Question 1

A student was supposed to multiply a number by 13/6 but instead multiplied it by 6/13. What is the percentage error in the result obtained, compared to the correct value?

Accepted Answer

Introduction / Context: This question tests understanding of percentage error when the wrong multiplicative factor is used in a calculation. Instead of multiplying by a fraction greater than one, the student uses the reciprocal, resulting in a significantly smaller value. The task is to measure how large this error is in percentage terms relative to the correct result. Given Data / Assumptions: The correct operation is to multiply a number N by 13/6. The student instead multiplies N by 6/13. We must find the percentage error in the student result compared to the correct result. We assume N is positive so that the direction of error is clear. Concept / Approach: Let N be the original number. Correct result = N * (13/6). Obtained result = N * (6/13). Percentage error is usually measured as: Error percent = [(Correct value - Obtained value) / Correct value] * 100. Since the student uses a much smaller fraction, the obtained value is less than the correct value, giving a positive error percentage. Step-by-Step Solution: Step 1: Correct result = N * (13/6). Step 2: Obtained result = N * (6/13). Step 3: Compute the ratio of obtained to correct result: (Obtained / Correct) = [N * (6/13)] / [N * (13/6)]. Step 4: Cancel N to get (6/13) / (13/6) = (6/13) * (6/13) = 36 / 169. Step 5: Thus, obtained value = (36/169) * correct value. Step 6: Error fraction = 1 - 36/169 = (169 - 36) / 169 = 133 / 169. Step 7: Error percent = (133 / 169) * 100. Step 8: Compute 133 / 169 approximately = 0.787, so error percent ≈ 78.7 percent. Verification / Alternative check: Take N = 1 for a simple numerical check. Correct result = 13/6 ≈ 2.1667. Obtained result = 6/13 ≈ 0.4615. Difference = 2.1667 - 0.4615 ≈ 1.7052. Error percent = (1.7052 / 2.1667) * 100 ≈ 78.7 percent. This numerical verification matches the algebraic computation. Why Other Options Are Wrong: 39.35 percent: This is about half of the correct error and arises if the fraction is mishandled or if the wrong reference value is used. 184.72 percent and 369.44 percent: These drastically overstate the error and may come from reversing the ratio or using obtained instead of correct value as the denominator. 21.3 percent: This is far too small and not consistent with the large difference between 13/6 and 6/13. Common Pitfalls: A common mistake is to compute error relative to the obtained value instead of the correct value. Others may simply take the numerical difference between 13/6 and 6/13 as a percentage without proper scaling. It is also easy to forget that when a fraction is inverted, its value changes drastically if it is not equal to one. Always form the ratio of obtained to correct, simplify it, and then use the standard percentage error formula relative to the correct value. Final Answer: The percentage error in the student's result is approximately 78.7 percent less than the correct value.

Question 2

In an examination, a student has to secure 36% marks to pass. He scores 53 marks but still fails by 37 marks. What are the maximum total marks of the examination?

Accepted Answer

Introduction / Context: This problem again uses the idea of passing percentage and total marks. It requires converting the information about how many marks the student falls short by into an equation involving the total marks of the examination. Given Data / Assumptions: Passing percentage = 36 percent. Marks obtained by the student = 53. The student fails by 37 marks. Maximum marks of the examination = unknown (let it be M). Passing marks are 36 percent of M. Concept / Approach: If a student fails by a certain number of marks, then: Passing marks = Marks obtained + Shortfall. These passing marks are also equal to the required percentage of the maximum marks: Passing marks = 36/100 * M. Thus, we set up an equation: Marks obtained + Shortfall = 0.36 * M. We then solve for M. Step-by-Step Solution: Step 1: Shortfall = 37 marks. Step 2: Passing marks = 53 + 37 = 90 marks. Step 3: Passing marks also equal 36 percent of M, so 0.36 * M = 90. Step 4: Solve for M: M = 90 / 0.36. Step 5: Compute 90 / 0.36 = 250. Step 6: Therefore, maximum marks M = 250 marks. Verification / Alternative check: Check with M = 250. Passing marks = 36 percent of 250 = 0.36 * 250 = 90. The student has 53 marks, so the shortfall is 90 - 53 = 37 marks, which matches the problem statement. Hence, the solution is consistent. Why Other Options Are Wrong: 275 marks: 36 percent of 275 is 99, which does not give a shortfall of 37 when compared with 53. 300 marks: 36 percent of 300 is 108, making the shortfall 55 marks, not 37. 325 marks: 36 percent of 325 is 117, again inconsistent with the given shortfall. 225 marks: 36 percent of 225 is 81, which does not match the required difference. Common Pitfalls: Some students mistakenly use 36 percent of marks obtained instead of 36 percent of total marks. Others misinterpret "fails by 37 marks" as 37 percent rather than a difference of 37 marks. Always write passing marks both as a sum (obtained plus shortfall) and as a percentage of total, then solve for the total marks from the resulting equation. Final Answer: The maximum marks of the examination are 250 marks.

Question 3

In an examination, a student must score 25% of the total marks to pass. He scores 47 marks and fails by 43 marks. What are the maximum marks of the examination?

Accepted Answer

Introduction / Context: This question is very similar to other passing-percentage problems and reinforces the technique of relating marks obtained, marks needed to pass, and the percentage requirement to find the total marks of the exam. Given Data / Assumptions: Passing percentage = 25 percent. Marks obtained by the student = 47. The student fails by 43 marks. Maximum marks of the examination = unknown (let it be M). Passing marks are 25 percent of M. Concept / Approach: As before, failing by 43 marks means: Passing marks = Marks obtained + Shortfall = 47 + 43. These passing marks must equal 25 percent of the total marks: Passing marks = 25/100 * M = 0.25 * M. We set up the equation 0.25 * M = 47 + 43 and solve for M. Step-by-Step Solution: Step 1: Shortfall in marks = 43. Step 2: Passing marks = 47 + 43 = 90. Step 3: Passing marks are 25 percent of M, so 0.25 * M = 90. Step 4: Solve for M: M = 90 / 0.25. Step 5: Compute 90 / 0.25 = 360. Step 6: Therefore, maximum marks M = 360 marks. Verification / Alternative check: Check the result using M = 360. Passing marks at 25 percent = 0.25 * 360 = 90. The student has 47 marks, and the shortfall is 90 - 47 = 43 marks, which exactly matches the given information. So the computed total marks are correct. Why Other Options Are Wrong: 385 marks: 25 percent of 385 is 96.25, giving a different shortfall. 410 marks: 25 percent of 410 is 102.5, not matching the provided data. 435 marks: 25 percent of 435 is 108.75, again inconsistent. 320 marks: 25 percent of 320 is 80, and the difference from 47 is 33, not 43. Common Pitfalls: Students sometimes mistake the shortfall as a percentage rather than a fixed number of marks. Others may apply the 25 percent to the marks obtained instead of the total marks. The safe method is always to express passing marks in two ways: as a numerical sum (obtained plus shortfall) and as the required percentage of the total, then solve for the total. Final Answer: The maximum marks of the examination are 360 marks.

Question 4

The selling price of an item is Rs 1150 after the shopkeeper grants a discount of 8% on its marked price. What is the original marked price of the item in rupees?

Accepted Answer

Introduction / Context: This is a direct application of reversing a simple discount. The question gives the selling price and the discount percent and asks for the marked price before the discount was applied. Given Data / Assumptions: Selling price after discount = Rs 1150. Discount on marked price = 8 percent. Marked price = unknown (let it be M). No taxes or extra discounts are involved beyond the given 8 percent. Concept / Approach: When a discount of d percent is applied, the selling price equals: Selling price = Marked price * (1 - d/100). Here d = 8, so factor = 1 - 8/100 = 0.92. Hence: 1150 = M * 0.92. We rearrange this equation to solve for M using division. Step-by-Step Solution: Step 1: Let the marked price be M. Step 2: After 8 percent discount, selling price = 0.92 * M. Step 3: Given selling price = Rs 1150, so 0.92 * M = 1150. Step 4: Solve for M: M = 1150 / 0.92. Step 5: Compute 1150 / 0.92 = 1250. Step 6: Therefore, the marked price of the item is Rs 1250. Verification / Alternative check: With marked price M = Rs 1250, discount at 8 percent = 1250 * 8/100 = Rs 100. Selling price after discount = 1250 - 100 = Rs 1150. This matches the given selling price exactly, confirming that the calculation is correct. Why Other Options Are Wrong: Rs 1242: Close but not exact; discount at 8 percent here would not yield Rs 1150. Rs 1058 and Rs 1064: These are far too low to be marked prices corresponding to 1150 after an 8 percent discount. Rs 1200: With 8 percent discount the selling price would be 1200 - 96 = 1104, not 1150. Common Pitfalls: A typical mistake is to subtract 8 percent of the selling price from 1150 instead of reversing the discount on the marked price. Another mistake is to consider 1150 as 92 percent of some number without using division and instead attempting approximate mental adjustments that produce rounding errors. The systematic method is to express the selling price as a fixed fraction of the marked price and then divide. Final Answer: The original marked price of the item is Rs 1250.

Percentage Questions