount} = 10\% \text{ of } \text{MP} = \frac{10}{100} \times 770 = 77 \]\[ \text{Selling Price (SP)} = \text{MP} - \text{Discount Amount} = 770 - 77 = 693 \]The person makes a 10% gain on the Cost Price. This means the Selling Price is 110% of the Cost Price.
\[ \text{SP} = \text{CP} + \text{Profit} \]\[ \text{SP} = \text{CP} + 10\% \text{ of } \text{CP} \]\[ \text{SP} = \text{CP} \left(1 + \frac{10}{100}\right) \]\[ \text{SP} = \text{CP} \left(\frac{110}{100}\right) \]\[ 693 = \text{CP} \times \frac{11}{10} \]\[ \text{CP} = \frac{693 \times 10}{11} \]\[ \text{CP} = 63 \times 10 \]\[ \text{CP} = 630 \]Therefore, the cost price of the toy is Rs. 630.
Correct Option: C) Rs. 630
