Question 1

If X = 600 and Y = 800, then X is how many percent less than Y?

Accepted Answer

Introduction / Context: This is a direct percentage comparison problem. The task is to find by what percentage one number is less than another given number. Such questions frequently appear as warm up questions in percentage sections. Given Data / Assumptions: X = 600. Y = 800. We need to find the percentage by which X is less than Y. Concept / Approach: To find how much X is less than Y in percentage terms, we compute the difference Y minus X and then divide that difference by Y, because Y is the reference quantity. Finally, we multiply by 100 to convert the fraction into a percentage. Step-by-Step Solution: Y = 800 and X = 600.Difference between Y and X = 800 - 600 = 200.Required percentage less = (difference / Y) * 100.So percentage = (200 / 800) * 100.Compute 200 / 800 = 0.25.Therefore, percentage less = 0.25 * 100 = 25 percent. Verification / Alternative check: We can think in terms of fractions. X / Y = 600 / 800 = 3/4. This means X is 75 percent of Y, so it is lacking 25 percent to reach Y. Thus X is 25 percent less than Y, confirming the computed result. Why Other Options Are Wrong: 33.33 would correspond to a one third reduction, which does not match the actual ratio. The value 75 is the fraction of Y that X represents, not how much it is less by. The value 35 is simply an arbitrary incorrect choice. Common Pitfalls: A typical mistake is to divide the difference by X instead of Y, which would answer a different question, namely how much Y is greater than X. Always pay attention to which number your percentage comparison is based on. Final Answer: X is 25 percent less than Y.

Question 2

The ticket price for an adult is Rs. 1600 and for a child is Rs. 1200. One child goes free with every two adults. If a group has 23 adults and 10 children, what is the discount percentage received by the group?

Accepted Answer

Introduction / Context: This question is based on a promotional offer scenario and tests percentage discount calculation. The key is to correctly understand how many child tickets become free under the given condition, then compare the amount saved with the original total ticket cost. Given Data / Assumptions: Adult ticket price = Rs. 1600. Child ticket price = Rs. 1200. Offer: one child ticket is free with every two adults. The group has 23 adults and 10 children. We must find the overall discount percentage for the group. Concept / Approach: First compute the total amount that would be paid without any offer. Next, apply the offer: for every pair of two adults, one child ticket becomes free, limited by the actual number of children. Then calculate the actual amount paid and use the formula discount percentage = (savings / original amount) * 100. Step-by-Step Solution: Original total without offer = 23 * 1600 + 10 * 1200.This equals 36800 + 12000 = Rs. 48800.Number of adult pairs = floor(23 / 2) = 11 pairs, but there are only 10 children.So all 10 children can travel free under the offer.Actual amount paid = payment for 23 adults only = 23 * 1600 = Rs. 36800.Savings = 48800 - 36800 = Rs. 12000.Discount percentage = (12000 / 48800) * 100 ≈ 24.59 percent. Verification / Alternative check: We can check by treating the free tickets as a direct discount on the original bill. Ten child tickets at Rs. 1200 would cost Rs. 12000. The discount percentage is simply 12000 divided by the original total 48800, leading again to approximately 24.59 percent. Why Other Options Are Wrong: Percentages like 25.95, 25.77 and 31.60 come from using wrong original totals, incorrect numbers of free children or mathematical errors in computing the ratio. Only 24.59 percent is consistent with the exact savings and original bill amount. Common Pitfalls: Some students assume that only full sets of two adults and one child count and mis-handle the leftover adults. Others apply the discount on adult tickets as well, which the offer does not allow. Always translate the offer wording into precise arithmetic steps. Final Answer: The group receives an overall discount of approximately 24.59 percent.

Question 3

The salary of Giridhar is 5/3 times the salary of Hariraj, and the salary of Shaunak is 2/3 times the salary of Hariraj. What is the ratio of the salary of Giridhar to the salary of Shaunak?

Accepted Answer

Introduction / Context: This problem deals with ratios generated from fractional relationships with a common base. It tests whether the student can correctly transform fractional salary multiples into a simple ratio between two individuals. Given Data / Assumptions: Giridhar's salary = 5/3 of Hariraj's salary. Shaunak's salary = 2/3 of Hariraj's salary. We have to find the ratio Giridhar salary : Shaunak salary. Concept / Approach: Since both salaries are expressed as fractions of the same base, Hariraj's salary, we can ignore the exact value of Hariraj's salary and work only with the fractional coefficients 5/3 and 2/3. Their ratio directly gives the required ratio of salaries after simplifying. Step-by-Step Solution: Let Hariraj's salary = H units.Giridhar's salary = (5/3) * H.Shaunak's salary = (2/3) * H.Required ratio = Giridhar : Shaunak = (5/3 * H) : (2/3 * H).Cancel the common factor (1/3 * H) from both sides.We get ratio = 5 : 2. Verification / Alternative check: We can assign a specific value to H, for example H = 3. Then Giridhar earns 5 units and Shaunak earns 2 units. The ratio 5 : 2 clearly matches the simplified result obtained algebraically. Why Other Options Are Wrong: Ratios like 9 : 10, 10 : 9 and 2 : 5 do not reflect the fractional relationship given in the question. They would require very different fractions such as 9/10 or 10/9 times H, which are not present here. Common Pitfalls: Some exam takers add or subtract the fractions instead of forming a ratio, or forget to cancel the common base H. Keeping track of which expressions are being compared is essential. The presence of a common factor in both numerator and denominator is a strong hint that the base cancels out. Final Answer: The ratio of the salary of Giridhar to the salary of Shaunak is 5 : 2.

Question 4

Tapas, Avi and Rishi share a cake. Tapas eats 1/2 of the cake and Rishi eats 1/3 of the cake. Avi eats the remaining part. What fraction of the cake does Avi eat?

Accepted Answer

Introduction / Context: This is a simple fraction and complement problem. It tests comfort with adding fractions and determining the remaining part as a fraction of the whole. Such questions appear in early quantitative aptitude and school level mathematics. Given Data / Assumptions: The total cake represents one whole unit. Tapas eats 1/2 of the cake. Rishi eats 1/3 of the cake. Avi eats whatever is left. Concept / Approach: The total fraction eaten by Tapas and Rishi together is the sum of their shares. The remaining fraction for Avi is obtained by subtracting that sum from 1 (the whole cake). The process involves adding and subtracting fractions with different denominators. Step-by-Step Solution: Tapas' share = 1/2.Rishi's share = 1/3.Sum of their shares = 1/2 + 1/3.Find common denominator 6: 1/2 = 3/6 and 1/3 = 2/6.So 1/2 + 1/3 = 3/6 + 2/6 = 5/6.Total cake = 1, so Avi's share = 1 - 5/6 = 1/6. Verification / Alternative check: We can imagine cutting the cake into 6 equal pieces. Tapas would eat 3 pieces, Rishi would eat 2 pieces, and that accounts for 5 pieces. The remaining 1 piece out of 6 belongs to Avi, confirming the fraction 1/6. Why Other Options Are Wrong: 3/5, 2/6 and 3/6 do not represent the remaining fraction correctly. In particular, 2/6 and 3/6 correspond to 1/3 and 1/2, which have already been assigned to Rishi and Tapas. The total shares would exceed the whole if any of these were given to Avi. Common Pitfalls: A typical mistake is to divide the remaining fraction 1/6 further or miscalculate the addition of fractions by adding numerators and denominators separately. Always work with a common denominator and then add or subtract numerators as appropriate. Final Answer: Avi's share of the cake is 1/6.

Question 5

An item is sold with a 5 percent discount on its marked price, and then the customer receives a 20 percent cashback on the amount actually paid. What is the effective discount percentage received by the customer?

Accepted Answer

Introduction / Context: This question combines percentage discount with cashback, which is a common marketing practice in online and retail sales. The effective discount must consider both the reduced price and the money returned to the customer afterward. Given Data / Assumptions: Initial discount on marked price = 5 percent. Cashback on the amount paid after discount = 20 percent. The customer's net effective payment is marked price minus discount minus cashback. Concept / Approach: Assume a convenient marked price, apply the percentage discount to find the amount paid, then compute the cashback as a percentage of that amount. Subtract cashback from the amount paid to get the net cost. Finally, compare net cost with the original marked price to find the effective discount percentage. Step-by-Step Solution: Let marked price = 100 units.After 5 percent discount, amount paid = 100 - 5 = 95 units.Cashback = 20 percent of 95 = 0.20 * 95 = 19 units.Net payment by customer = 95 - 19 = 76 units.Effective discount = 100 - 76 = 24 units.Effective discount percentage = (24 / 100) * 100 = 24 percent. Verification / Alternative check: For a different marked price, say Rs. 5000, after 5 percent discount the customer pays Rs. 4750. Cashback is 20 percent of 4750, which is Rs. 950. Net payment is 4750 - 950 = Rs. 3800. Effective discount is 5000 - 3800 = 1200, which is 24 percent of 5000, confirming the same result. Why Other Options Are Wrong: 28.8 percent arises if someone mistakenly applies cashback on marked price. Values like 25 percent or 21 percent reflect approximate or stepwise reasoning that ignores the correct bases for each percentage. Common Pitfalls: Many candidates wrongly apply the cashback to the marked price instead of the discounted price or mistakenly add the percentages 5 and 20 to get 25. Percentages need to be applied to the right intermediate amounts, not simply added. Final Answer: The effective discount received by the customer is 24 percent.

Question 6

An item is sold with a 25 percent discount on its marked price, and then the customer receives an 8 percent cashback on the amount paid after discount. What is the effective discount percentage?

Accepted Answer

Introduction / Context: This question also combines discount and cashback, but with different percentages. The aim is to compute the single equivalent discount on the original marked price that results from both the discount and the cashback. Given Data / Assumptions: Discount on marked price = 25 percent. Cashback on post discount amount = 8 percent. We assume standard commercial arithmetic without taxes. Concept / Approach: As before, we assume a convenient marked price and compute the amount paid after discount, then compute cashback based on that reduced amount. Subtracting the cashback from the amount paid yields the net effective payment. Comparing this with the original marked price gives the effective discount percentage. Step-by-Step Solution: Let marked price = 100 units.After 25 percent discount, amount paid = 100 - 25 = 75 units.Cashback = 8 percent of 75 = 0.08 * 75 = 6 units.Net payment = 75 - 6 = 69 units.Effective discount amount = 100 - 69 = 31 units.Effective discount percentage = (31 / 100) * 100 = 31 percent. Verification / Alternative check: Take a marked price of Rs. 2000. After 25 percent discount, the price becomes 1500. Cashback is 8 percent of 1500, which is Rs. 120. Net payment is 1500 - 120 = 1380. The effective discount is 2000 - 1380 = 620, which is 31 percent of 2000. So the answer is consistent. Why Other Options Are Wrong: 35.75 percent and 35 percent come from adding percentages incorrectly or misapplying cashback to the wrong base. The value 12.5 percent is much too small and ignores the normal discount entirely. Common Pitfalls: The main mistake is adding 25 percent and 8 percent directly and treating 33 percent as the effective discount, or applying both on the marked price. Cashback is always calculated on the amount that the customer actually pays, which here is the discounted price, not the original marked price. Final Answer: The effective discount percentage is 31 percent.

Question 7

The value of a company stock drops from Rs. 25 per share to Rs. 21 per share. What is the percentage decrease per share?

Accepted Answer

Introduction / Context: This is a simple percentage decrease question in a financial context. It checks whether the learner can calculate the percentage drop in price when the old and new prices are given. Given Data / Assumptions: Initial price per share = Rs. 25. New price per share = Rs. 21. We must find the percentage decrease. Concept / Approach: Percentage decrease is calculated as (decrease / original value) * 100. Here the original value is the initial stock price of Rs. 25 and the decrease is the difference between the initial and new prices. Step-by-Step Solution: Original price = 25.New price = 21.Decrease = 25 - 21 = 4.Percentage decrease = (4 / 25) * 100.Compute 4 / 25 = 0.16.So percentage decrease = 0.16 * 100 = 16 percent. Verification / Alternative check: If the stock price had decreased by 10 percent, the new price would be 22.50, and if it had decreased by 20 percent, it would be 20. A 16 percent decrease lies between these values and more exactly gives 25 - 0.16 * 25 = 21, confirming the accuracy. Why Other Options Are Wrong: 4 percent, 8 percent and 12 percent do not produce a new price of 21 when applied to 25. For example, 10 percent decrease would yield 22.5, not 21. Only a 16 percent decrease matches the provided numbers. Common Pitfalls: Sometimes students accidentally divide by the new price instead of the old one or mistakenly treat the absolute decrease as a percentage directly. Always use the original value as the base for percentage decrease calculations. Final Answer: The percentage decrease in the stock price is 16 percent.

Question 8

If the price of kerosene is increased by 11 percent, by what percentage should a householder reduce consumption so that the total expenditure on kerosene remains unchanged?

Accepted Answer

Introduction / Context: This question is a typical price and consumption adjustment problem. It tests understanding of inverse proportion: when price rises, consumption must fall for expenditure to remain constant. The relationship is slightly more subtle because the required reduction is not equal to the price increase percentage. Given Data / Assumptions: Price of kerosene increases by 11 percent. Total expenditure on kerosene must remain the same. We need the percentage reduction in consumption. Concept / Approach: Expenditure = price * quantity. If expenditure is constant, then new price * new quantity = old price * old quantity. Hence, when price increases by a factor, quantity must decrease by the reciprocal factor. We express both in terms of the original price and consumption and then solve for the required reduction percentage. Step-by-Step Solution: Let original price = P and original consumption = Q.Original expenditure = P * Q.New price after 11 percent increase = 1.11 * P.Let new consumption = q.Condition for same expenditure: 1.11 * P * q = P * Q.Cancel P from both sides to get 1.11 * q = Q.So q = Q / 1.11.Reduction in consumption = Q - q = Q - Q / 1.11 = Q * (1 - 1 / 1.11).Fractional reduction = (1.11 - 1) / 1.11 = 0.11 / 1.11 = 1 / 10.09 approximately.More exactly, 0.11 / 1.11 = 1 / 10.09 ≈ 0.099, or about 9.09 percent. Verification / Alternative check: Take original price = Rs. 100 per unit and quantity = 100 units, so expenditure = Rs. 10,000. New price = 111 per unit. If consumption is reduced by 9.09 percent, new quantity ≈ 90.91 units. New expenditure = 111 * 90.91 ≈ 10,000, confirming that 9.09 percent reduction keeps expenditure constant. Why Other Options Are Wrong: Reducing consumption by 11 percent would make the expenditure drop below the original value. Values like 11.09 percent or 8.25 percent do not satisfy the condition of equal expenditure when computed precisely. Common Pitfalls: Many candidates incorrectly subtract the same percentage as the price increase from consumption. Others forget that the base for the percentage reduction is the new consumption, not the original expenditure. Always set up the product price * quantity explicitly. Final Answer: The consumption of kerosene must be reduced by approximately 9.09 percent.

Question 9

A shopkeeper marks his goods 40 percent above cost price and then allows a discount of 25 percent on the marked price. What is his overall percentage gain?

Accepted Answer

Introduction / Context: This is another markup and discount combination problem. The task is to find the net profit or loss percentage when goods are marked up from cost and then sold after a percentage discount on the marked price. Given Data / Assumptions: Goods are marked 40 percent above cost price. A discount of 25 percent is given on the marked price. We assume no other expenses. We must find the overall gain percentage. Concept / Approach: First use a convenient assumed cost price and calculate the marked price using the markup. Then apply the discount to find the selling price. Profit is selling price minus cost price, and gain percentage is profit divided by cost price multiplied by 100. Step-by-Step Solution: Let cost price (CP) = 100 units.Marked price (MP) = 100 + 40 percent of 100 = 100 + 40 = 140 units.Discount of 25 percent on marked price = 25 percent of 140 = 35 units.Selling price (SP) = MP - discount = 140 - 35 = 105 units.Profit = SP - CP = 105 - 100 = 5 units.Gain percentage = (profit / CP) * 100 = (5 / 100) * 100 = 5 percent. Verification / Alternative check: If cost price were Rs. 200 instead, marked price would be Rs. 280. A 25 percent discount reduces this to Rs. 210. Profit = 210 - 200 = 10, which is again 5 percent of 200. So the result is independent of the chosen cost price, confirming 5 percent gain. Why Other Options Are Wrong: Values like 10 percent, 15 percent or 20 percent result from incorrect combination of percentages, often by direct subtraction or adding discounts and markups. These do not match the actual difference between selling price and cost price obtained by correct computation. Common Pitfalls: A very common error is to subtract the discount percentage directly from the markup percentage and conclude 15 percent profit. This ignores that markup is on cost while discount is on marked price. Always convert percentages into actual rupee or unit figures before combining them. Final Answer: The shopkeeper makes an overall gain of 5 percent.

Question 10

The marked price of a sofa set is Rs. 4800. It is sold for Rs. 3672 after two successive discounts, the first being 10 percent. What is the second discount percentage?

Accepted Answer

Introduction / Context: This question is about successive discounts. Knowing how to work step by step from marked price to final selling price helps in determining unknown discount percentages when one discount and final price are given. Given Data / Assumptions: Marked price = Rs. 4800. First discount = 10 percent. Final selling price after two discounts = Rs. 3672. Second discount percentage is unknown. Concept / Approach: Apply the first discount on the marked price to obtain an intermediate price. Then assume the second discount as d percent on this intermediate price, and set the resulting expression equal to the given selling price. Solving for d gives the second discount percentage. Step-by-Step Solution: Marked price MP = 4800.After first discount of 10 percent, price becomes 4800 * (1 - 0.10) = 4800 * 0.9 = 4320.Let second discount be d percent.Selling price SP = 4320 * (1 - d/100).Given SP = 3672.So 4320 * (1 - d/100) = 3672.Divide both sides by 4320: 1 - d/100 = 3672 / 4320 = 0.85.Therefore, d/100 = 1 - 0.85 = 0.15.So d = 15 percent. Verification / Alternative check: First discount: 10 percent on 4800 gives 480 reduction, new price 4320. Second discount: 15 percent on 4320 gives 0.15 * 4320 = 648 reduction. Final price is 4320 - 648 = 3672, which matches the given figure exactly. Why Other Options Are Wrong: Discounts such as 13 percent, 14 percent or 17 percent do not convert 4320 into 3672. They would give selling prices that are either too high or too low compared with the required final price. Common Pitfalls: A common mistake is to treat successive discounts as a single discount equal to the sum of percentages. This does not hold because each discount applies to a different base. Always apply each percentage stepwise in the correct order. Final Answer: The second discount percentage is 15 percent.

Question 11

At what simple interest rate per annum will a sum of money become five times of itself in 8 years?

Accepted Answer

Introduction / Context: This question focuses on simple interest, where the interest is calculated on the original principal throughout the entire period. The situation where an amount becomes a multiple of the principal is very common in aptitude tests and helps in quickly finding the rate of interest. Given Data / Assumptions: Time period = 8 years. Amount after 8 years = 5 times the principal. Interest is simple interest, not compound. We must find the simple interest rate per annum. Concept / Approach: Under simple interest, Amount = Principal + Interest. If the amount is five times the principal, then interest equals four times the principal. The simple interest formula is I = (P * R * T) / 100, where P is principal, R is rate per annum and T is time in years. Using the relationship between interest and principal, we can solve for R. Step-by-Step Solution: Let principal = P.Given that amount after 8 years is 5P.So interest I = amount - principal = 5P - P = 4P.Simple interest formula: I = (P * R * T) / 100.Substitute I = 4P, T = 8 years.4P = (P * R * 8) / 100.Cancel P from both sides: 4 = (R * 8) / 100.Thus R * 8 = 400 and R = 400 / 8 = 50 percent per annum. Verification / Alternative check: If P = Rs. 100, then at 50 percent simple interest, yearly interest is Rs. 50. Over 8 years, total interest = 8 * 50 = Rs. 400. Amount = 100 + 400 = 500, which is exactly 5 times the principal, confirming our calculation. Why Other Options Are Wrong: Rates like 30 percent, 40 percent or 55 percent do not yield an amount that is 5P in 8 years. For example, at 40 percent, total interest in 8 years would be 3.2P, not 4P, giving an amount of 4.2P instead of 5P. Common Pitfalls: Students sometimes confuse simple interest with compound interest or misinterpret "five times" as an interest of 5P rather than an amount of 5P. Keep clear that amount equals principal plus interest, so the extra portion beyond the principal is the interest. Final Answer: The required simple interest rate is 50 percent per annum.

Question 12

One third of a number is 45. What is 45 percent of that number?

Accepted Answer

Introduction / Context: This problem connects fractions and percentages for the same unknown number. It tests the ability to translate given fractional information into the full number and then use that to compute a specified percentage of the number. Given Data / Assumptions: One third of a number is 45. We must find 45 percent of that number. Concept / Approach: If 1/3 of a number is known, we can recover the full number by multiplying by 3. After that, 45 percent of the number can be computed by multiplying the number by 45/100. Both steps are straightforward applications of fraction and percentage operations. Step-by-Step Solution: Let the number be N.Given that (1/3) * N = 45.So N = 45 * 3 = 135.We need to find 45 percent of N.45 percent of N = (45 / 100) * 135.Compute (45 / 100) * 135 = 0.45 * 135.0.45 * 135 = 60.75. Verification / Alternative check: We can think of 45 percent as 9/20. Then 9/20 of 135 equals (9 * 135) / 20 = 1215 / 20 = 60.75, which matches the decimal method and confirms the result. Why Other Options Are Wrong: Values 58.25, 61.25 and 70.75 arise from arithmetic errors like using the wrong fraction or confusing 45 percent with 45 out of 135 without scaling. Only 60.75 is the correct 45 percent of 135. Common Pitfalls: Some candidates mistakenly treat 45 as the original number rather than one third of it, or they compute 45 percent of 45 instead of 135. Carefully follow the order: first reconstruct the number, then apply the percentage. Final Answer: 45 percent of the number is 60.75.

Question 13

In a school, 60 percent of the students are girls. If the number of boys is 480, what is the total number of students in the school?

Accepted Answer

Introduction / Context: This is a straightforward percentage composition question. It checks the ability to relate the percentage of one group to the whole and deduce the total when the count of another group is known. Given Data / Assumptions: Girls form 60 percent of the total students. Boys form the remaining 40 percent. Number of boys = 480. We need to find the total number of students. Concept / Approach: If 40 percent of the total equals 480, then the total can be obtained by dividing 480 by 0.40. This uses the definition percentage = (part / whole) * 100 reversed to find the whole from a known part and percentage. Step-by-Step Solution: Let total number of students = T.Boys are 40 percent of T, so 0.40 * T = 480.Therefore, T = 480 / 0.40.Compute T = 480 / 0.4 = 480 * (10 / 4) = 480 * 2.5 = 1200.So total students = 1200. Verification / Alternative check: Check the girl count: 60 percent of 1200 = 0.6 * 1200 = 720 girls. Boys are 40 percent of 1200 = 480, which matches the given number. This validates the total of 1200 students. Why Other Options Are Wrong: Values 840, 96 and 1320 are not consistent with 480 representing 40 percent. For example, 40 percent of 840 is 336, not 480, so 840 cannot be the total. Common Pitfalls: Some students mistakenly divide 480 by 0.60 or treat 480 as 60 percent rather than 40 percent. Others confuse the direction of the percentage relationship. Carefully identify which group corresponds to which percentage before setting up the equation. Final Answer: The total number of students in the school is 1200.

Question 14

The marked price of an article is Rs. 960 and it is sold at a discount of Rs. 192. What is the discount percentage?

Accepted Answer

Introduction / Context: This is a basic discount calculation based on a known marked price and a known absolute discount in rupees. Such questions form the foundation for more complex profit, loss and discount scenarios. Given Data / Assumptions: Marked price (MP) = Rs. 960. Discount amount = Rs. 192. We need to calculate discount percentage. Concept / Approach: Discount percentage is calculated by dividing the discount amount by the marked price and multiplying by 100, because the discount is given on the marked price. Step-by-Step Solution: Marked price MP = 960.Discount amount = 192.Discount percentage = (discount / MP) * 100.So discount percentage = (192 / 960) * 100.Compute 192 / 960 = 0.20.Therefore discount percentage = 0.20 * 100 = 20 percent. Verification / Alternative check: If the discount is 20 percent on 960, then discount amount = 0.20 * 960 = 192. So the selling price would be 960 - 192 = 768, and the numbers fit the given data, confirming 20 percent. Why Other Options Are Wrong: Values like 15 percent, 25 percent and 30 percent would correspond to discount amounts of 144, 240 and 288 respectively when applied to 960, none of which match the given discount of 192. Common Pitfalls: Errors usually arise from dividing by the selling price instead of the marked price or from miscalculating the fraction 192 / 960. Always check that the denominator in the discount formula is the marked price. Final Answer: The discount percentage on the article is 20 percent.

Question 15

In an election between two candidates, the winning candidate receives 70 percent of the votes polled and wins by 15,400 votes. How many votes were polled for the losing candidate?

Accepted Answer

Introduction / Context: This is a standard election question involving percentages and vote margins. It evaluates the ability to translate information about winning margins into totals and then compute the votes for each candidate. Given Data / Assumptions: There are only two candidates. Winner receives 70 percent of the votes polled. Winner wins by 15,400 votes. We need the number of votes polled for the losing candidate. Concept / Approach: Let total votes be V. Then the winner gets 70 percent of V and the loser gets 30 percent of V since together they must account for 100 percent of the votes. The margin of victory equals difference between 70 percent of V and 30 percent of V, which is 40 percent of V. Setting this equal to 15,400 allows us to find V and then the loser votes. Step-by-Step Solution: Let total votes polled = V.Winner's votes = 0.70 * V.Loser's votes = 0.30 * V.Margin = winner - loser = (0.70V - 0.30V) = 0.40V.Given margin = 15,400, so 0.40V = 15400.Thus V = 15400 / 0.40 = 15400 * (10 / 4) = 15400 * 2.5 = 38,500.Loser's votes = 0.30 * 38,500 = 11,550. Verification / Alternative check: Winner's votes = 0.70 * 38,500 = 26,950. Difference between winner and loser votes = 26,950 - 11,550 = 15,400, which matches the margin given. This confirms that 11,550 is correct for the losing candidate. Why Other Options Are Wrong: 38,500 is the total votes, not the loser votes. Other options such as 26,950 or 13,550 are incorrect because they either correspond to the winner's votes or to arbitrary figures that do not satisfy the margin condition. Common Pitfalls: People sometimes treat 70 percent directly as the margin or forget that the loser also has a share of the total. Another common mistake is to take the margin as 30 percent instead of 40 percent of the total. Always compute the difference between the two percentage shares to find the margin portion. Final Answer: The losing candidate received 11,550 votes.

Question 16

In an examination, 65 percent of students pass in History and 55 percent of students pass in Hindi. If 5 percent of students fail in both subjects, what percentage of students pass in both History and Hindi?

Accepted Answer

Introduction / Context: This is another set theory question using percentages, now in the context of examination results. It focuses on the relationship between passing in individual subjects and in both subjects using complements and the inclusion exclusion principle. Given Data / Assumptions: 65 percent of students pass in History. 55 percent of students pass in Hindi. 5 percent of students fail in both subjects. We need the percentage of students who pass in both subjects. Concept / Approach: If 5 percent fail both, then 95 percent pass in at least one subject. Passing in History or Hindi or both is the union of the passing sets. Using the formula for the union of two sets, we can find the intersection, that is, those who pass both subjects. Step-by-Step Solution: Let H be the set of students passing History.Let D be the set of students passing Hindi.P(H) = 65 percent, P(D) = 55 percent.Failing in both subjects = 5 percent.So passing in at least one subject = 100 - 5 = 95 percent.By inclusion exclusion, P(H ∪ D) = P(H) + P(D) - P(H ∩ D).Thus 95 = 65 + 55 - P(H ∩ D).So P(H ∩ D) = 65 + 55 - 95 = 25 percent. Verification / Alternative check: Assume 100 students. Then 25 students pass both, 40 pass only History, 30 pass only Hindi and 5 fail in both. Totals: History passes = 25 + 40 = 65, Hindi passes = 25 + 30 = 55, and fail both = 5. All conditions are satisfied, confirming that 25 percent pass both subjects. Why Other Options Are Wrong: Values 15, 20 and 30 percent do not satisfy the inclusion exclusion equation once the given individual percentages and total failing percentage are accounted for. They break the consistency between the sets or the complement. Common Pitfalls: Some candidates mistakenly use 5 percent as pass in both or misinterpret it as intersection instead of the complement. Others forget to subtract the intersection term when using P(H ∪ D) = P(H) + P(D) - P(H ∩ D). Always align each percentage with the correct region of the Venn diagram. Final Answer: 25 percent of the students pass in both History and Hindi.

Question 17

Priya scores 9 marks more in History than in Geography. Her History marks are 56 percent of the sum of her History and Geography marks. What are her Geography marks?

Accepted Answer

Introduction / Context: This is a percentage and algebra problem involving two related scores. It tests the ability to translate verbal conditions into algebraic equations and then solve for one of the unknowns using percentage information. Given Data / Assumptions: Priya scores 9 marks more in History than in Geography. History marks are 56 percent of the sum of History and Geography marks. We need to find Geography marks. Concept / Approach: Let Geography marks be G and History marks be G + 9. The sum of the marks is (G + 9) + G = 2G + 9. The percentage condition says that History marks equal 56 percent of this sum. We form an equation based on this and solve for G. Step-by-Step Solution: Let Geography marks = G.Then History marks = G + 9.Sum of marks = G + 9 + G = 2G + 9.Given that History marks = 56 percent of sum.So G + 9 = 0.56 * (2G + 9).Write 0.56 as 56 / 100 = 14 / 25 for exact arithmetic.Thus G + 9 = (14 / 25) * (2G + 9).Multiply both sides by 25: 25G + 225 = 14(2G + 9) = 28G + 126.Rearrange: 25G + 225 = 28G + 126 → 225 - 126 = 28G - 25G.So 99 = 3G and G = 33. Verification / Alternative check: If Geography marks are 33, History marks are 33 + 9 = 42. Sum is 33 + 42 = 75. Check the percentage: 56 percent of 75 = 0.56 * 75 = 42, which exactly equals the History marks, confirming the solution. Why Other Options Are Wrong: Values 42, 65 and 53 do not satisfy the condition that History marks are 9 more than Geography marks and at the same time 56 percent of the sum. Substituting any of them for G fails the equation G + 9 = 0.56(2G + 9). Common Pitfalls: Common mistakes include mixing up which subject has the higher marks or incorrectly forming the percentage equation, for example taking 56 percent of only one subject. It is crucial to define variables clearly and express all conditions algebraically before solving. Final Answer: Priya's Geography marks are 33.

Question 18

When a number is increased by 69, it becomes 103 percent of itself. What is the original number?

Accepted Answer

Introduction / Context: This question is a simple application of percentage increase expressed in terms of an absolute increase. It tests algebraic manipulation involving converting a percentage expression into an equation with an unknown number. Given Data / Assumptions: When the number is increased by 69, it becomes 103 percent of the original. We need to find the original number. Concept / Approach: If the original number is N, then 103 percent of N is 1.03 * N. The statement says that N + 69 equals 1.03 * N. Setting up this equation and solving for N gives the original number. Step-by-Step Solution: Let the original number = N.Increased by 69 gives N + 69.Given that N + 69 = 103 percent of N = 1.03 * N.So N + 69 = 1.03N.Rearrange: 1.03N - N = 69.Thus 0.03N = 69.So N = 69 / 0.03 = 69 * (100 / 3) = 2300. Verification / Alternative check: Check with N = 2300. Then 103 percent of 2300 is 1.03 * 2300 = 2369. Increasing the original number by 69 yields 2300 + 69 = 2369, exactly matching the stated condition, confirming the result. Why Other Options Are Wrong: Numbers like 1300, 3300 or 4300 do not satisfy the relationship N + 69 = 1.03N. Substituting them leads to inconsistencies where the required increase is not 69 or the percentage is not 103 percent. Common Pitfalls: Some learners confuse 103 percent with 1.3 or mistakenly divide by 103 instead of 3 in the calculation. Remember that 103 percent means 103/100, so the factor is 1.03, and the extra part beyond the original 100 percent is 3 percent. Final Answer: The original number is 2300.

Question 19

Two students appear for an examination. One scores 24 marks more than the other, and his marks are 65 percent of the sum of their marks. What are the marks obtained by the two students?

Accepted Answer

Introduction / Context: This is a classic algebra and percentage question involving two unknowns linked by both a difference and a percentage condition. It checks the ability to set up and solve simultaneous equations based on a word problem statement. Given Data / Assumptions: Let the higher scoring student have marks x and the other have marks y. x is 24 more than y. x is 65 percent of the sum (x + y). We must find x and y. Concept / Approach: We translate the verbal conditions into two equations: x = y + 24 and x = 0.65(x + y). Solving this system gives unique values for x and y that satisfy both relationships. Step-by-Step Solution: Let the higher marks = x and the lower marks = y.From the problem, x = y + 24.Also, x = 65 percent of (x + y) = 0.65(x + y).Write 0.65 as 65 / 100 = 13 / 20.So x = (13 / 20)(x + y).Multiply by 20: 20x = 13x + 13y.So 7x = 13y and x = (13/7)y.But x = y + 24, so y + 24 = (13/7)y.Multiply both sides by 7: 7y + 168 = 13y.Thus 168 = 6y and y = 28.Then x = y + 24 = 28 + 24 = 52. Verification / Alternative check: Check the percentage condition: sum = 52 + 28 = 80. Sixty five percent of 80 is 0.65 * 80 = 52, which indeed equals the marks of the higher student, confirming the solution pair (52, 28). Why Other Options Are Wrong: Pairs such as 78 and 54 or 85 and 61 do not satisfy the condition that the higher score is both 24 more than the lower score and 65 percent of the total. Substituting them breaks one or both equations. Common Pitfalls: Misidentifying which student has higher marks or misplacing the 65 percent on the wrong side of the equation are common mistakes. It is important to define variables clearly and ensure that each textual condition is precisely captured in algebraic form. Final Answer: The two students score 52 marks and 28 marks respectively.

Question 20

The marked price of an item is Rs. 200. On the purchase of 1 item, the discount is 22 percent; on the purchase of 4 items, the discount is 33 percent. Rabia buys 5 items together. What is the effective discount percentage on her total purchase?

Accepted Answer

Introduction / Context: This question mixes quantity based slab discounts with overall effective discount calculation. It tests how well a candidate can interpret conditions where discounts depend on how many items are purchased and then compute a single net discount for the entire transaction. Given Data / Assumptions: Marked price per item = Rs. 200. Discount on purchase of 1 item = 22 percent. Discount on purchase of 4 items = 33 percent. Rabia buys 5 items in total. We assume that 4 items receive the 33 percent discount and 1 item receives the 22 percent discount. Concept / Approach: First, compute the total bill without any discount. Next, calculate the actual amount Rabia pays given that four items are billed with 33 percent discount and one item with 22 percent discount. The difference between the original total and the amount actually paid is the total discount, which can then be expressed as a percentage of the original total. Step-by-Step Solution: Marked price per item = 200.Total marked price for 5 items = 5 * 200 = 1000.For 4 items at 33 percent discount, price per item = 200 * (1 - 0.33) = 200 * 0.67 = 134.Total for these 4 items = 4 * 134 = 536.For 1 item at 22 percent discount, price = 200 * (1 - 0.22) = 200 * 0.78 = 156.Amount actually paid = 536 + 156 = 692.Total discount in rupees = 1000 - 692 = 308.Effective discount percentage = (308 / 1000) * 100 = 30.8 percent. Verification / Alternative check: If we redistributed the discounts incorrectly, the total discount would change and no longer correspond to 30.8 percent. The arrangement 4 items at 33 percent and 1 at 22 percent is the only interpretation consistent with the given slab structure for a 5 item purchase. Why Other Options Are Wrong: Values like 35 percent or 34 percent assume that all 5 items receive the higher discount, while 20.4 percent assumes something like averaging discounts incorrectly. Only 30.8 percent corresponds to the accurate weighted calculation described above. Common Pitfalls: Students often average 22 percent and 33 percent directly to get a wrong answer or assume that the best discount applies to all items. Carefully interpret the offer and count how many items receive each discount before computing amounts. Final Answer: The effective discount on Rabia's total purchase is 30.8 percent.

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