Question 1

In an examination, $65\%$ of the total examinees passed. If the number of failures is $420$, the total number of examinees is

Accepted Answer

Concept & Logic: Similar to the previous problem, the sum of passed and failed examinees equals 100% of the total examinees. Find the percentage of failures to match the given absolute number. Step-by-Step Solution: Given: • Percentage of passed examinees = 65% • Number of failures = 420 Calculation: • Determine the percentage of students who failed: % Failures = 100% - 65% = 35% • Let the total number of examinees be x. 35% of x = 420 (35/100) × x = 420 • Solve for x: x = (420 × 100/35) • Simplify by dividing 420 by 35: x = 12 × 100 = 1200 Exam Strategy & Shortcut: Use the unitary method directly. 35% corresponds to 420. Notice that 35 × 2 = 70, and 70 × 6 = 420. So, 35 × 12 = 420. This means 1% = 12. Therefore, 100% = 1200. Common Pitfall: Students sometimes mistake the 420 failures as the passed students and equate 65% = 420. Carefully read whether the absolute number given represents the "passed" or "failed" group. Final Answer: Therefore, the correct answer is 1200.

Question 2

If a bucket is $80\%$ full, then it contains $2$ litres more water than when it is $66\frac{2}{3}\%$ full. What is the capacity of the bucket?

Accepted Answer

Concept & Logic: The absolute difference in water volume corresponds directly to the difference in the percentage full. Step-by-Step Solution: Given: • Scenario 1: Bucket is 80% full. • Scenario 2: Bucket is 66(2/3)% full. • Difference in volume = 2 litres. Calculation: • Let the total capacity of the bucket be x litres. • Find the difference in percentages: Difference = 80% - 66(2/3)% = 80% - (200/3)% = (240 - 200/3)% = (40/3)% • This percentage difference equals 2 litres of the total capacity x: (40/3)% of x = 2 ((40/3 × 100)) × x = 2 (40/300) × x = 2 (2/15) × x = 2 • Solve for x: x = (2 × 15/2) = 15 Exam Strategy & Shortcut: Work with fractions instead of percentages. 80% = (4/5) 66(2/3)% = (2/3) The difference is: (4/5) - (2/3) = (12 - 10/15) = (2/15) So, (2/15) of the bucket equals 2 litres. This means (1/15) of the bucket is 1 litre, making the total capacity (15 × 1) 15 litres. Fractional equivalents make the subtraction much cleaner. Common Pitfall: Handling mixed fraction percentages like 66(2/3)% as decimals (e.g., 66.67%) leads to imprecise calculations and rounding errors. Always convert them to improper fractions (like (200/3)%) or their base fractional equivalents (like (2/3)). Final Answer: Therefore, the correct answer is 15 litres.

Question 3

Vinay decided to donate 5% of his salary. On the day of donation he changed his mind and donated ₹ 1687.50, which was 75% of what he had decided earlier. How much is Vinay's salary?

Accepted Answer

Concept & Logic: The problem requires calculating the original base amount (salary) when a percentage of a percentage is given. The final donated amount is 75% of the 5% that was initially decided. Final Amount = Salary × 5% × 75% Step-by-Step Solution: • Given: Initially decided donation = 5% of salary. Actual donation = ₹ 1687.50, which is 75% of the decided amount. • Let Vinay's total salary be x. • Amount decided to donate = 5% of x = 0.05x. • Amount actually donated = 75% of 0.05x. • Set up the equation based on the actual donation: 0.75 × 0.05x = 1687.50 • Simplify the percentages to fractions for easier calculation: ((75/100)) × ((5/100)) × x = 1687.50 ((3/4)) × ((1/20)) × x = 1687.50 (3/80) × x = 1687.50 • Solve for x: x = (1687.50 × 80/3) x = 562.50 × 80 = 45000 Exam Strategy & Shortcut: Use the fraction equivalent method to avoid messy decimals. 5% = (1/20) and 75% = (3/4). The effective donation is (1/20) × (3/4) = (3/80) of the salary. If 3 units = 1687.50, then 1 unit = 562.50. Total salary (80 units) = 80 × 562.50 = 45000. Common Pitfall: A frequent mistake is assuming the ₹ 1687.50 is 75% of the total salary, or calculating 75% of 5% by adding/subtracting them instead of multiplying. Always remember "of" implies multiplication. Final Answer: Therefore, the correct answer is ₹ 45000.

Question 4

One-fourth of sixty percent of a number is equal to two-fifths of twenty percent of another number. What is the respective ratio of the first number to the second number?

Accepted Answer

Concept & Formula: To find the ratio between two unknown numbers, translate the given percentage and fraction conditions into an algebraic equation and simplify to find the ratio (A/B). Fraction_1 × Percentage_1 × A = Fraction_2 × Percentage_2 × B Step-by-Step Solution: • Let the first number be A and the second number be B. • Translate the first condition: "One-fourth of sixty percent of a number" (1/4) × 60% × A • Translate the second condition: "two-fifths of twenty percent of another number" (2/5) × 20% × B • Equate both sides: (1/4) × ((60/100)) × A = (2/5) × ((20/100)) × B • Simplify the percentages: (1/4) × ((3/5)) × A = (2/5) × ((1/5)) × B (3/20) × A = (2/25) × B • Rearrange to find the ratio (A/B): (A/B) = (\frac{2/25)}{(3/20)} (A/B) = (2/25) × (20/3) = (40/75) • Simplify the fraction by dividing the numerator and denominator by 5: (A/B) = (8/15) • The required ratio is 8:15. Since this is not among the options (a), (b), or (c), the answer is "None of these". Exam Strategy & Shortcut: Ignore the percentage signs on both sides since they cancel out (effectively dividing both sides by 100). The equation immediately becomes: (1/4) × 60 × A = (2/5) × 20 × B 15A = 8B (A/B) = (8/15) This directly yields the ratio 8:15 in seconds. Common Pitfall: Students often incorrectly match the ratio values, calculating (B/A) instead of (A/B), or they get lost simplifying fractions on opposite sides of the equation. Also, since 8:15 is not in the options, a common mistake is second-guessing and choosing a close option instead of confidently marking "None of these". Final Answer: Therefore, the correct answer is None of these.

Question 5

If the monthly salary of an employee is increased by $2 \frac{2}{3}\%$, he gets ₹ 72 more. His monthly salary (in ₹) is

Accepted Answer

Concept & Formula: The percentage increase corresponds directly to the absolute increase in value. If a quantity x increases by R%, and the absolute increase is V, then R% of x = V. (R/100) × x = V Step-by-Step Solution: • Given: The increase in salary is 2 (2/3)%. The monetary value of this increase is ₹ 72. • Let the original monthly salary be x. • Convert the mixed fraction percentage into an improper fraction: 2 (2/3)% = (3 × 2 + 2/3)% = (8/3)% • Set up the equation based on the given condition: (8/3)% of x = 72 • Convert the percentage to a mathematical operation (divide by 100): ((8/3 × 100)) × x = 72 (8/300) × x = 72 • Solve for x: x = (72 × 300/8) x = 9 × 300 = 2700 Exam Strategy & Shortcut: Treat the percentage purely as a fraction. (8/3)% = (8/300) = (2/75). This means an increase of 2 parts on a base of 75 parts. We are given that 2 parts = 72. Therefore, 1 part = 36. The total salary (75 parts) = 75 × 36. To multiply 75 × 36 quickly: (100 - 25) × 36 = 3600 - 900 = 2700. Common Pitfall: Students often forget to divide by 100 when dealing with fraction percentages, erroneously solving (8/3) × x = 72, which leads to x = 27 (a clearly incorrect salary). Always remember the `%` symbol means × (1/100). Final Answer: Therefore, the correct answer is 2700.

Question 6

A store raised the price of an item by exactly 10 percent. Which of the following could not be the resulting price of the item?

Accepted Answer

Concept & Logic: When an item's price is raised by 10%, the new price is 110% of the original price. This means the new price is the original price multiplied by 1.1 (or (11/10)). Assuming the original price was a standard currency amount (like exact cents or rupees without infinite repeating decimals), the new price, when expressed in the smallest currency unit (cents/paise), must be perfectly divisible by 11. New Price = Old Price × (11/10) Step-by-Step Solution: • Let the original price in the smallest unit (paise) be x. • The new price is x × (11/10). • For the original price x to be an integer (a valid exact currency amount), the new price multiplied by 10 must be divisible by 11. Effectively, the new price without the decimal point should be a multiple of 11. • Let's convert the options to integer values (paise) and check for divisibility by 11: • (a) ₹ 5.50 \rightarrow 550. 550 / 11 = 50. (Original price was ₹ 5.00. Valid.) • (b) ₹ 7.60 \rightarrow 760. 760 / 11 = 69.09... (Original price would be a non-terminating fraction. Invalid.) • (c) ₹ 11.00 \rightarrow 1100. 1100 / 11 = 100. (Original price was ₹ 10.00. Valid.) • (d) ₹ 12.10 \rightarrow 1210. 1210 / 11 = 110. (Original price was ₹ 11.00. Valid.) • Since 7.60 does not yield a clean original price, it cannot be the resulting price. Exam Strategy & Shortcut: Use the divisibility rule for 11! Discard the decimal point and check the digits. The alternating sum of digits must be 0 or a multiple of 11. (a) 550 \rightarrow 5 - 5 + 0 = 0 (Divisible) (b) 760 \rightarrow 7 - 6 + 0 = 1 (Not Divisible) \leftarrow This is the answer! (c) 1100 \rightarrow 1 - 1 + 0 - 0 = 0 (Divisible) (d) 1210 \rightarrow 1 - 2 + 1 - 0 = 0 (Divisible) Common Pitfall: Attempting to find 10% of the *options* and subtracting it. The 10% increase was applied to the *original* unknown price, not the final price. Calculating 10% of 5.50 gives 0.55, which is an incorrect reverse calculation method. You must divide by 1.1, not multiply by 0.9. Final Answer: Therefore, the correct answer is ₹ 7.60.

Question 7

A number increased by $37 \frac{1}{2}\%$ gives 33. The number is

Accepted Answer

Concept & Formula: When a number is increased by a certain percentage, you add that percentage to 100% of the original number. Converting complex percentages to their fractional equivalents simplifies the calculation immensely. Final Value = Original Value × (1 + Fractional Equivalent) Step-by-Step Solution: • Given: Percentage increase = 37 (1/2)%. Final value = 33. • First, convert 37 (1/2)% to a standard fraction. • We know that 12.5% = (1/8). • Since 37.5% = 3 × 12.5%, the fractional equivalent of 37 (1/2)% is (3/8). • Let the original number be x. • The number is increased by (3/8) of itself. New Number = x + (3/8)x = (11/8)x • We are given that the new number is 33. (11/8)x = 33 • Solve for x: x = 33 × (8/11) x = 3 × 8 = 24 Exam Strategy & Shortcut: Memorize standard fraction-percentage pairs. Recognizing 37.5% = (3/8) is key. An increase of (3/8) means the new value represents 11 parts, while the original represents 8 parts. 11 parts = 33 \implies 1 part = 3. Original number (8 parts) = 8 × 3 = 24. This can be done mentally in under 5 seconds. Common Pitfall: A common error is setting the equation as just (3/8)x = 33 (ignoring the words "increased by") and solving for x = 88, which severely misinterprets the question. Final Answer: Therefore, the correct answer is 24.

Question 8

Three-fourths of a number is equal to 60% of another number and the difference between these two numbers is 20. What is the sum of these two numbers?

Accepted Answer

Concept & Logic: Equating a fraction of one variable to a percentage of another establishes a fixed ratio between the two variables. Once the ratio is found, the given difference allows us to calculate the exact multiplier, leading to the sum. Fraction × A = Percentage × B \implies Ratio of A:B Step-by-Step Solution: • Let the first number be A and the second number be B. • Given: (3/4) of A is equal to 60% of B. • Convert 60% to a fraction: 60% = (60/100) = (3/5). • Set up the equation: (3/4) × A = (3/5) × B • Find the ratio of A to B by isolating the variables on one side: (A/B) = (\frac{3/5)}{(3/4)} = (3/5) × (4/3) = (4/5) • This means A is 4k and B is 5k for some constant multiplier k. • We are given that the difference between the numbers is 20. B - A = 20 5k - 4k = 20 \implies k = 20 • Now, calculate the sum of the two numbers. Sum = A + B = 4k + 5k = 9k • Substitute the value of k: Sum = 9 × 20 = 180 Exam Strategy & Shortcut: Once you find the ratio (A/B) = (4/5), you know the numbers are multiples of 4 and 5. The sum of the numbers is 4 + 5 = 9 parts. Therefore, the correct answer MUST be a multiple of 9. Look at the options: (a) 170 (Not divisible by 9) (b) 180 (Divisible by 9) (c) 220 (Not divisible by 9) Without calculating the exact values, you can confidently mark 180. Common Pitfall: Setting up the difference backwards as A - B = 20 (which leads to negative numbers) or getting confused between the fractions and percentages while crossing over terms. Relying heavily on equations without noticing the "multiple of 9" shortcut costs valuable time. Final Answer: Therefore, the correct answer is 180.

Question 9

The number which exceeds 16% of it by 42 is

Accepted Answer

Concept & Formula: Every number represents 100% of itself. If a number "exceeds" a percentage of itself by a specific value, that value represents the remaining percentage gap between 100% and the given percentage. Number (100%) - Percentage (P%) = Remaining Percentage ((100-P)%) Step-by-Step Solution: • Let the required number be x. • "16% of it" can be written as 0.16x or 16% of x. • The phrase "exceeds... by 42" means the difference between the number and its 16% is 42. • Set up the algebraic equation: x - 16% of x = 42 100% of x - 16% of x = 42 84% of x = 42 • Convert the percentage to a fraction to solve: (84/100) × x = 42 • Solve for x: x = 42 × (100/84) • Notice that 42 is exactly half of 84. x = (1/2) × 100 = 50 Exam Strategy & Shortcut: Think entirely in percentages. The number itself is 100%. It exceeds its 16% by (100% - 16%) = 84%. So, 84% of the value is equal to 42. Notice the simple math: 42 is exactly half of 84. If 84% = 42, then 100% must be exactly half of 100, which is 50. Common Pitfall: The phrasing can be tricky. Students often misread this as "16% of a number is 42" and calculate x = (42/0.16), or they write 0.16x + 42 = x and make an arithmetic error. Translating the English phrase "exceeds... by" directly to a subtraction equation (A - B = difference) prevents this. Final Answer: Therefore, the correct answer is 50.

Question 10

If a number $x$ is 10% less than another number $y$ and $y$ is 10% more than 125, then $x$ is equal to

Accepted Answer

Concept & Logic: This is a problem of successive percentage changes. You can trace the values sequentially or use successive multipliers to find the final value in one step. x = Base × (1 + (R_1/100)) × (1 - (R_2/100)) Step-by-Step Solution: • Step 1: Find the value of y. We are given that y is 10% more than 125. y = 125 + (10% of 125) y = 125 + 12.5 = 137.5 • Step 2: Find the value of x. We are given that x is 10% less than y. x = 137.5 - (10% of 137.5) x = 137.5 - 13.75 = 123.75 Exam Strategy & Shortcut: Use decimal multipliers for a faster, one-line calculation. "10% more" means multiply by 1.1. "10% less" means multiply by 0.9. x = 125 × 1.1 × 0.9 x = 125 × 0.99 Calculating 99% of a number is just subtracting 1% of it from the original value: x = 125 - 1.25 = 123.75 Common Pitfall: A very common intuition trap is assuming that a 10% increase followed by a 10% decrease cancels out, bringing the value back to the original 125. Successive identical increases and decreases always result in a net decrease (specifically, a loss of (x^2/100)%). Final Answer: Therefore, the correct answer is 123.75.

Question 11

Which of the following multipliers will cause a number to be increased by 29.7%?

Accepted Answer

Concept & Logic: To increase a number by a certain percentage, you add that percentage of the number to the original number itself (100%). The final multiplier is simply the decimal equivalent of the final percentage. Final Multiplier = 1 + \frac{Percentage Increase}{100} Step-by-Step Solution: • Let the original number be x. The original number represents 100% of itself, or 1 \cdot x. • We want to increase this number by 29.7%. • The increase amount is 29.7% of x, which can be written as 0.297x. • The new number is the sum of the original number and the increase: New Number = x + 0.297x • Factor out x: New Number = x(1 + 0.297) = 1.297x • Therefore, the required multiplier is 1.297. Exam Strategy & Shortcut: A 29.7% increase means the final value is 129.7% of the original. To convert any percentage to a decimal multiplier, just divide by 100 (move the decimal point two places to the left). 129.7 / 100 = 1.297. Common Pitfall: A very common mistake is confusing the percentage value with the decimal multiplier (e.g., choosing 129.7 or 12.97) by failing to divide by 100 or moving the decimal point the wrong number of spaces. Final Answer: Therefore, the correct answer is 1.297.

Question 12

The sum of two numbers is $\frac{28}{25}$ of the first number.  The second number is what percent of the first?

Accepted Answer

Concept & Formula: When a relationship between variables is given as a fraction, setting up a direct algebraic equation lets you find the ratio between the variables. Once you have the ratio, calculating the percentage is straightforward. Percentage = ( \frac{Second Number}{First Number} ) × 100 Step-by-Step Solution: • Let the first number be x and the second number be y. • Given: The sum of the two numbers (x + y) is (28/25) of the first number (x). • Set up the equation: x + y = (28/25)x • Isolate y to find its relationship with x: y = (28/25)x - x y = ( (28/25) - 1 )x y = ( (28 - 25/25) )x y = (3/25)x • The question asks for the second number (y) as a percentage of the first number (x). Required Percentage = ( (y/x) ) × 100 Required Percentage = ( (\frac{3/25)x}{x} ) × 100 Required Percentage = (3/25) × 100 Required Percentage = 3 × 4 = 12% Exam Strategy & Shortcut: Use ratio units directly. The phrase "sum ... is (28/25) of the first" means if the first number is 25 units, the total sum is 28 units. Therefore, the second number is simply the difference: 28 - 25 = 3 units. The required percentage is (3/25). Since (1/25) = 4%, then (3/25) = 3 × 4% = 12%. Common Pitfall: Students sometimes calculate the second number as a percentage of the *sum* instead of the *first number*, leading to (3/28) \approx 10.7%, which is incorrect based on the phrasing "what percent of the first". Final Answer: Therefore, the correct answer is 12%.

Question 13

A number reduced by 25% becomes 225. What percent should it be increased so that it becomes 390?

Accepted Answer

Concept & Logic: This is a two-step problem. First, find the original base number (100%) by reversing the initial percentage reduction. Second, calculate the percentage increase required to go from the original base number to the new target value. Percentage Increase = ( \frac{Final Value - Original Base}{Original Base} ) × 100 Step-by-Step Solution: • Step 1: Find the original number. Let the original number be x. A reduction of 25% means the number becomes 100% - 25% = 75% of itself. 75% of x = 225 (75/100)x = 225 \implies (3/4)x = 225 x = 225 × (4/3) = 75 × 4 = 300 The original number is 300. • Step 2: Find the required percentage increase. The new target value is 390. We need to find the increase from the original number (300) to the target (390). Absolute Increase = 390 - 300 = 90 Percentage Increase = ( (90/300) ) × 100 Percentage Increase = (9/30) × 100 = 30% Exam Strategy & Shortcut: Work purely in ratios or percentages. 75% \equiv 225. Divide both sides by 75: 1% \equiv 3. So, 100% \equiv 300 (This is your base). The target is 390, which is 90 more than 300. Since 1% = 3, then 90 is exactly 30 × 3, meaning it represents 30%. Common Pitfall: The most frequent error is applying the second percentage calculation on the intermediate value (225) instead of the original base number (300), or mistakenly trying to find the percentage directly from the difference between 390 and 225. Final Answer: Therefore, the correct answer is 30%.

Question 14

If 25% of a number is subtracted from a second number, the second number reduces to its five-sixths. What is the ratio of the first number to the second number?

Accepted Answer

Concept & Formula: When subtracting a part of one variable from another causes a proportional reduction in the second, you can equate the subtracted amount directly to the amount lost. Amount Subtracted = Initial Value - Final Value Step-by-Step Solution: • Let the first number be A and the second number be B. • Given: Subtracting 25% of A from B results in (5/6) of B. • Translate this into an algebraic equation: B - 25% of A = (5/6)B • Convert the percentage to a fraction (25% = (1/4)): B - (1/4)A = (5/6)B • Group the B terms on one side and the A terms on the other: B - (5/6)B = (1/4)A • Factor out B on the left side: B (1 - (5/6)) = (1/4)A (1/6)B = (1/4)A • We need the ratio of the first number to the second number, which is (A/B). • Rearrange the equation to isolate (A/B): (A/B) = (\frac{1/6)}{(1/4)} (A/B) = (1/6) × 4 = (4/6) = (2/3) Exam Strategy & Shortcut: Read carefully: "reduces to its five-sixths". This implies it lost (1/6) of its value (1 - (5/6) = (1/6)). What caused this loss of (1/6) of the second number? The subtraction of 25% ((1/4)) of the first number. Therefore, directly equate them: (1/4) of First = (1/6) of Second. \frac{First}{Second} = (4/6) = 2:3. You can solve this mentally in seconds without writing the full equation. Common Pitfall: A very common mistake is confusing "reduces to its five-sixths" with "reduces by its five-sixths", which would lead to a completely different (and incorrect) equation: B - (1/4)A = B - (5/6)B. Always pay close attention to prepositions in word problems. Final Answer: Therefore, the correct answer is 2: 3.

Question 15

When any number is divided by 12, then dividend becomes $ \frac{1}{4} $th of the other number.  By how much percent first number is greater than the second number?

Accepted Answer

Concept & Formula: This problem requires translating a word problem into a simple algebraic equation to find the ratio between two unknown numbers. The core formula for finding how much percent one value (x) is greater than another (y) is: Percentage Greater = (x - y/y) × 100 Step-by-step Solution: Let the first number be x and the second (other) number be y. According to the question, when the first number is divided by 12, the result (often mistranslated as "dividend" in local textbooks, but contextually meaning the quotient here) is equal to (1/4) of the second number. Set up the equation: (x/12) = (y/4) Cross-multiply to find the relationship between x and y: 4x = 12y x = 3y This means the first number is exactly 3 times the second number. Now, find how much percent x is greater than y: Difference = x - y = 3y - y = 2y Percentage greater: ( (2y/y) ) × 100 = 200% Exam Strategy & Shortcut: Use the Assumption Method to solve this instantly without algebra. Assume the second number is a convenient multiple of 4, say 4. Then (1/4) of the second number is 1. So, the first number divided by 12 equals 1, making the first number 12. Now compare 12 and 4. 12 is 8 units greater than 4. (8/4) × 100 = 200% . Common Pitfall: The most common mistake is confusing "percentage of" with "percentage greater than". Since x = 3y, the first number is 300% *of* the second number. Students often hastily pick 300 (Option C). Always subtract the base value before calculating the percentage difference. Final Answer: Therefore, the correct answer is 200.

Question 16

Three candidates contested an election and received 1136, 7636 and 11628 votes respectively.  What percentage of the total votes did the winning candidate get?

Accepted Answer

Concept & Formula: This problem requires calculating a simple percentage share. The core formula to find the percentage share of a specific part out of a whole is: Percentage = ( \frac{Part Value}{Total Value} ) × 100 Step-by-step Solution: Given: Votes for Candidate 1 = 1136 Votes for Candidate 2 = 7636 Votes for Candidate 3 (Winner) = 11628 First, calculate the total number of votes polled in the election: Total Votes = 1136 + 7636 + 11628 Total Votes = 20400 The winning candidate is the one with the maximum votes, which is 11628. Now, calculate the winning candidate's percentage of the total votes: Winning Percentage = ( (11628/20400) ) × 100 Simplify the fraction by canceling the zeros: = (11628/204) Perform the division: 11628 \div 204 = 57 Exam Strategy & Shortcut: Use the Approximation Method to avoid tedious long division. Total votes \approx 20,400 . Winner's votes \approx 11,600 . Fraction \approx (11.6/20.4) \approx (11.6/20) . (11.6/20) = (116/200) = (58/100) = 58% . Looking at the options, 57% is extremely close to 58%, making it the obvious choice without doing exact arithmetic. Common Pitfall: Students sometimes calculate the percentage based only on the combined votes of the losing candidates rather than the grand total of all votes cast. Always ensure your denominator represents the total ecosystem. Final Answer: Therefore, the correct answer is 57%.

Question 17

A student multiplied a number by $ \frac{3}{5} $ instead of $ \frac{5}{3} $.  What is the percentage error in the calculation?

Accepted Answer

Concept & Formula: This problem requires finding the percentage error caused by using an incorrect multiplier. The formula for percentage error is: Percentage Error = ( \frac{True Value - Measured Value}{True Value} ) × 100 Alternatively, it can be written as: Percentage Error = ( \frac{Error}{True Value} ) × 100 Step-by-step Solution: Given: The correct multiplier should have been (5/3) . The incorrect multiplier used was (3/5) . Let the number to be multiplied be x. The true (correct) value = (5/3)x The measured (incorrect) value = (3/5)x Calculate the error (difference between the true value and the measured value): Error = (5/3)x - (3/5)x Find a common denominator to subtract the fractions: Error = (25x - 9x/15) = (16x/15) Now, calculate the percentage error using the formula: Percentage Error = (\frac{16x/15)}{(5/3)x} × 100 The x variables cancel out. Simplify the complex fraction: Percentage Error = ( (16/15) × (3/5) ) × 100 Percentage Error = (16/25) × 100 Calculate the final percentage: 16 × 4 = 64% Exam Strategy & Shortcut: Use the LCM Assumption Method to avoid variables entirely. Look at the denominators of the two fractions: 3 and 5. Assume the original number is their Least Common Multiple (LCM), which is 15. Correct calculation: 15 × (5/3) = 25 . Incorrect calculation: 15 × (3/5) = 9 . Error: 25 - 9 = 16 . Percentage error: (16/25) × 100 = 64% . This method is significantly faster and less prone to algebraic mistakes. Common Pitfall: A very common mistake is calculating the percentage error against the *incorrect* value (the measured value) instead of the *true* value. If you divide by 9 instead of 25, you get (16/9) × 100 \approx 177.7% , which is totally wrong. Always use the intended correct result as the base denominator. Final Answer: Therefore, the correct answer is 64%.

Question 18

A tempo is insured to the extent of $ \frac{4}{5} $ of its original value.  If the premium on it at the rate of 1.3 percent amounts to ₹ 910, the original value of the tempo is

Accepted Answer

Concept & Strategy: This question tests consecutive percentage applications. The premium is calculated based on the *insured value*, not the original value. The insured value is a fraction of the original value. Premium = Premium Rate × Insured Value Insured Value = Fraction Insured × Original Value Step-by-step Solution: Given: Insured portion = (4/5) of original value. Premium rate = 1.3% = (1.3/100) . Total premium paid = ₹ 910. Let the original value of the tempo be x. Determine the insured value in terms of x: Insured Value = (4/5)x Set up the equation based on the premium calculation: 1.3% of Insured Value = 910 (1.3/100) × ( (4/5)x ) = 910 Simplify the equation to solve for x: Convert 1.3 to (13/10) to remove decimals: ( (13/1000) ) × ( (4/5)x ) = 910 (52/5000)x = 910 Isolate x: x = (910 × 5000/52) Simplify the fraction. Notice that 910 and 52 share common factors: x = (910/52) × 5000 Divide 910 by 13 to get 70, and 52 by 13 to get 4: x = (70/4) × 5000 x = 17.5 × 5000 x = 87500 Exam Strategy & Shortcut: Instead of complex fractions, use a reverse multiplier approach mentally. The premium is 1.3%, meaning the insured value is 910 \div 0.013 . 910 \div 0.013 = 910000 \div 13 = 70000 . So, the insured value is 70,000. We know (4/5) of original = 70,000. This means 4 parts = 70,000. 1 part = 70,000 \div 4 = 17,500 . 5 parts (Original value) = 17,500 × 5 = 87,500 . Common Pitfall: The most common trap is assuming the 1.3% premium applies to the *original* value of the tempo directly, completely ignoring the (4/5) insurance constraint. This would lead to incorrectly solving 1.3% × x = 910 , yielding 70,000 (which isn't even an option, leading to panic). Final Answer: Therefore, the correct answer is ₹ 87,500.

Question 19

When 15% is lost in grinding wheat, a country can export 30 lakh tons of wheat.  On the other hand, if 10% is lost in grinding, it can export 40 lakh tons of wheat.  The production of wheat in the country is:

Accepted Answer

Concept & Logic: This problem utilizes the concept of constant totals and absolute differences. The total production of wheat remains constant. The difference in the amount available for export is directly and solely caused by the difference in the grinding loss percentages. Difference in Export = Difference in Loss % × Total Production Step-by-step Solution: Given: Scenario 1: 15% loss results in 30 lakh tons export. Scenario 2: 10% loss results in 40 lakh tons export. Let the total production of wheat be x lakh tons. Determine the difference in the loss percentages: 15% - 10% = 5% This 5% difference in loss directly accounts for the difference in the export volumes. Determine the difference in export volumes: 40 lakh tons - 30 lakh tons = 10 lakh tons Set up the relationship: 5% of Total Production = Difference in Exports 5% of x = 10 Solve for x: (5/100) × x = 10 (1/20) × x = 10 x = 10 × 20 = 200 Exam Strategy & Shortcut: Use the Percentage Gap Strategy. The gap in percentages is 5% (15% vs 10%). The gap in real values is 10 lakh tons (40 vs 30). Therefore, 5% of the total = 10. To find 100% (the total production), simply multiply both sides by 20. 5% × 20 = 100% . 10 × 20 = 200 . The total production is 200 lakh tons. You can solve this entirely in your head in 5 seconds. Common Pitfall: A tedious misstep is trying to set up individual complex equations for each scenario like x - 0.15x = 30 + domestic consumption . The problem doesn't mention domestic consumption, but the difference between production and export implies it exists. Focus strictly on the *change* between scenarios to bypass unknown variables. Final Answer: Therefore, the correct answer is 200 lakh tons.

Question 20

In a competitive examination in State A, 6% candidates got selected from the total appeared candidates.  State B had an equal number of candidates appeared and 7% candidates got selected with 80 more candidates got selected than A.  What was the num…

Accepted Answer

Concept & Logic: This problem is solved using the absolute difference method between percentages on a common base. Because the total number of appeared candidates is identical for both State A and State B, we can directly subtract their selection percentages. The difference in these percentages will equal the absolute difference in the number of selected candidates. Step-by-step Solution: Given: State A selection rate = 6% State B selection rate = 7% State B has 80 more selected candidates than State A. The number of appeared candidates is equal in both states. Let the total number of appeared candidates from each state be x. Calculate the number of selected candidates from each state in terms of x: Selected from State A = 6% of x = 0.06x Selected from State B = 7% of x = 0.07x We are given that State B had 80 more selections than State A. Set up the equation: 0.07x - 0.06x = 80 Simplify and solve for x: 0.01x = 80 (1/100) × x = 80 x = 80 × 100 = 8000 Exam Strategy & Shortcut: Recognize the common base immediately. Since candidates in A = candidates in B, the 1% difference in selection rate (7% - 6%) directly corresponds to the 80 student difference. If 1% = 80 students, then 100% = 8000 students. No algebraic equations are necessary; this is a purely mental calculation. Common Pitfall: A common error is overcomplicating the setup by introducing two separate variables (like x and y) for the states before realizing the problem states they are equal. Always leverage the "equal number" constraint immediately to keep it a single-variable mental math problem. Final Answer: Therefore, the correct answer is 8000.

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