Question 1

Approximate the value of 74.99% of 1299 plus 8.98% of 1899. (Round your answer to the nearest whole number.)

Accepted Answer

Introduction / Context: This question tests the ability to handle percentages with non-integer values and perform approximate calculations accurately. It is a common type of problem in banking, finance and competitive exams, where values like 74.99% and 8.98% appear to reflect near rounded percentages. The focus is on systematic computation and correct rounding rather than mental shortcuts alone. Given Data / Assumptions: We need to compute 74.99% of 1299. We also need to compute 8.98% of 1899. Then we add both results and round to the nearest whole number. All percentages are treated as exact decimal values and calculations use standard arithmetic. Concept / Approach: First, convert each percentage into its decimal form by dividing by 100. Then multiply each decimal by its corresponding number. After obtaining the two products, add them together to get the combined value. Finally, round the result to the nearest integer. Using approximate mental checks is helpful, but a structured calculation ensures accuracy. Step-by-Step Solution: Step 1: Convert 74.99% to decimal form. 74.99% = 74.99 / 100 = 0.7499. Step 2: Compute 0.7499 * 1299. 0.7499 * 1299 is approximately 974.18 (keeping a few decimal places). Step 3: Convert 8.98% to decimal form. 8.98% = 8.98 / 100 = 0.0898. Step 4: Compute 0.0898 * 1899. 0.0898 * 1899 is approximately 170.47. Step 5: Add both partial results. Combined value ≈ 974.18 + 170.47 = 1,144.65. Step 6: Round 1,144.65 to the nearest whole number. This rounds to 1,145. So the required approximate value is 1,145. Verification / Alternative check: As a rough mental check, we can approximate 74.99% as 75% and 8.98% as 9%. Then 75% of 1300 is about 975, and 9% of 1900 is about 171. Adding 975 and 171 gives 1,146. This quick estimate is very close to the more accurate calculated value of about 1,145, which supports the correctness of our more precise computation. Why Other Options Are Wrong: 1,120 and 1,100 are too small compared to both the precise and rough calculations, implying a significant underestimation of the percentages. 1,190 and 1,210 are too large and would require much higher effective percentages than those given. Among the options, only 1,145 properly matches the accurate and approximate computations. Common Pitfalls: A common mistake is to round the percentages to 75% and 9% at the very beginning and then round again at intermediate steps, which can introduce larger errors. Another issue is misplacing the decimal when converting from percentage to decimal, for example writing 0.7499 as 7.499. Learners may also incorrectly add the percentages directly to the numbers instead of multiplying. Always remember to convert percentages to decimals, multiply by the base number, and then add the results. Final Answer: The approximate value of the expression is 1,145 (to the nearest whole number).

Question 2

Arun's weight is 140% of Akhil's weight. Shreyon's weight is twice Akhil's weight, and Nani's weight is 90% of Shreyon's weight. Nani's weight is what percentage of Arun's weight?

Accepted Answer

Introduction / Context: This question deals with chained percentage relationships between the weights of four people. Such problems are common in percentage comparison and ratio topics, where each quantity is defined in terms of another, and we need to express one final quantity as a percentage of another. The key is to choose a base variable, express all weights in terms of that, and then compare the two required values. Given Data / Assumptions: Let Akhil's weight be A. Arun's weight is 140% of Akhil's weight, so Arun = 1.4A. Shreyon's weight is twice Akhil's weight, so Shreyon = 2A. Nani's weight is 90% of Shreyon's weight, so Nani = 0.9 * 2A. We need Nani's weight as a percentage of Arun's weight. Concept / Approach: All weights are expressed as multiples of Akhil's weight A. Once these expressions are set up, we only need the ratio of Nani to Arun, because the base A cancels out. The formula we use is: percentage of Arun represented by Nani = (Nani / Arun) * 100 This method avoids the need for actual numerical weights and works with pure ratios and percentages. Step-by-Step Solution: Step 1: Express all weights in terms of A. Arun = 1.4A. Shreyon = 2A. Nani = 90% of Shreyon = 0.90 * 2A = 1.8A. Step 2: Form the ratio Nani / Arun. Nani / Arun = (1.8A) / (1.4A). Step 3: Cancel A from numerator and denominator. Nani / Arun = 1.8 / 1.4. Step 4: Simplify 1.8 / 1.4. 1.8 / 1.4 = 18 / 14 = 9 / 7. Step 5: Convert this ratio to a percentage. (9 / 7) * 100 ≈ 128.5714%. Step 6: Rounded to one decimal place this is about 128.6%. Therefore, Nani's weight is approximately 128.6% of Arun's weight. Verification / Alternative check: For a quick check, we can assign a convenient value to A, say Akhil weighs 50 kg. Then Arun weighs 1.4 * 50 = 70 kg. Shreyon weighs 2 * 50 = 100 kg. Nani weighs 0.9 * 100 = 90 kg. Now compute Nani as a percentage of Arun: (90 / 70) * 100 = (9 / 7) * 100 ≈ 128.57%. This again confirms that Nani is about 128.6% of Arun in weight. Why Other Options Are Wrong: 120% and 125% are both less than the accurate value of about 128.6%, so they underestimate Nani's weight relative to Arun. 130% and 135% both overestimate the ratio, with 135% being significantly higher than the exact value of about 128.6%. Only 128.6% matches the computed ratio closely. Common Pitfalls: Many learners accidentally compare Nani to Akhil or Shreyon instead of to Arun, leading to different percentages. Another typical error is to add or subtract the percentages linearly instead of expressing them as multipliers. Confusing which person is the base for each percentage (for example, mixing up 140% of Akhil and 90% of Shreyon) can also lead to mistakes. Always express each quantity cleanly in terms of a single base variable to avoid confusion. Final Answer: Nani's weight is approximately 128.6% of Arun's weight.

Question 3

What value should replace the question mark in the expression: 52.5% of 800 + 30.5% of 2800 = ? + 87.30 ?

Accepted Answer

Introduction / Context: This question combines percentage calculations with a simple linear equation. We are given an expression where two percentage terms are added and the sum is split into an unknown part plus a constant. The task is to compute the actual numerical value of the percentage expressions and then isolate the missing term. This kind of question is common in quantitative aptitude and helps build accuracy with multi-step arithmetic expressions. Given Data / Assumptions: First term: 52.5% of 800. Second term: 30.5% of 2800. Their sum equals an unknown value plus 87.30. We need to find the value of the unknown term represented by question mark. Percentages are interpreted as simple hundred-based percentages with standard arithmetic. Concept / Approach: First compute each percentage value by converting the percentage into decimal and multiplying by the base number. Then add the two results to obtain the left side total. According to the given equation, that total equals the unknown term plus 87.30. We then subtract 87.30 from the total to get the unknown. This is a straightforward use of percentage calculation followed by a simple rearrangement of an equation. Step-by-Step Solution: Step 1: Compute 52.5% of 800. 52.5% = 52.5 / 100 = 0.525. 0.525 * 800 = 420. Step 2: Compute 30.5% of 2800. 30.5% = 30.5 / 100 = 0.305. 0.305 * 2800 = 854. Step 3: Add both results to get the left side of the equation. Left side total = 420 + 854 = 1,274. Step 4: Use the given relationship. 1,274 = ? + 87.30. Step 5: Subtract 87.30 from both sides to solve for the unknown. ? = 1,274 - 87.30. ? = 1,186.70. Hence, the missing value is 1,186.70. Verification / Alternative check: To verify, substitute 1,186.70 back into the original expression. The right side becomes 1,186.70 + 87.30 = 1,274. This matches the left side value calculated as 420 + 854 = 1,274. Since both sides are equal after substitution, the value 1,186.70 satisfies the equation and is correct. Why Other Options Are Wrong: 1,096.30 and 1,124.20 would give right side totals of 1,183.60 and 1,211.50 respectively, both less than 1,274. 1,226.70 would give a total of 1,314, which is greater than 1,274. 1,274.00 would imply the other term is zero, but 87.30 is clearly present on the right side, so this cannot be correct. Common Pitfalls: Some learners misread 52.5% or 30.5% as 0.52 or 0.30 instead of 0.525 and 0.305. Others might forget to multiply by 100 when converting to decimal or misplace the decimal point. Another frequent error is to add the 87.30 instead of subtracting it when solving for the unknown term. Carefully converting percentages and keeping track of the equation structure prevents these mistakes. Final Answer: The value that should replace the question mark is 1,186.70.

Question 4

In a competitive examination, the average mark for all applicants is 60. If 20% of the applicants scored 85 marks and 25% scored 95 marks, what is the average mark of the remaining applicants?

Accepted Answer

Introduction / Context: This is a weighted average question. We are told the overall average mark of all candidates and the marks scored by two specific groups. We need to find the average for the remaining group of candidates. Such problems are very common in exam analysis, data interpretation and statistics, where we break a population into subgroups and relate their averages to the overall mean. Given Data / Assumptions: Overall average for all candidates = 60 marks. 20% of candidates scored 85 marks each. 25% of candidates scored 95 marks each. The remaining 55% scored some common average that we need to find. All candidates are counted once and marks are in the same subject or test. Concept / Approach: Let the total number of candidates be N. Then 20% of N is 0.20N, and 25% of N is 0.25N. The remaining candidates are 0.55N. We use the formula for the overall average as total marks divided by total candidates, and express total marks as the sum of contributions from each group. Then we set this equal to 60N and solve for the unknown average of the remaining 55% group. Step-by-Step Solution: Step 1: Let the total number of candidates be N. Candidates scoring 85 marks: 0.20N. Candidates scoring 95 marks: 0.25N. Remaining candidates: N - 0.20N - 0.25N = 0.55N. Let the average mark of the remaining group be x. Step 2: Compute the total marks from each group. Group 1: total marks = 0.20N * 85 = 17N. Group 2: total marks = 0.25N * 95 = 23.75N. Remaining group: total marks = 0.55N * x. Step 3: Use the overall average condition. Overall total marks = average * total candidates = 60 * N. So, 17N + 23.75N + 0.55N * x = 60N. Step 4: Combine like terms. 17N + 23.75N = 40.75N. So, 40.75N + 0.55N * x = 60N. Step 5: Subtract 40.75N from both sides. 0.55N * x = 60N - 40.75N = 19.25N. Step 6: Divide both sides by 0.55N. x = 19.25 / 0.55. x = 35. Thus, the average of the remaining applicants is 35 marks. Verification / Alternative check: As a simple check, suppose N = 100 for convenience. Then 20 candidates scored 85, 25 scored 95, and 55 scored x. Total marks from first two groups are 20 * 85 = 1,700 and 25 * 95 = 2,375, giving 4,075. If x is 35, the remaining group contributes 55 * 35 = 1,925 marks. Total marks = 4,075 + 1,925 = 6,000. Average = 6,000 / 100 = 60, which matches the given overall average. Why Other Options Are Wrong: 60 cannot be the remaining group average because that would keep the overall average at 60 only if the other groups also averaged 60, which they do not. 52 and 45 produce total marks that do not match the required overall average of 60. 55 also leads to a higher total average than 60 when combined with 85 and 95 on the given proportions. Common Pitfalls: One common mistake is to compute the simple average of 85 and 95 and ignore the overall average and proportions. Another error is to treat the percentages as additive weights without carefully converting them into the number of candidates. Some may set up the equation but forget to multiply the averages by the respective number of candidates, thereby mixing averages and totals incorrectly. Always convert averages to total marks using number of candidates and then work with those totals. Final Answer: The average marks of the remaining applicants is 35.

Question 5

In a 3-hour examination containing 150 questions, 50 questions are of type A and each type A question requires twice as much time as each type B question. How many minutes should a candidate spend on type A questions to complete the exam in the allo…

Accepted Answer

Introduction / Context: This question is about time management in an examination. We are given total available time, total number of questions and a relationship between the time required for two types of questions. The goal is to determine how much time should be devoted to one type of question. This is a typical exam-planning and average-time-per-question problem that also reinforces the idea of relative time requirements. Given Data / Assumptions: Total exam duration = 3 hours = 180 minutes. Total number of questions = 150. Number of type A questions = 50. Number of type B questions = 150 - 50 = 100. Each type A question takes twice as much time as each type B question. We assume the candidate uses the full 180 minutes efficiently. Concept / Approach: Let the time taken per type B question be t minutes. Then, by the given condition, the time taken per type A question is 2t minutes. We can express the total exam time as the sum of the time spent on all type A and type B questions in terms of t. Then equate that total to 180 minutes, solve for t, and finally compute the total time spent on all type A questions, which is 50 * 2t. Step-by-Step Solution: Step 1: Let time per type B question be t minutes. Step 2: Time per type A question is twice that, so it is 2t minutes. Step 3: Compute total time in terms of t. There are 50 type A questions, so time for them = 50 * 2t = 100t. There are 100 type B questions, so time for them = 100 * t = 100t. Step 4: Total exam time = time for type A + time for type B. Total time = 100t + 100t = 200t. Step 5: Set this equal to the given total time of 180 minutes. 200t = 180. Step 6: Solve for t. t = 180 / 200 = 0.9 minutes per type B question. Step 7: Time per type A question = 2t = 2 * 0.9 = 1.8 minutes. Step 8: Total time for all type A questions = 50 * 1.8 = 90 minutes. Verification / Alternative check: If 90 minutes are spent on type A questions, then 180 - 90 = 90 minutes remain for type B questions. With 100 type B questions, that gives 90 / 100 = 0.9 minutes for each type B question. Type A questions require twice this time: 1.8 minutes each. For 50 type A questions, this is 90 minutes, and for 100 type B questions, it is 90 minutes, summing to 180 minutes, which matches the total exam duration. This confirms our answer. Why Other Options Are Wrong: 75 minutes for type A would leave 105 minutes for type B, which would imply inconsistent time per question compared with the twice-as-much condition. 82 minutes or 101 minutes also fail to satisfy both the total time and the exact 2:1 time ratio between question types. 60 minutes for type A would leave 120 minutes for type B, making type B questions much longer than allowed by the 2:1 condition. Common Pitfalls: Students often try to assign arbitrary times to each question without using the ratio information correctly. Another mistake is to divide total time by total number of questions directly and ignore the fact that type A questions take double the time. Some may also misinterpret twice as much as adding a fixed amount instead of multiplying. Always set a variable for the smaller time and then use the ratio to express the larger time before writing the total time equation. Final Answer: The candidate should spend 90 minutes on type A questions.

Question 6

Two numbers are respectively 35% and 42% less than a third number. By what percentage is the second number less than the first number?

Accepted Answer

Introduction / Context: This question is about comparing two numbers that are both defined relative to a common base number. Each number is given as a certain percentage less than a third number. We are asked to find how much smaller the second number is compared with the first one, expressed as a percentage of the first. This type of problem is common in percentage comparison and relative change topics. Given Data / Assumptions: Let the third number be N. First number is 35% less than N. Second number is 42% less than N. We need the percentage by which the second number is less than the first number. All percentages are simple and based on the same third number N. Concept / Approach: We express both numbers in terms of N, then compare them. If the first number is 35% less than N, it is 65% of N. If the second number is 42% less than N, it is 58% of N. Once both are expressed as multiples of N, we find the difference between them and express this difference as a percentage of the first number. The key formula is: percentage difference = (difference / reference) * 100 where the reference here is the first number. Step-by-Step Solution: Step 1: Express the first number in terms of N. 35% less than N means it is 65% of N. First number = 65% of N = 0.65N. Step 2: Express the second number in terms of N. 42% less than N means it is 58% of N. Second number = 58% of N = 0.58N. Step 3: Find the absolute difference between the first and second numbers. Difference = 0.65N - 0.58N = 0.07N. Step 4: Express this difference as a percentage of the first number. Percentage by which second is less than first = (difference / first) * 100. = (0.07N / 0.65N) * 100. Step 5: Cancel N from numerator and denominator. = (0.07 / 0.65) * 100. Step 6: Compute 0.07 / 0.65. 0.07 / 0.65 ≈ 0.1076923. Step 7: Multiply by 100 to convert to percent. 0.1076923 * 100 ≈ 10.769%. Thus, the second number is approximately 10.769% less than the first number. Verification / Alternative check: As a check, we can choose a convenient value for N, say 100. Then the first number is 65 and the second is 58. The difference is 65 - 58 = 7. Now compute the percentage of 7 relative to 65: (7 / 65) * 100 ≈ 10.769%. This matches the earlier result and confirms the calculation. Why Other Options Are Wrong: Values like 11.23% or 10.4% are close but not equal to the exact fraction 7 / 65 when converted to percentage. 9.87% and 8.5% are clearly smaller than the correct value around 10.769%, and they correspond to different ratios than the one produced by 65 and 58. Common Pitfalls: A common mistake is to subtract the percentages 42% and 35% directly and report 7% as the answer, without referencing the correct base. The 7% represents a difference relative to the third number N, not relative to the first number. Another error is to express the difference as a percentage of the second number instead of the first, which would give a slightly different value. Always pay close attention to which number is being used as the reference when expressing a percentage difference. Final Answer: The second number is approximately 10.769% less than the first number.

Question 7

Express 6.72% as a simplified ratio.

Accepted Answer

Introduction / Context: This question asks to convert a percentage into a ratio. Conversions between percentages, fractions and ratios are very common in quantitative aptitude, as they allow a value to be interpreted in different forms depending on the problem. Expressing a percentage as a ratio often requires writing it as a fraction and then simplifying to lowest terms. Given Data / Assumptions: We are given the percentage 6.72%. We need to express 6.72% as a ratio of two integers in simplest form. Ratios are usually written as a : b, where a and b are whole numbers with no common factor other than 1. Concept / Approach: First convert the percentage to a fraction over 100. Then remove the decimal by multiplying numerator and denominator by an appropriate power of 10. After that, simplify the resulting fraction by dividing numerator and denominator by their greatest common divisor. Finally, interpret the simplified fraction as a ratio in the form a : b. Step-by-Step Solution: Step 1: Write 6.72% as a fraction. 6.72% = 6.72 / 100. Step 2: Remove the decimal by multiplying numerator and denominator by 100. 6.72 / 100 = 672 / 10,000. Step 3: Simplify the fraction 672 / 10,000. First, divide both numerator and denominator by 16. 672 / 16 = 42, and 10,000 / 16 = 625. So 672 / 10,000 simplifies to 42 / 625. Step 4: Check if 42 / 625 can be reduced further. 42 factors into 2 * 3 * 7. 625 factors into 5 * 5 * 5 * 5. There is no common factor except 1, so the fraction is in simplest form. Step 5: Write the simplified fraction as a ratio. 42 / 625 corresponds to the ratio 42 : 625. Verification / Alternative check: To verify, convert the ratio 42 : 625 back into a percentage. First interpret it as the fraction 42 / 625. Compute 42 / 625, which equals 0.0672. Now convert this decimal to a percentage by multiplying by 100: 0.0672 * 100 = 6.72%. This matches the original percentage given in the question, confirming that the ratio 42 : 625 is correct. Why Other Options Are Wrong: Ratios like 65 : 124 or 21 : 125 correspond to different fractions and therefore different percentages when converted back. 37 : 442 and 168 : 2500 also simplify to values that are not equal to 0.0672 and hence do not represent 6.72%. Only 42 : 625 converts exactly back to 6.72%. Common Pitfalls: Learners sometimes treat 6.72% as 0.672 or 6.72 directly without dividing by 100, which gives incorrect fractions. Another error is to stop simplifying too early, leaving the fraction in non-simplest form. Some may also incorrectly cancel digits instead of factors when simplifying. Always convert the percentage to a fraction over 100, clear the decimals properly, and then systematically reduce numerator and denominator by their greatest common divisor. Final Answer: 6.72% expressed as a ratio is 42 : 625.

Question 8

What percentage is equivalent to the fraction 1/2?

Accepted Answer

Introduction / Context: Converting a fraction to a percentage is a fundamental skill in mathematics. Fractions, decimals and percentages are different ways of expressing the same part-whole relationships. The fraction 1/2 is one of the most common fractions, and knowing its percentage form is useful in many everyday situations, including discounts, scores and probability problems. Given Data / Assumptions: We are given the fraction 1 / 2. We need to express 1 / 2 as a percentage. All calculations use the standard relationship between fractions, decimals and percentages. Concept / Approach: To convert a fraction into a percentage, we first convert the fraction to a decimal by performing the division, and then multiply the result by 100 to obtain the equivalent percent. In general: percentage = (fraction value) * 100 For 1/2, this approach is very direct because 1 divided by 2 is a well known decimal. Step-by-Step Solution: Step 1: Convert 1 / 2 to decimal form. 1 / 2 = 0.5. Step 2: Convert the decimal 0.5 into a percentage. Percentage = 0.5 * 100. Step 3: Multiply 0.5 by 100. 0.5 * 100 = 50. Step 4: Attach the percent symbol. So 1 / 2 = 50%. Verification / Alternative check: We can also think conceptually. The fraction 1 / 2 represents half of a whole. Half of 100% is 50%, so 1 / 2 must correspond to 50%. Another check is to convert the percentage back to a fraction: 50% = 50 / 100. Simplifying 50 / 100 by dividing numerator and denominator by 50 gives 1 / 2, which matches the original fraction. Why Other Options Are Wrong: 5% corresponds to 5 / 100 = 1 / 20, a much smaller fraction than 1 / 2. 0.5% is 0.5 / 100 = 1 / 200, which is even smaller. 0.05% is 0.05 / 100 = 1 / 2000, extremely small compared to 1 / 2. 25% equals 25 / 100 = 1 / 4, which is half of the required fraction. Common Pitfalls: Some learners confuse 0.5 with 0.5%, forgetting that a percent already includes division by 100. Others may incorrectly divide 2 by 1 instead of 1 by 2 when converting to decimal, which reverses the fraction. It is also possible to mistakenly multiply 1/2 directly by 100 without first interpreting the fraction as 0.5, though in this particular case that still gives 50, but this habit may cause errors in more complex fractions. Keeping the general rule clear helps avoid such confusion. Final Answer: The fraction 1/2 is equivalent to 50%.

Question 9

Rs. 69 is divided among 115 family members so that each lady receives 50 paise less than each gentleman, and each gentleman receives twice as many paise as each lady. How many gentlemen are there in the family?

Accepted Answer

Introduction / Context: This question combines basic algebra with a real life style distribution scenario. The total money is given, along with conditions relating how much each gentleman and each lady receives. We need to determine how many gentlemen are in the family. The problem is solved by converting rupees to paise, setting up equations based on the payment conditions and the total number of members, and then solving for the unknown counts. Given Data / Assumptions: Total money to be distributed = Rs. 69. Total family members = 115. Each gentleman receives twice as many paise as each lady. Each lady receives 50 paise less than each gentleman. We are asked to find the number of gentlemen. All calculations are easier if we work in paise instead of rupees. Concept / Approach: First convert 69 rupees into paise to avoid decimals: 1 rupee = 100 paise, so 69 rupees = 6,900 paise. Let the amount received by each lady be L paise and by each gentleman be G paise. Then G is given as twice L, and at the same time, G is also 50 paise more than L. Using these relationships, we find L and G. Then we use the total amount equation in terms of the number of ladies and gentlemen and the total count of 115 to solve for how many gentlemen there are. Step-by-Step Solution: Step 1: Convert total amount to paise. Total money = 69 rupees = 69 * 100 = 6,900 paise. Step 2: Let each lady receive L paise, each gentleman receive G paise. Given: Each gentleman receives twice as many paise as each lady, so G = 2L. Also given: Each lady receives 50 paise less than each gentleman, so G - L = 50. Step 3: Use G = 2L in G - L = 50. 2L - L = 50. So L = 50 paise. Then G = 2L = 2 * 50 = 100 paise. Step 4: Let the number of gentlemen be g and the number of ladies be l. Total members: g + l = 115. Total money in paise: 100g + 50l = 6,900. Step 5: Use l = 115 - g in the money equation. 100g + 50(115 - g) = 6,900. 100g + 5,750 - 50g = 6,900. (100g - 50g) + 5,750 = 6,900. 50g + 5,750 = 6,900. Step 6: Subtract 5,750 from both sides. 50g = 1,150. g = 1,150 / 50 = 23. So there are 23 gentlemen in the family. Verification / Alternative check: If there are 23 gentlemen, then the number of ladies is 115 - 23 = 92. Total money given to gentlemen is 23 * 100 = 2,300 paise. Total money given to ladies is 92 * 50 = 4,600 paise. Combined, this is 2,300 + 4,600 = 6,900 paise, which equals 69 rupees. This matches the original total, so the distribution is consistent and the count of 23 gentlemen is correct. Why Other Options Are Wrong: If there were 92 gentlemen instead, the number of ladies would be 23, and the total payout would not equal 6,900 paise. Similarly, choices like 48, 52 or 27 for the count of gentlemen do not satisfy both the total members equation and the total money equation when tested. Common Pitfalls: Some learners forget to convert rupees to paise and work with decimals incorrectly. Others misinterpret the phrase that each gentleman receives twice as many paise as each lady and may set up the relationships incorrectly. It is also easy to mix up the conditions about who gets more and by how much. Carefully setting up both the value equations (G = 2L and G - L = 50) and the count equations ensures a correct and straightforward solution. Final Answer: The number of gentlemen in the family is 23.

Question 10

The number 3 is what percent of 5?

Accepted Answer

Introduction / Context: This is a simple percentage comparison problem. We are asked to find what percent one number (3) is of another number (5). Understanding this basic conversion helps in many areas such as ratio comparison, test scores, and everyday percentage calculations like discount and success rates. Given Data / Assumptions: Part (numerator) = 3. Whole (denominator) = 5. We need 3 as a percentage of 5. We use the standard percent formula without any approximations because the numbers are small and exact. Concept / Approach: To find what percent a number is of another, we use the formula: percentage = (part / whole) * 100 Here, the part is 3 and the whole is 5. We first compute the fraction 3 / 5 and then multiply by 100 to obtain the percentage. Since 3 and 5 are small, the fraction and percentage are straightforward. Step-by-Step Solution: Step 1: Form the fraction of part to whole: 3 / 5. Step 2: Convert this fraction to percentage using the formula. percentage = (3 / 5) * 100. Step 3: Compute 3 / 5. 3 / 5 = 0.6. Step 4: Multiply 0.6 by 100. 0.6 * 100 = 60. Step 5: Attach the percent symbol. So 3 is 60% of 5. Verification / Alternative check: We can also think in reverse. If 60% of 5 is computed, does it give 3? Calculate (60 / 100) * 5 = 0.6 * 5 = 3. This matches the original part, confirming that 60% is the correct percentage. Another way is to see that 50% of 5 is 2.5 and an extra 10% of 5 is 0.5, giving 3 in total, which corresponds to 60%. Why Other Options Are Wrong: 45% of 5 is (45 / 100) * 5 = 2.25, which is less than 3. 72% of 5 equals 3.6, which is more than 3. 80% of 5 equals 4, also more than 3. 55% of 5 equals 2.75, still less than 3. Common Pitfalls: A frequent mistake is to invert the fraction and compute 5 / 3 instead of 3 / 5, which gives a value greater than 1 and leads to a percentage above 100%. Another error is to forget to multiply by 100 after computing the fraction. Some learners may also round too early when dealing with more complicated fractions, but in this simple case 3 / 5 is exactly 0.6, so rounding is not necessary. Keeping the formula clear and applying it in the correct order avoids these issues. Final Answer: The number 3 is 60% of 5.

Question 11

In a class there are 80 students and 5 teachers. Each student receives sweets equal to 15% of the total number of students and each teacher receives sweets equal to 25% of the total number of students. How many sweets are distributed in total?

Accepted Answer

Introduction / Context:This percentage based aptitude question checks whether you can correctly convert percentage information into actual quantities and then aggregate them. The situation describes a school with students and teachers, where sweets are distributed according to a rule based on the total number of students. Such questions are very common in quantitative aptitude tests, banking exams and campus recruitment tests, because they test conceptual clarity of percentages and careful reading of the conditions.Given Data / Assumptions:There are 80 students in the class.There are 5 teachers in the class.Each student gets sweets equal to 15% of the total number of students.Each teacher gets sweets equal to 25% of the total number of students.The total number of sweets distributed equals sweets given to all students plus sweets given to all teachers.Concept / Approach:The key idea is that the number of sweets per person is defined as a percentage of the total number of students, not of the total number of people. First we convert the given percentages into actual counts of sweets for one student and for one teacher. Then we multiply by the number of students and teachers respectively. Finally, we add both results to obtain the overall total. This uses the basic percentage formula value = (percentage / 100) * base.Step-by-Step Solution:Total number of students = 80.Total number of teachers = 5.Sweets given to one student = 15% of 80 = (15 / 100) * 80 = 12 sweets.Sweets given to one teacher = 25% of 80 = (25 / 100) * 80 = 20 sweets.Total sweets for all students = 80 * 12 = 960 sweets.Total sweets for all teachers = 5 * 20 = 100 sweets.Total sweets distributed = 960 + 100 = 1060 sweets.Verification / Alternative check:We can quickly check the reasonableness of the answer. Since each student gets 12 sweets and there are 80 students, that part alone is 960 sweets. Teachers each receive more sweets (20), but there are only 5 of them, so that adds 100 sweets. The overall total is in the same magnitude as 1000, which is reasonable for these numbers. No arithmetic step gives a fractional sweet, so the result 1060 is consistent and clean.Why Other Options Are Wrong:960 ignores sweets given to teachers and counts only student sweets, so it is incomplete.860 is lower than the student total itself and cannot be correct once teacher sweets are added.760 also ignores a large part of the distribution and does not match any logical combination of counts.1000 is a rounded guess and does not come from the exact calculations based on 15% and 25% of 80.Common Pitfalls:A very common mistake is to assume that the percentages are taken on the total number of people (students plus teachers) instead of only on the number of students. Another error is to miscalculate 15% or 25% of 80 or to forget to multiply by the number of persons in each group. Some learners also stop after calculating the sweets for students and forget to include teachers, leading to an underestimation.Final Answer:Using correct percentage calculations and adding sweets for both students and teachers, the total number of sweets distributed is 1060.

Question 12

Of the total length of a pencil, one eighth is black, one half of the remaining length is white, and the remaining 3.5 cm is blue. What is the total length of the pencil?

Accepted Answer

Introduction / Context:This question is a classic example of fraction and percentage style reasoning in quantitative aptitude. The pencil is divided into three coloured segments: black, white and blue. The fractions for the first two parts are given in terms of the total or remaining length, while the last part is given as an absolute length in centimetres. The goal is to translate these verbal conditions into equations and solve for the total length of the pencil.Given Data / Assumptions:Let the total length of the pencil be L cm.Black portion = 1 / 8 of the total length L.White portion = 1 / 2 of the remaining length after the black part is removed.Blue portion equals the remaining length after black and white parts and is given to be 3.5 cm.All parts together add up to the total length L.Concept / Approach:The idea is to apply fractional operations step by step. First we remove 1 / 8 of the total for the black part. Then from what is left, we remove half as the white part. Whatever still remains is the blue segment, whose length is given numerically. By expressing the remaining fractions of L after each stage, we can equate the last part to 3.5 cm and solve for L using basic algebra with fractions.Step-by-Step Solution:Let total length = L cm.Black part = L / 8.Length remaining after black part = L - L / 8 = (7L) / 8.White part = 1 / 2 of the remaining length = (1 / 2) * (7L / 8) = 7L / 16.Remaining length after black and white parts = (7L / 8) - (7L / 16) = 7L / 16.This remaining length is the blue part, which equals 3.5 cm.Therefore 7L / 16 = 3.5.So L = 3.5 * 16 / 7 = 8 cm.Verification / Alternative check:Check the result by reconstructing the pencil lengths. For L = 8 cm: black part = 8 / 8 = 1 cm, remaining = 7 cm. White part = half of 7 cm = 3.5 cm, remaining = 3.5 cm. This final remaining part is the blue section, which matches the given 3.5 cm. The sum of all parts is 1 + 3.5 + 3.5 = 8 cm, which is consistent with the assumed total length.Why Other Options Are Wrong:7 cm does not satisfy the condition; plugging L = 7 gives a different blue length than 3.5 cm.9 cm produces fractional parts that do not match the given final blue portion of 3.5 cm.11 cm likewise leads to a final remaining segment that is not equal to 3.5 cm.10 cm also fails when the step by step fraction calculations are applied, so it cannot be the correct total length.Common Pitfalls:Learners often misinterpret phrases like one half of the remaining as one half of the total, which completely changes the fractions. Another common error is to forget that the blue part corresponds to the remaining length only after both black and white parts are removed. Some students also confuse 3.5 with 3 / 5 or interpret 3.5 as 3 / 2.2, which is incorrect. Careful reading and clear algebraic representation prevent these mistakes.Final Answer:After correctly applying the fractional conditions to the coloured parts of the pencil, the total length of the pencil is 8 cm.

Question 13

A man reduces his travelling time by a fraction of 2/5, and as a result the distance covered by him increases by 20%. By what percentage does the speed of the man increase?

Accepted Answer

Introduction / Context:This problem combines the core concepts of speed, distance and time with percentage changes. The question states that a man reduces his travel time and simultaneously increases the distance he covers. The task is to determine the resulting percentage increase in speed. Such questions are common in aptitude tests because they check understanding of the proportional relationship between speed, distance and time, as well as the correct handling of successive percentage changes.Given Data / Assumptions:Let the initial time taken be T hours.Let the initial distance travelled be D kilometres (or any unit).Initial speed v = D / T.Travelling time is reduced by a fraction of 2 / 5, so new time = (1 - 2 / 5) * T = 3T / 5.Distance travelled increases by 20%, so new distance = 1.2D.New speed v new = new distance / new time.Concept / Approach:The key relationship is speed = distance / time. When time and distance change together, speed changes according to their combined effect. We express the new time and new distance as multiples of the original values, then form the ratio new speed divided by old speed. Finally we convert this ratio into a percentage increase. Handling the fraction 2 / 5 and 20% carefully is crucial for arriving at the correct result.Step-by-Step Solution:Initial speed v = D / T.New time = 3T / 5 (because time is reduced by 2 / 5 of T).New distance = 1.2D (a 20% increase over D).New speed v new = (1.2D) / (3T / 5).Simplify: v new = 1.2D * (5 / 3T) = (1.2 * 5 / 3) * (D / T).Compute the numeric factor: 1.2 * 5 = 6, and 6 / 3 = 2.Therefore v new = 2 * (D / T) = 2v.So the new speed is twice the old speed, which is a 100% increase.Verification / Alternative check:We can test with simple numbers. Suppose initially D = 100 km and T = 5 hours, so v = 20 km per hour. New time = 3 / 5 of 5 = 3 hours. New distance = 1.2 * 100 = 120 km. New speed = 120 / 3 = 40 km per hour. Since 40 is exactly double 20, the speed has increased by 20 km per hour, that is 100%. This numerical example confirms our algebraic result.Why Other Options Are Wrong:77% and 88% correspond to incorrect combinations of the percentage changes and usually come from partial or incorrect multiplication of factors.99% is a near miss and often appears when rounding mistakes are made, but the exact calculation shows a clean doubling of speed, not a 99% increase.60% does not match any logical calculation from the given data and reflects a misunderstanding of how speed depends on both distance and time.Common Pitfalls:Many learners subtract or add percentages directly, or they treat time and distance changes as if they affect speed independently. Another mistake is to apply the 2 / 5 reduction as 2 / 5 of the new time instead of the original time. Forgetting that a reduction in time increases speed and an increase in distance also increases speed can lead to sign errors. Always express new quantities as clear multiples of the old values before forming the speed ratio.Final Answer:The new speed is twice the original speed, so the percentage increase in the speed of the man is 100%.

Question 14

The price of a car is increased by 25%. By what percentage must this new price be reduced to restore the original price of the car?

Accepted Answer

Introduction / Context:This question illustrates a very important concept in percentages: an increase of a certain percentage followed by a decrease of the same percentage does not bring a value back to its original amount. Instead, the required decrease must be calculated from the increased value. Here, the price of a car first goes up by 25%, and we are asked to find the percentage decrease needed to return to the original price. This idea appears frequently in discount, profit and loss, and price revision problems.Given Data / Assumptions:Let the original price of the car be P rupees.The price is increased by 25%, so the new price becomes 1.25P.We want to find a percentage decrease x% applied to the new price that will bring it back to P.After decreasing, the price should again equal P.Concept / Approach:When a quantity is increased by a percentage and then decreased, the base for the second operation is the new value, not the original value. Therefore, we must set up an equation where the new price after decrease equals the original price. Algebraically, we multiply the increased price by (1 - x / 100) and set the result equal to the original price P. Solving this equation gives the required percentage decrease. This method generalizes to any similar successive percentage change problem.Step-by-Step Solution:Let original price = P.After a 25% increase, new price = P * (1 + 25 / 100) = 1.25P.Let the required percentage decrease be x%.After decreasing, price = 1.25P * (1 - x / 100).This must equal the original price P.So, 1.25P * (1 - x / 100) = P.Divide both sides by P: 1.25 * (1 - x / 100) = 1.Therefore, 1 - x / 100 = 1 / 1.25 = 0.8.So x / 100 = 1 - 0.8 = 0.2.Hence x = 0.2 * 100 = 20%.Verification / Alternative check:Take a simple assumed original price, for example P = 100 rupees. After a 25% increase, the new price becomes 125 rupees. If we now decrease 125 by 20%, we subtract 20% of 125 which is 25 rupees. So the new price after reduction is 125 - 25 = 100 rupees, exactly back to the original price. This confirms that the required decrease is 20% and not 25%.Why Other Options Are Wrong:24% and 21% would produce final prices different from the original and so do not restore the price exactly.25% decrease from the increased price of 125 would give 93.75, which is less than the original 100 and therefore incorrect.18% also fails to bring the price back to the original level and arises from incorrect algebra or mental approximation.Common Pitfalls:A very common misconception is to think that a 25% increase followed by a 25% decrease returns to the starting value. This is not true because the base values for the percentage operations are different. Another pitfall is performing calculations directly on percentages without converting them to multiplication factors. Always work with the idea that new value = old value * (1 ± percentage / 100) and carefully set up the equation.Final Answer:To restore the original price of the car after a 25% increase, the new price must be reduced by 20%.

Question 15

What is the value of 40% of 80?

Accepted Answer

Introduction / Context:This is a straightforward question on basic percentage calculation. Although it looks very simple, such questions appear frequently in the beginning section of aptitude tests to check whether candidates are comfortable converting percentages into numerical values. Mastery of these fundamentals is essential for tackling more advanced questions that combine multiple percentage operations.Given Data / Assumptions:The base quantity is 80.We need to find 40% of 80.Percent literally means per hundred, so 40% means 40 out of every 100 parts.No additional conditions or hidden steps are involved.Concept / Approach:The general formula for finding a percentage of a number is value = (percentage / 100) * base. Here, percentage = 40 and base = 80. There are two convenient ways to evaluate this: either apply the formula directly, or first convert 40% into its fractional form 40 / 100 = 2 / 5 and then multiply this fraction by 80. Both methods arrive at the same numeric result and reinforce understanding of how percentages relate to fractions.Step-by-Step Solution:We want 40% of 80.Using the percentage formula: value = (40 / 100) * 80.Compute 40 / 100 = 0.4.So value = 0.4 * 80.0.4 * 80 = 32.Therefore, 40% of 80 is 32.Verification / Alternative check:Convert 40% into fraction form. 40% = 40 / 100 = 2 / 5. Now compute (2 / 5) of 80. First divide 80 by 5 to get 16. Then multiply 16 by 2 to get 32. This again confirms that 40% of 80 equals 32. The agreement of both decimal and fractional methods gives confidence in the result and the calculation steps.Why Other Options Are Wrong:36 results from incorrect multiplication or mistakenly taking a different percentage of 80.42 may come from mixing up numbers or adding instead of multiplying.44 does not correspond to 40% of 80 and is simply a distractor.30 is a rounded guess that ignores the exact definition of 40%.Common Pitfalls:Some learners confuse 40% with 4% or 0.4% and misplace the decimal point. Others may attempt to take 40% of a wrong base due to reading errors. A few students also try to add 40 directly to 80 instead of multiplying by 40 / 100. To avoid these mistakes, always remember that percent is a multiplication factor equal to percentage divided by 100, and apply this consistently.Final Answer:The value of 40% of 80 is 32.

Question 16

What is the value of 5% of 40,000?

Accepted Answer

Introduction / Context:This question checks your comfort level with calculating a small percentage of a relatively large number. Finding 5% of a value is an extremely common operation in real life situations such as discounts, commissions, interest and tax calculations. A clear, quick method for this kind of problem is very useful during competitive exams where time is limited.Given Data / Assumptions:The base amount is 40,000.We must find 5% of 40,000.No additional charges, changes or extra conditions are involved.The answer should be a single exact number.Concept / Approach:Conceptually, 5% means 5 out of every 100 parts. The general formula is value = (percentage / 100) * base. Another fast mental method for 5% is to first find 10% of the number (which is simple) and then take half of that, because 5% is half of 10%. Both methods are correct and lead to the same result. Practising both builds flexibility and speed in percentage problems.Step-by-Step Solution:We want 5% of 40,000.Using the formula: value = (5 / 100) * 40,000.Compute 5 / 100 = 0.05.So value = 0.05 * 40,000.0.05 * 40,000 = 2,000.Therefore, 5% of 40,000 is 2,000.Verification / Alternative check:Using the 10% method: First find 10% of 40,000. Ten percent of any number is that number divided by 10, so 10% of 40,000 is 4,000. Now, since 5% is half of 10%, take half of 4,000. Half of 4,000 is 2,000. This again confirms that 5% of 40,000 equals 2,000. Agreement of both methods verifies the correctness of the result.Why Other Options Are Wrong:20,000 corresponds to 50% of 40,000, not 5%, so it is ten times too large.200 corresponds to only 0.5% of 40,000 and is much smaller than the correct value.20 is extremely small and corresponds to 0.05% of 40,000.4,000 is 10% of 40,000, not 5%, so it is double the correct answer.Common Pitfalls:A common error is misplacing the decimal point when converting percent to a decimal. Some learners treat 5% as 5 instead of 0.05 and thus multiply incorrectly. Others confuse 5% with 50% or 0.5%, leading to large magnitude errors. To avoid these mistakes, remember that you always divide the percentage by 100 to convert it into a factor for multiplication.Final Answer:The value of 5% of 40,000 is 2000.

Question 17

Express the decimal number 0.125 as a percentage.

Accepted Answer

Introduction / Context:This question tests the basic but important skill of converting a decimal number into a percentage. Being able to move comfortably between fractions, decimals and percentages is fundamental in quantitative aptitude, finance, data interpretation and many everyday calculations. Here, the decimal 0.125 must be written in percentage form.Given Data / Assumptions:The decimal given is 0.125.We need to express this value as an equivalent percentage.No rounding is required because 0.125 converts neatly to a terminating percentage.The percentage symbol % will be included in the final answer.Concept / Approach:To convert a decimal to a percentage, we use the rule: percentage = decimal * 100. This is because percent literally means per hundred. Another way is to first convert the decimal into a fraction and then into a percentage. For 0.125, it is easy to recognize that it equals 1 / 8. The fraction 1 / 8 can then be converted to a percentage by multiplying by 100. Both approaches lead to the same numerical result and strengthen the understanding of how decimals, fractions and percentages relate.Step-by-Step Solution:We are given the decimal 0.125.Method 1 (direct decimal to percent): Percentage = 0.125 * 100.0.125 * 100 = 12.5.So the required percentage is 12.5%.Method 2 (via fraction): 0.125 = 125 / 1000.Simplify 125 / 1000 by dividing numerator and denominator by 125 to get 1 / 8.Now multiply by 100 to convert to percent: (1 / 8) * 100 = 12.5.Therefore, the decimal 0.125 equals 12.5%.Verification / Alternative check:We can check the consistency by reversing the process. Take 12.5% and convert it back to decimal. First write 12.5% as 12.5 / 100. Now 12.5 / 100 = 0.125. This matches the original decimal given in the problem, so our conversion is confirmed to be correct. This forward and backward check is a good habit for exam practice.Why Other Options Are Wrong:12% is slightly smaller than 12.5% and corresponds to 0.12, not 0.125.13% equals 0.13 as a decimal and therefore is larger than 0.125.13.5% equals 0.135 and again does not match 0.125.1.25% equals 0.0125, which is ten times smaller than 0.125.Common Pitfalls:Some students forget to multiply by 100 and instead divide by 100, ending up with 0.00125. Others may misplace the decimal point and write 1.25% instead of 12.5%. Another pitfall is rounding too early or incorrectly simplifying the fraction representation. To avoid these errors, always follow the simple rule: to convert a decimal to a percentage, multiply by 100 and attach the percent symbol at the end.Final Answer:The decimal number 0.125 expressed as a percentage is 12.5%.

Question 18

Compute the value of 40% of 265 plus 35% of 180, and find the number whose 50% equals this sum.

Accepted Answer

Introduction / Context:This question involves a combination of percentage calculations and an equation. We are asked to find 40% of one quantity and 35% of another, add them, and then determine the number whose 50% (half) is equal to this combined value. Such problems test the ability to perform sequential percentage calculations and to set up and solve a simple linear equation in one variable.Given Data / Assumptions:We need to compute 40% of 265.We also need to compute 35% of 180.The sum of these two results equals 50% of an unknown number X.We must find the value of this unknown number X.Concept / Approach:The approach is to handle the problem in two phases. First calculate the numeric value of each percentage using the formula value = (percentage / 100) * base. Then add the two values together. Second, interpret the equation 0.5 * X = sum and solve for X by dividing the sum by 0.5. This is equivalent to multiplying the sum by 2. Careful arithmetic is important to avoid simple calculation mistakes.Step-by-Step Solution:Step 1: Find 40% of 265.40% of 265 = (40 / 100) * 265 = 0.4 * 265.0.4 * 265 = 106.Step 2: Find 35% of 180.35% of 180 = (35 / 100) * 180 = 0.35 * 180.0.35 * 180 = 63.Step 3: Add both results.Sum = 106 + 63 = 169.Step 4: Let the required number be X, such that 50% of X equals 169.50% of X = (50 / 100) * X = 0.5 * X.So, 0.5 * X = 169.Therefore, X = 169 / 0.5 = 338.Verification / Alternative check:We can check by working backward. Take X = 338. Then 50% of 338 is (50 / 100) * 338 = 0.5 * 338 = 169. We already calculated that 40% of 265 plus 35% of 180 equals 169. Since both sides match, X = 338 is confirmed to be correct. This double check reduces the chance of having carried an arithmetic error unnoticed.Why Other Options Are Wrong:383, 253.5 and 300 are all inconsistent with the equation 0.5 * X = 169, because half of these numbers is not 169.84.5 is actually half of 169 and represents 25% of 338, not 338 itself.Any value other than 338 fails to satisfy the relationship that 50% of X must equal the computed sum of 169.Common Pitfalls:Some learners mistakenly add the percentages before applying them, or they may take 40% and 35% of the same number. Others compute the sum correctly but incorrectly solve 0.5 * X = 169 by subtracting or using the wrong inverse operation. Always remember that when a coefficient multiplies a variable, solving for the variable requires dividing by that coefficient. In this case, divide by 0.5, which is the same as multiplying by 2.Final Answer:The number whose 50% is equal to 40% of 265 plus 35% of 180 is 338.

Question 19

In a municipality election, two candidates contest. One of them receives 54% of the total votes polled and wins by a margin of 1080 votes. What is the total number of votes polled in the election?

Accepted Answer

Introduction / Context:This problem is a standard election percentage question. There are two candidates, one of whom secures a known percentage of the total votes. The margin of victory in terms of votes is also known. We are asked to find the total number of votes polled. Questions of this type are common in entrance exams and competitive tests, as they combine percentage understanding with basic algebra.Given Data / Assumptions:Only two candidates contest the election.The winning candidate gets 54% of the total votes.The losing candidate therefore gets the remaining 46% of the total votes.The margin of victory, that is the difference between their vote counts, is 1080 votes.We need to find the total number of votes polled.Concept / Approach:If the total number of votes is V, then the winner receives 0.54V votes and the loser receives 0.46V votes. The difference between these two quantities equals the given margin of 1080 votes. So we form the equation (0.54V - 0.46V) = 1080. The left side simplifies to a certain percentage of V, and solving this equation gives the total number of votes. This direct approach avoids unnecessary steps and is very efficient.Step-by-Step Solution:Let the total number of votes polled be V.Votes for winner = 54% of V = 0.54V.Votes for loser = 46% of V = 0.46V.Margin of victory = votes for winner minus votes for loser.So, margin = 0.54V - 0.46V = 0.08V.We are given that margin = 1080 votes.Therefore, 0.08V = 1080.So V = 1080 / 0.08.Compute 1080 / 0.08 = 1080 * (100 / 8) / 100 = 13500.Thus, the total number of votes polled is 13,500.Verification / Alternative check:Check by computing the individual vote counts. With V = 13,500, winner's votes = 54% of 13,500 = 0.54 * 13,500 = 7,290. Loser's votes = 46% of 13,500 = 0.46 * 13,500 = 6,210. The difference is 7,290 - 6,210 = 1,080 votes, which matches the given margin. Therefore, V = 13,500 is correct.Why Other Options Are Wrong:12,000 would give a margin of 8% of 12,000, which is 960, not 1,080.15,400 would give a margin of 8% of 15,400, which is 1,232, not 1,080.11,000 would give a margin of 8% of 11,000, which is 880, also incorrect.14,500 would give a margin of 8% of 14,500, equal to 1,160, again not matching 1,080.Common Pitfalls:Some students mistakenly equate 54% directly to 1080 votes and then scale incorrectly. Others forget that the margin corresponds to the difference between the two percentages, that is 54% minus 46% = 8%. Another error is to miscalculate 8% of a number when working backwards. Always clearly define the total as V, express both candidates' vote counts in terms of V, and set their difference equal to the given margin.Final Answer:The total number of votes polled in the election is 13500.

Question 20

In an election with only two candidates, a candidate secures 70% of the valid votes and wins by a majority of 172 votes. What is the total number of valid votes?

Accepted Answer

Introduction / Context:This is another election based percentage question, focusing on valid votes. Two candidates contest, and one candidate receives 70% of the valid votes. The margin of victory in votes is given, and we are asked to find the total number of valid votes. This tests the understanding that the sum of both candidates' vote shares is 100% and that the majority is the difference between their vote counts.Given Data / Assumptions:There are exactly two candidates in the election.The winning candidate secures 70% of the valid votes.The losing candidate receives the remaining 30% of the valid votes.The majority (difference between their vote counts) is 172 votes.We must determine the total number of valid votes.Concept / Approach:If the total valid votes are V, the winner gets 0.70V and the loser gets 0.30V. The majority is the difference 0.70V - 0.30V = 0.40V. This difference is equal to the given margin of 172 votes. So we set up the equation 0.40V = 172 and solve for V. This is a direct proportionality relationship, very similar in structure to other percentage difference problems.Step-by-Step Solution:Let the total number of valid votes be V.Votes for winner = 70% of V = 0.70V.Votes for loser = 30% of V = 0.30V.Majority = votes for winner minus votes for loser.Majority = 0.70V - 0.30V = 0.40V.Given majority = 172 votes.Hence 0.40V = 172.So V = 172 / 0.40.Compute: 172 / 0.40 = 1720 / 4 = 430.Thus, total number of valid votes is 430.Verification / Alternative check:Let us verify with V = 430. Winner's votes = 70% of 430 = 0.70 * 430 = 301. Loser's votes = 30% of 430 = 0.30 * 430 = 129. The difference between them is 301 - 129 = 172 votes, which exactly matches the given majority. Therefore, 430 valid votes is the correct total.Why Other Options Are Wrong:446 would give majority = 40% of 446 = 178.4, which is not an integer and not equal to 172.415 would give majority = 40% of 415 = 166, different from the required margin.404 would give majority = 40% of 404 = 161.6, again not 172.400 would give majority = 40% of 400 = 160, which is also incorrect.Common Pitfalls:Some learners mistakenly calculate 70% of V as the majority itself instead of the winner's total. Others forget that the loser receives 30% and do not compute the difference correctly. A few may also misinterpret the majority as 70% minus 50% or make arithmetic errors while dividing 172 by 0.40. Always express both candidates' vote counts as percentages of the same base and then form a difference equation with the given majority.Final Answer:The total number of valid votes cast in the election is 430.

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