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.

Question 5

How is $\frac{3}{4}$ expressed as percentage?

Accepted Answer

Concept & Formula: To convert any fraction to a percentage, multiply it by 100 and add the percentage symbol (%). Percentage = ( Fraction × 100 )% Step-by-Step Solution: • Given: The fraction is (3/4) • Calculation: Multiply the fraction by 100. (3/4) × 100 = 3 × 25 = 75% Exam Strategy & Shortcut: Memorize standard fraction-to-percentage conversions. Knowing that (1/4) = 25% immediately tells you that (3/4) = 3 × 25% = 75%. No manual calculation is needed. Common Pitfall: Converting the fraction to a decimal (0.75) and forgetting to multiply by 100 to get the percentage, leading to the incorrect selection of 0.75%. Final Answer: Therefore, the correct answer is 75%.

Question 6

$3.5$ can be expressed in terms of percentage as

Accepted Answer

Concept & Formula: To convert a decimal number to a percentage, you must multiply the number by 100 and add the percentage sign (%). Percentage = ( Decimal × 100 )% Step-by-Step Solution: • Given: The decimal number is 3.5. • Calculation: Multiply the given number by 100. 3.5 × 100 • Shift the decimal point two places to the right: = 350% Exam Strategy & Shortcut: Remember that the whole number 1 represents 100%. Therefore, 3 represents 300%. By basic logic, the number 3.5 must clearly be 350%. This can be done instantly by observation without writing anything down. Common Pitfall: Mistaking the decimal itself as the percentage (choosing 3.5%) or shifting the decimal in the wrong direction to get 0.35%. Always remember the fundamental rule: "Decimal to Percent = Multiply by 100". Final Answer: Therefore, the correct answer is 350%.

Question 7

When expressed as a fraction $64\%$ would mean

Accepted Answer

Concept & Formula: To convert a percentage into a fraction, divide the percentage value by 100 and simplify the resulting fraction to its lowest terms. Fraction = \frac{Percentage}{100} Step-by-Step Solution: • Given: The percentage is 64%. • Calculation: Express the percentage as a fraction with a denominator of 100. (64/100) • Divide both the numerator and the denominator by their greatest common divisor, which is 4. (64 \div 4/100 \div 4) = (16/25) Exam Strategy & Shortcut: Memorize base percentage fractions. Knowing that 4% = (1/25), you can quickly scale it up. Since 64 is 16 × 4, the fraction is simply 16 × (1/25) = (16/25). Common Pitfall: Failing to simplify the fraction fully. Students might divide by 2 to get (32/50) and stop, assuming it is the final answer, which will not match the multiple-choice options provided. Final Answer: Therefore, the correct answer is (16/25).

Question 8

Half of $1$ percent written as a decimal is

Accepted Answer

Concept & Formula: The term "percent" translates to "per one hundred". To convert a percentage to a decimal, divide the numerical value by 100, which effectively shifts the decimal point two places to the left. Decimal = \frac{Percentage}{100} Step-by-Step Solution: • Given: The value requested is "Half of 1 percent". • Calculation: First, write "Half of 1 percent" as a numerical percentage. 0.5% • Convert this percentage to a decimal by dividing by 100. (0.5/100) = 0.005 Exam Strategy & Shortcut: Keep standard conversions in mind: 1% = 0.01. Half of that value is simply 0.01 \div 2 = 0.005. This logical step can be done instantly in your head without writing out the division. Common Pitfall: Misinterpreting "Half of 1 percent" as just "Half" (which is 50% or 0.5) or "Half a tenth" (which is 5% or 0.05). Pay close attention to the word "percent" to ensure you divide the half by 100. Final Answer: Therefore, the correct answer is 0.005.

Question 9

$12\%$ of $5000 = $x$

Accepted Answer

Concept & Formula: To find a percentage of a given number, convert the percentage to a fraction (or decimal) and multiply it by that number. Value = ( (P/100) ) × N Step-by-Step Solution: • Given: We need to find 12% of the number 5000. • Calculation: Set up the multiplication using the percentage formula. (12/100) × 5000 • Cancel out two zeros from the numerator and denominator to simplify the expression. 12 × 50 = 600 Exam Strategy & Shortcut: Use the split method for percentages to avoid complex multiplication. Find 10% and 2% separately. 10% of 5000 = 500. 1% of 5000 = 50, which means 2% = 100. Add them together: 500 + 100 = 600. This is much faster and less error-prone. Common Pitfall: Making a simple calculation error by missing a zero, such as calculating 12 × 5 = 60 instead of 12 × 50 = 600. Always double-check your zero counts when cancelling out fractions. Final Answer: Therefore, the correct answer is 600.

Question 10

$38\%$ of $341 = $x$

Accepted Answer

Concept & Formula: To find a specific percentage of a number, convert the percentage into a decimal (or fraction) and multiply it by the given number. Value = (P/100) × N Step-by-Step Solution: • Given: We need to calculate 38% of the number 341. • Calculation: Convert 38% to the decimal 0.38. 0.38 × 341 = (38/100) × 341 = (12958/100) = 129.58 Exam Strategy & Shortcut: Use the unit digit technique for quick verification. Ignore the decimals temporarily and look at the last digits of the multiplication: 8 × 1 = 8. The final answer must end in .x8. While multiple options end in .8, you can use approximation to narrow it down: 40% of 340 is 4 × 34 = 136. Since 38% is slightly less than 40%, the answer must be slightly less than 136. Therefore, 129.58 is the only logical choice. Common Pitfall: Performing full manual multiplication without estimating first, which consumes valuable exam time. Always approximate first to see if you can eliminate options immediately. Final Answer: Therefore, the correct answer is 129.58.

Question 11

$(550\%$ of $250) \div 275 = $x$

Accepted Answer

Concept & Formula: According to the order of operations (BODMAS/PEMDAS), operations enclosed in brackets must be solved first. The term "of" indicates multiplication. Result = ( (P/100) × N ) \div D Step-by-Step Solution: • Given: The expression is (550% of 250) \div 275. • Calculation: First, solve the bracket portion. 550% of 250 = (550/100) × 250 = 5.5 × 250 = 1375 • Next, divide this result by the remaining divisor. 1375 \div 275 = 5 • Checking the options, the value 5 is not listed among (a), (b), (c), or (d). Exam Strategy & Shortcut: Instead of doing massive multiplication, simplify the fractions first. Rearrange the terms: (5.5 × 250) \div 275 = 5.5 × ( (250/275) ) Divide the fraction by 25 to get (10/11). = 5.5 × (10/11) = (55/10) × (10/11) = (55/11) = 5 This entirely bypasses the large multiplication step of finding 1375. Common Pitfall: Misplacing the decimal point during the 550% multiplication step, leading to an incorrect bracket result, or forcing an incorrect choice instead of confidently selecting "None of these". Final Answer: Therefore, the correct answer is None of these.

Question 12

$92.5\%$ of $550 = $x$

Accepted Answer

Concept & Formula: Finding large percentages can often be simplified by calculating the difference from 100%, a technique known as negative percentage calculation. P% of N = (100% - (100-P)%) of N Step-by-Step Solution: • Given: Find 92.5% of 550. • Calculation: Represent 92.5% as (100% - 7.5%). 100% of 550 = 550 • Now, calculate 7.5% of 550. Let's break down 7.5% into 5% + 2.5%. 10% of 550 = 55 5% of 550 = 27.5 2.5% of 550 = 13.75 • Add the parts together to find 7.5%. 7.5% = 27.5 + 13.75 = 41.25 • Subtract this from the 100% value. 550 - 41.25 = 508.75 Exam Strategy & Shortcut: The subtraction method shown above is by far the most efficient exam shortcut for percentages near 100. It replaces heavy multiplication involving 0.925 with simple mental additions and one final subtraction, drastically reducing calculation errors. Common Pitfall: Multiplying 0.925 directly by 550. While technically correct, this brute-force method relies on keeping track of three decimal places during a lengthy multiplication, which often results in arithmetic mistakes under time pressure. Final Answer: Therefore, the correct answer is 508.75.

Question 13

$2\%$ of $2 = $x$

Accepted Answer

Concept & Formula: Percentage conversion requires dividing the percentage figure by 100 and multiplying it by the target value. Value = (P/100) × N Step-by-Step Solution: • Given: We must find 2% of 2. • Calculation: Convert 2% into its fractional equivalent. (2/100) × 2 • Multiply the numerators together. = (4/100) • Convert the resulting fraction back into a decimal format. = 0.04 Exam Strategy & Shortcut: For single digit percentages of single digits, multiply the two numbers first (2 × 2 = 4), and then strictly move the decimal point two places to the left (resulting in 0.04). This can be done instantly without pen and paper. Common Pitfall: Incorrectly shifting the decimal place. A very common error is answering 0.4 (which would be 20% of 2) or 0.02 (which ignores the multiplication step entirely). Final Answer: Therefore, the correct answer is 0.04.

Question 14

$15\%$ of $578 + 22.5\%$ of $644 = $x$

Accepted Answer

Concept & Formula: Break down difficult decimal percentages into summable, manageable fractions (like 10%, 5%, and 2.5%) to perform calculations without needing a calculator. Step-by-Step Solution: • Given: We must evaluate 15% of 578 + 22.5% of 644. • Calculation: Calculate the first part using the split method (10% + 5%). 10% of 578 = 57.8 5% of 578 = (57.8/2) = 28.9 15% of 578 = 57.8 + 28.9 = 86.7 • Calculate the second part. 22.5% is equivalent to the fraction (22.5/100) = (45/200) = (9/40). (9/40) × 644 = (9 × 161/10) = (1449/10) = 144.9 • Add the two results together. 86.7 + 144.9 = 231.6 Exam Strategy & Shortcut: Converting 22.5% to its fractional equivalent (9/40) is a massive time-saver. By turning a complex decimal multiplication into a simple division by 4 and multiplication by 9, you bypass the risk of misplaced decimal points and can quickly arrive at 144.9. Common Pitfall: Attempting to brute-force the multiplication of 0.225 × 644. This takes significantly longer and makes it highly likely you will misplace the decimal point during the multi-line addition step of standard multiplication. Final Answer: Therefore, the correct answer is 231.6.

Question 15

$(7.9\%$ of $134) - (3.4\%$ of $79) = $x$

Accepted Answer

Concept & Formula: This problem can be drastically simplified by using algebraic factoring rather than brute-force calculation. The key is to find common numerical elements to pull out. Value = (P/100) × N Step-by-Step Solution: • Given: (7.9% of 134) - (3.4% of 79) • Calculation: Express both terms as decimals. (0.079 × 134) - (0.034 × 79) • Notice the relationship between 0.079 and 79. We can rewrite 0.034 × 79 by shifting decimals to match the 7.9 pattern. 0.034 × 79 = 0.34 × 7.9 • Also, write the first term using 7.9. 0.079 × 134 = 7.9 × 1.34 • Substitute these back into the expression. (7.9 × 1.34) - (7.9 × 0.34) • Factor out the common 7.9. 7.9 × (1.34 - 0.34) = 7.9 × 1.00 = 7.9 • The value 7.9 is not among the specific numeric options (a, b, c, d). Exam Strategy & Shortcut: The factoring method demonstrated above is the ultimate shortcut. Spotting that 7.9 × 134 and 3.4 × 79 share the base digits '79' allows you to completely avoid three-digit decimal multiplication. 7.9 × (1.34 - 0.34) = 7.9 can be solved mentally in under 10 seconds. Common Pitfall: Manually calculating 0.079 × 134 = 10.586 and 0.034 × 79 = 2.686. While this will eventually lead to the correct answer of 7.9, the heavy decimal arithmetic is a massive time drain and prone to carrying errors. Final Answer: Therefore, the correct answer is None of these.

Question 16

$(23.6\%$ of $1254) - (16.6\%$ of $834) = $x$

Accepted Answer

Concept & Formula: When dealing with heavy multi-digit decimal multiplication in multiple-choice exams, utilizing the Digital Sum (also known as digit root or casting out nines) is the most effective way to identify the correct answer. Digital Sum(A × B) = Digital Sum(A) × Digital Sum(B) Step-by-Step Solution: • Given: Evaluate (23.6% of 1254) - (16.6% of 834) • Calculation: Let's use the Digital Sum (DS) method. Find the DS of the first part: 23.6 × 1254 DS(23.6) = 2 + 3 + 6 = 11 \rightarrow 1 + 1 = 2 DS(1254) = 1 + 2 + 5 + 4 = 12 \rightarrow 1 + 2 = 3 DS(Part 1) = 2 × 3 = 6 • Find the DS of the second part: 16.6 × 834 DS(16.6) = 1 + 6 + 6 = 13 \rightarrow 1 + 3 = 4 DS(834) = 8 + 3 + 4 = 15 \rightarrow 1 + 5 = 6 DS(Part 2) = 4 × 6 = 24 \rightarrow 2 + 4 = 6 • Subtract the digital sums: Total DS = 6 - 6 = 0 (which is equivalent to 9) • Check the Digital Sum of the options: (a) 153.5 \rightarrow 1+5+3+5 = 14 \rightarrow 5 (b) 155.5 \rightarrow 1+5+5+5 = 16 \rightarrow 7 (c) 157.5 \rightarrow 1+5+7+5 = 18 \rightarrow 9 (Matches!) (d) 159.5 \rightarrow 1+5+9+5 = 20 \rightarrow 2 Exam Strategy & Shortcut: As demonstrated, calculating the digital sum takes less than 20 seconds and cleanly singles out 157.5 as the only mathematically viable option, entirely bypassing the horrifying manual calculation of 0.236 × 1254 and 0.166 × 834. Common Pitfall: Attempting to solve this using standard multiplication. Even with approximation, 0.24 × 1250 - 0.17 × 830 = 300 - 141 = 159. Approximation leads you to guess 159.5, which is incorrect. Only digital sum provides exact verification here. Final Answer: Therefore, the correct answer is 157.5.

Question 17

What is 45% of 25% of $\frac{4}{5}$th of 850?

Accepted Answer

Concept & Strategy: This is a sequential percentage and fraction problem. The most effective strategy is to translate the English statement directly into a single continuous mathematical equation, replacing "of" with multiplication, and converting standard percentages into their simplest fraction forms before calculating. Step-by-Step Solution: Given: • Equation: 45% of 25% of (4/5) of 850 Calculation / Deduction: • Convert the percentages to fractions for easier cancellation: 25% = (1/4) 45% = (45/100) = (9/20) • Rewrite the entire expression using these fractions: (9/20) × (1/4) × (4/5) × 850 • Cancel out the 4 in the numerator and denominator: (9/20) × (1/5) × 850 • Combine the denominators (20 × 5 = 100): (9/100) × 850 • Multiply 9 × 850: 7650 • Divide by 100: (7650/100) = 76.5 Exam Strategy & Shortcut: Memorizing basic percentage-to-fraction conversions is crucial. Instantly seeing 25% as (1/4) allows you to immediately cancel it with the numerator in (4/5), leaving just (1/5). Then you simply find (1/5) of 850 (170) and take 45% of 170. 45% = 50% - 5%. 50% of 170 = 85. 5% is 8.5. 85 - 8.5 = 76.5. This mental math completely bypasses heavy multiplication. Common Pitfall: Students often try to calculate the values step-by-step from left to right (e.g., finding 45% of 25% first). This leads to ugly decimals. Always expand the entire expression and cancel terms vertically and diagonally before doing any actual multiplication. Final Answer: Therefore, the correct answer is 76.5.

Question 18

What is 28% of 36% of $\frac{5}{7}$th of 5000?

Accepted Answer

Concept & Strategy: Like other chain-percentage problems, the fundamental concept relies on translating the sequence of operations into a single product. The primary strategy is to utilize the commutative property of multiplication to rearrange the terms and cancel out large numbers (like 5000) with the denominators (100) before proceeding to the complex multiplication. Step-by-Step Solution: Given: • Expression: 28% of 36% of (5/7) of 5000 Calculation / Deduction: • Translate the expression into standard mathematical form: (28/100) × (36/100) × (5/7) × 5000 • Group the denominators together: 100 × 100 = 10000 • Simplify the zeroes by dividing 5000 by 10000: (5000/10000) = (1/2) • Rewrite the expression with the simplified terms: 28 × 36 × (5/7) × (1/2) • Rearrange to cancel out the denominator 7 with the 28 (since 28 \div 7 = 4): 4 × 36 × 5 × (1/2) • Multiply 4 × 5 to get 20: 20 × 36 × (1/2) • Cancel the 20 with the (1/2) to get 10: 10 × 36 • Final calculation: 360 Exam Strategy & Shortcut: Look for synergies between the fractions. You have a denominator of 7. Look at the numerators: 28 is a multiple of 7. Cancel them immediately to get 4. You have two percentage signs meaning division by 10000. You have a multiplier of 5000. 5000 \div 10000 = (1/2). Now you are just left with 4 × 36 × 5 × 0.5. Since 4 × 5 × 0.5 = 10, the answer is instantly 36 × 10 = 360. Common Pitfall: A major time-waster is multiplying the numerators (28 × 36) before simplifying the denominators. Never multiply two-digit numbers if there are still denominators left that can be cancelled. Final Answer: Therefore, the correct answer is 360.

Question 19

(0.56% of 225) $	imes$ (3.25% of 430) = $x$

Accepted Answer

Concept & Strategy: This problem tests your ability to handle the multiplication of two independent decimal-percentage expressions. The core concept is isolating the calculation into two distinct parts, simplifying each to a decimal, and then multiplying the final results. Step-by-Step Solution: Given: • Part A: 0.56% of 225 • Part B: 3.25% of 430 Calculation / Deduction: • Solve Part A first. Convert the percentage to a fraction: (0.56/100) × 225 = 0.0056 × 225 • To multiply easily, think of it as 56 × 225 and then shift the decimal four places: 56 × 225 = 12600 Shift decimal 4 places left: 1.26 • Solve Part B. Convert the percentage to a fraction: (3.25/100) × 430 = 0.0325 × 430 • Think of it as 325 × 43 and shift the decimal three places: 325 × 43 = 13975 Shift decimal 3 places left: 13.975 • Multiply the two results (Part A × Part B): 1.26 × 13.975 • To perform this manually, calculate 126 × 13975 = 1760850. • Since 1.26 has two decimal places and 13.975 has three, the final answer needs five decimal places: 17.60850, which is 17.6085. Exam Strategy & Shortcut: Use the unit digit and decimal placement to verify the options without doing the full massive multiplication. Look at the final step: 1.26 × 13.975. The last significant digits are 6 and 5. 6 × 5 = 30, so the final number (before trailing zeroes are dropped) ends in a 0. 126 × 13975. We have 1 × 13 \approx 13, plus a quarter of 13 is about 3, so the answer must be around 16 or 17. Options C and D are in the 17 range. Looking at exact digits requires the full math, but splitting 1.26 into (1 + (1/4)) works wonderfully: 13.975 + (13.975/4) \approx 13.975 + 3.493 = 17.46. Because 0.26 is slightly larger than 0.25, the answer must be slightly larger than 17.46. The only option that fits is 17.6085. Common Pitfall: Losing track of decimal places during the final multiplication is the most common error. When multiplying 1.26 by 13.975, carefully count the decimal places in the factors (2 + 3 = 5) and apply them to the final integer product. Final Answer: Therefore, the correct answer is 17.6085.

Question 20

An agent sells goods of value of ₹ 15000. The commission which he receives at the rate of $12\frac{1}{2}\%$ is

Accepted Answer

Concept & Logic: This is a direct application of percentages in a real-world commercial context. The key to solving this instantly is converting the mixed fraction percentage into a simple ratio. The equivalent fraction for 12.5% is a standard value that should be memorized. 12(1/2)% = 12.5% = (1/8) Step-by-Step Solution: Given: • Total Sales Value = ₹ 15000 • Commission Rate = 12(1/2)% Calculation / Deduction: • Identify the fractional equivalent of the given commission rate to bypass decimal division: 12(1/2)% = (12.5/100) = (1/8) • Set up the equation to find the total commission amount: Commission = (1/8) × 15000 • Perform the division. You can divide by 2 three times to divide by 8: First division by 2: 15000 \div 2 = 7500 Second division by 2: 7500 \div 2 = 3750 Third division by 2: 3750 \div 2 = 1875 • The resulting value is the commission earned by the agent. Exam Strategy & Shortcut: Never use 12.5 \div 100 when calculating manually. Always use the fraction (1/8). If you cannot divide 15000 by 8 quickly in your head, break it down: 15000 is 8000 + 7000. (1/8) of 8000 is 1000. (1/8) of 7000 is 875. Adding them together gives 1875. Common Pitfall: A common error is converting 12(1/2)% to 12.05% or making arithmetic mistakes when multiplying 15000 by 12.5 and then dividing by 100. Relying on the fractional equivalent entirely eliminates decimal errors. Final Answer: Therefore, the correct answer is ₹ 1875.

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