To find the single equivalent discount for two successive discounts, we can assume an initial marked price (e.g., 100 units). Then, we apply each discount sequentially to find the final price. The difference between the initial marked price and the final price will give us the total discount, which can then be expressed as a percentage of the marked price.
Assume an initial Marked Price (MP):
Let the original Marked Price be \( \$100 \).
Apply the first discount:
The first discount is 65%.
\[ \text{Price after first discount} = \text{MP} - (\text{MP} \times \text{Discount}_1) \] \[ \text{Price after first discount} = \\(100 - (\\)100 \times 0.65) \] \[ \text{Price after first discount} = \\(100 - \\)65 = \$35 \]Apply the second discount:
The second discount is 24%, applied to the price after the first discount.
\[ \text{Price after second discount} = \text{Price after first discount} - (\text{Price after first discount} \times \text{Discount}_2) \] \[ \text{Price after second discount} = \\(35 - (\\)35 \times 0.24) \] \[ \text{Price after second discount} = \\(35 - \\)8.40 = \$26.60 \]Calculate the total discount:
The total discount is the difference between the original Marked Price and the final price.
\[ \text{Total Discount Amount} = \text{Original MP} - \text{Final Price} \] \[ \text{Total Discount Amount} = \\(100 - \\)26.60 = \$73.40 \]Express the total discount as a percentage:
Since we started with a Marked Price of \$100, the total discount amount directly represents the percentage discount.
\[ \text{Equivalent Single Discount} = \frac{\text{Total Discount Amount}}{\text{Original MP}} \times 100\% \] \[ \text{Equivalent Single Discount} = \frac{\\(73.40}{\\)100} \times 100\% = 73.4\% \]Alternatively, using the formula for two successive discounts \(d_1\) and \(d_2\):
\[ \text{Equivalent Discount} = \left(d_1 + d_2 - \frac{d_1 \times d_2}{100}\right)\% \]Given \(d_1 = 65\%\) and \(d_2 = 24\%\):
\[ \text{Equivalent Discount} = \left(65 + 24 - \frac{65 \times 24}{100}\right)\% \] \[ \text{Equivalent Discount} = \left(89 - \frac{1560}{100}\right)\% \] \[ \text{Equivalent Discount} = (89 - 15.6)\% \] \[ \text{Equivalent Discount} = 73.4\% \]The calculation shows that two successive discounts of 65% and 24% are equivalent to a single discount of 73.4%.
A) 77% — This would be the result if the second discount was applied to the original price, not the discounted price, or if there was an error in calculation.
B) 74.5% — This value is close but incorrect. It might arise from a calculation error, for example, if the product term \(\frac{d_1 \times d_2}{100}\) was miscalculated or incorrectly subtracted.
C) 76.6% — This value is incorrect. It might be obtained by some miscalculation, such as adding the discounts and then subtracting a different value, or an arithmetic error during the sequential discount calculation.
The problem asks for the ratio of two numbers given a relationship between a percentage of the first number and a fraction of the second number. We will set up an equation based on the given information and then rearrange it to find the ratio of the first number to the second number.
Let the first number be \(A\) and the second number be \(B\).
According to the problem statement, "38% of first number is equal to two-sevenths of second number". We can write this as an equation:
\[ 38\% \times A = \frac{2}{7} \times B \]Convert the percentage to a fraction or decimal:
\[ \frac{38}{100} \times A = \frac{2}{7} \times B \]Simplify the fraction \(\frac{38}{100}\):
\[ \frac{19}{50} \times A = \frac{2}{7} \times B \]To find the ratio of the first number to the second number, \(A:B\), we need to express \(\frac{A}{B}\). Rearrange the equation:
\[ \frac{A}{B} = \frac{2}{7} \div \frac{19}{50} \]\[ \frac{A}{B} = \frac{2}{7} \times \frac{50}{19} \]Multiply the fractions:
\[ \frac{A}{B} = \frac{2 \times 50}{7 \times 19} \]\[ \frac{A}{B} = \frac{100}{133} \]Therefore, the ratio of the first number to the second number is \(100:133\).
C) 100 : 133 is the correct ratio derived from the given conditions.
The selling price of 50 books is equal to the cost price of 39 books. Find the loss or gain percentage.
\( \frac{100}{11} \)% gain
\( \frac{100}{11} \) % loss
22% loss
22% gain
To find the loss or gain percentage, we need to compare the total selling price (SP) with the total cost price (CP). The problem states a relationship between the selling price of a certain number of books and the cost price of another number of books. We can use this relationship to find the ratio of SP to CP for a single book, or to determine the overall profit/loss if we consider a common number of books.
Let the Cost Price of one book be \( \text{CP} \).
Let the Selling Price of one book be \( \text{SP} \).
According to the problem statement, the selling price of 50 books is equal to the cost price of 39 books. We can write this as an equation:
\[ 50 \times \text{SP} = 39 \times \text{CP} \]From this equation, we can find the ratio of SP to CP:
\[ \frac{\text{SP}}{\text{CP}} = \frac{39}{50} \]Since \( \text{SP} < \text{CP} \) (because \( 39 < 50 \)), there is a loss.
To calculate the loss percentage, we first find the loss amount. If we consider the CP to be 50 units and SP to be 39 units (for a common number of books, or per unit), the loss is:
\[ \text{Loss} = \text{CP} - \text{SP} = 50 - 39 = 11 \text{ units} \]The loss percentage is calculated with respect to the Cost Price:
\[ \text{Loss Percentage} = \frac{\text{Loss}}{\text{CP}} \times 100 \% \] \[ \text{Loss Percentage} = \frac{11}{50} \times 100 \% \] \[ \text{Loss Percentage} = 11 \times 2 \% \] \[ \text{Loss Percentage} = 22 \% \]Therefore, there is a 22% loss.
C) 22% loss
A) \( \frac{100}{11} \)% gain — This is incorrect because there is a loss, not a gain, as the selling price of 50 books is less than their cost price (when compared to the cost of 39 books). The value \( \frac{100}{11} \) is also incorrect.
B) \( \frac{100}{11} \)% loss — While it correctly identifies a loss, the magnitude of the loss percentage is calculated incorrectly. \( \frac{100}{11} \% \approx 9.09 \% \), which is not 22%.
D) 22% gain — This is incorrect because there is a loss, not a gain. The calculation of 22 is correct in magnitude but the nature (gain vs. loss) is wrong.
To solve this problem, we need to represent two consecutive odd natural numbers algebraically. Then, we will form an equation based on the given condition (the sum of their squares is 34) and solve for the numbers. Finally, we will find the sum of these numbers.
The first number is \(x = 3\).
The second number is \(x+2 = 3+2 = 5\).
Let's verify: \(3^2 + 5^2 = 9 + 25 = 34\). This matches the given condition.
A) The sum of the numbers is \(3+5=8\).
The problem involves simple interest calculations for two different investments. We are given the total principal amount, two different interest rates, and that the interest earned from both investments is equal after a certain period. We need to find the amount invested at one of the rates.
Let the total investment be \(P = \text{₹}22,800\).
Let the amount invested at 14% per annum be \(P_1\). Then the amount invested at 7% per annum will be \(P_2 = P - P_1 = 22,800 - P_1\).
The time period for both investments is \(T = 6\) years.
The formula for simple interest (SI) is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]For the first investment (at 14%):
\[ \text{SI}_1 = \frac{P_1 \times 14 \times 6}{100} = \frac{84 P_1}{100} \]For the second investment (at 7%):
\[ \text{SI}_2 = \frac{P_2 \times 7 \times 6}{100} = \frac{42 P_2}{100} \]According to the problem, the interests earned from the two investments are equal:
\[ \text{SI}_1 = \text{SI}_2 \]\[ \frac{84 P_1}{100} = \frac{42 P_2}{100} \]We can simplify this equation:
\[ 84 P_1 = 42 P_2 \]\[ 2 P_1 = P_2 \]Now substitute \(P_2 = 22,800 - P_1\) into the equation:
\[ 2 P_1 = 22,800 - P_1 \]Solve for \(P_1\):
\[ 2 P_1 + P_1 = 22,800 \]\[ 3 P_1 = 22,800 \]\[ P_1 = \frac{22,800}{3} \]\[ P_1 = 7,600 \]Thus, the sum invested at 14% per annum is ₹7,600.
Correct Option: A) 7,600
The calculation shows that the amount invested at 14% per annum is ₹7,600, which matches option A.
The problem involves calculating the cost price for the shopkeeper after successive discounts and then determining a new list price to achieve a desired profit margin after allowing another discount. We will first calculate the effective discount, then the cost price, and finally work backward from the desired profit and discount to find the new list price.
The wholesaler offers two successive discounts of 35% and 5%.
Let the initial list price be \(P\).
After the first discount of 35%, the price becomes:
\[ P_1 = P \times \left(1 - \frac{35}{100}\right) = P \times 0.65 \]After the second discount of 5% on \(P_1\), the final price becomes:
\[ P_2 = P_1 \times \left(1 - \frac{5}{100}\right) = P_1 \times 0.95 \]Substitute \(P_1\):
\[ P_2 = (P \times 0.65) \times 0.95 = P \times 0.6175 \]So, the effective discount is \(1 - 0.6175 = 0.3825\) or 38.25%.
The item's listed price from the wholesaler was ₹1,600.
The shopkeeper purchased it after an effective discount of 38.25%.
\[ \text{CP} = 1600 \times \left(1 - \frac{38.25}{100}\right) \] \[ \text{CP} = 1600 \times 0.6175 \] \[ \text{CP} = \text{₹}988 \]The shopkeeper wants to gain 25% on the cost price.
\[ \text{SP} = \text{CP} \times \left(1 + \frac{25}{100}\right) \] \[ \text{SP} = 988 \times 1.25 \] \[ \text{SP} = \text{₹}1235 \]The shopkeeper allows a 5% discount on the new list price to achieve the desired selling price.
Let the new list price be LP.
\[ \text{SP} = \text{LP} \times \left(1 - \frac{5}{100}\right) \] \[ 1235 = \text{LP} \times 0.95 \] \[ \text{LP} = \frac{1235}{0.95} \] \[ \text{LP} = \text{₹}1300 \]We need to find the total number of registered voters. We are given information about the percentage of voters who cast their vote, the percentage of invalid votes, the winning candidate's share of valid votes, and the margin of victory. We will work backward from the margin of victory to find the total registered voters.
Let the total number of registered voters be \(V\).
Percentage of registered voters who cast their vote = 87%.
Number of votes polled = \(0.87 \times V\).
Percentage of votes polled that were invalid = 5%.
Percentage of votes polled that were valid = \(100\% - 5\% = 95\%\).
Number of valid votes = \(0.95 \times (\text{votes polled})\)
Number of valid votes = \(0.95 \times 0.87 \times V\).
The winning candidate got 52% of the valid votes.
The losing candidate got \(100\% - 52\% = 48\%\) of the valid votes.
The margin of victory is the difference between the votes received by the winning candidate and the losing candidate.
Margin of victory = (Winning candidate's percentage - Losing candidate's percentage) \(\times\) Valid votes
Margin of victory = \((52\% - 48\%) \times (\text{valid votes})\)
Margin of victory = \(4\% \times (\text{valid votes})\)
We are given that the margin of victory is 1653 votes.
So, \(0.04 \times (\text{valid votes}) = 1653\).
Substitute the expression for valid votes from step 3:
\[ 0.04 \times (0.95 \times 0.87 \times V) = 1653 \]Now, solve for \(V\):
\[ 0.04 \times 0.95 \times 0.87 \times V = 1653 \]\[ 0.038 \times 0.87 \times V = 1653 \]\[ 0.03306 \times V = 1653 \]\[ V = \frac{1653}{0.03306} \]\[ V = 50000 \]C) 50000 is the correct answer. The calculations show that the total number of registered voters is 50000.
Find the curved surface area of a sphere whose radius is 2.8 cm. (Use π = \( \frac{22}{7} \))
To find the curved surface area of a sphere, we use the standard formula for the surface area of a sphere, which is \(4 \pi r^2\), where \(r\) is the radius. We are given the radius and the value of \(\pi\).
The profit in a business is distributed among partners in the ratio of their investments. We are given the total investment, total profit, and one partner's share of the profit. We need to find the individual investments and then their difference.
Total profit = ₹15,600
Sanjay's share = ₹1,900
Ritik's share = Total profit - Sanjay's share
\[ \text{Ritik's share} = \text{₹}15,600 - \text{₹}1,900 = \text{₹}13,700 \]The ratio of their profits is equal to the ratio of their investments.
\[ \text{Ratio of profits (Ritik : Sanjay)} = 13,700 : 1,900 = 137 : 19 \]Total investment = ₹62,400
\[ \text{Ritik's investment} = \frac{\text{Ritik's ratio part}}{\text{Total parts}} \times \text{Total investment} \] \[ \text{Ritik's investment} = \frac{137}{156} \times \text{₹}62,400 \] \[ \text{Ritik's investment} = 137 \times \frac{62400}{156} = 137 \times 400 = \text{₹}54,800 \]Alternatively, Sanjay's investment = Total investment - Ritik's investment
\[ \text{Sanjay's investment} = \text{₹}62,400 - \text{₹}54,800 = \text{₹}7,600 \]Correct Option: B) ₹47,200
We are given that a number, when increased by 100%, results in 2740. We need to find the original number. We can set up an algebraic equation to represent this relationship.
Let the unknown number be \(x\).
An increase of 100% means adding 100% of the number to the original number. In decimal form, 100% is \(100/100 = 1\).
So, increasing the number \(x\) by 100% of \(x\) can be written as:
\[ x + (100\% \text{ of } x) \] \[ x + (1 \times x) \] \[ x + x \] \[ 2x \]We are given that this increased value is 2740. So, we can set up the equation:
\[ 2x = 2740 \]To find the value of \(x\), divide both sides of the equation by 2:
\[ x = \frac{2740}{2} \] \[ x = 1370 \]Thus, the original number is 1370.
Correct Option: B) 1370