Question 1

Rajeev buys goods worth ₹ 6650. He gets a rebate of 6% on it. After getting the rebate, he pays sales tax @ 10%. Find the amount he will have to pay for the goods.

Accepted Answer

Concept & Logic: When both a discount (rebate) and a tax are applied, they must be calculated sequentially, not simultaneously. The rebate decreases the initial cost, and the sales tax is then applied to that newly reduced amount. Final Amount = Initial Price × (1 - \frac{Rebate}{100}) × (1 + \frac{Tax}{100}) Step-by-Step Solution: • Given: Initial value of goods = ₹ 6650 Rebate (Discount) = 6% Sales Tax = 10% • Calculation: Step 1: Calculate the amount after the 6% rebate. Price after rebate = 6650 × (1 - 0.06) = 6650 × 0.94 Price after rebate = ₹ 6251 Step 2: Calculate the final amount by adding the 10% sales tax to the rebated price. Final Amount = 6251 × (1 + 0.10) = 6251 × 1.10 Final Amount = ₹ 6876.10 Exam Strategy & Shortcut: Use successive percentage multipliers to solve it in a single line. Final Price = 6650 × 0.94 × 1.10 . Instead of multiplying 0.94 by 6650 first, multiply 0.94 × 1.10 = 1.034 . Now, calculate 6650 × 1.034 . To do this mentally: 6650 × 1 = 6650 . Plus 3% of 6650 ( 199.5 ) plus 0.4% ( 26.6 ). 6650 + 199.5 + 26.6 = 6876.10 . Common Pitfall: A major mistake is combining the percentages by simply taking the net difference (e.g., 10% tax - 6% rebate = +4% ). Applying a flat 4% increase to ₹ 6650 yields ₹ 6916, which is incorrect because the base amount changes after the rebate. Final Answer: Therefore, the correct answer is ₹ 6876.10.

Question 2

A, B, C and D have ₹ 40, 50, 60 and 70 respectively when they go to visit a fair. A spends ₹ 18, B spends ₹ 21, C spends ₹ 24 and D spends ₹ 27.   Who has done the highest expenditure proportionate to his resources?

Accepted Answer

Concept & Logic: To find the highest expenditure proportionate to resources, we must calculate the ratio (or percentage) of the amount spent to the total initial amount for each person. The highest fraction represents the highest proportionate expenditure. Proportionate Expenditure = \frac{Amount Spent}{Total Resources} Step-by-Step Solution: • Given: Resources: A = ₹ 40, B = ₹ 50, C = ₹ 60, D = ₹ 70 Expenditure: A = ₹ 18, B = ₹ 21, C = ₹ 24, D = ₹ 27 • Calculation: Let us determine the fraction of resources spent by each person: Person A: (18/40) = (9/20) = 45% Person B: (21/50) = (42/100) = 42% Person C: (24/60) = (4/10) = 40% Person D: (27/70) \approx 38.57% Comparing the percentages: 45% (A), 42% (B), 40% (C), and 38.57% (D). Person A has the highest percentage. Exam Strategy & Shortcut: Instead of calculating exact percentages, convert the denominators to a common baseline or use fractional comparison techniques. Notice B is exactly (42/100) (42%). A is (18/40) . If A had ₹ 50, proportionally they would have spent (18/4) × 5 = 22.5 , which is greater than B's 21. C is clearly (4/10) = 40% , which is less than A and B. For D, (27/70) is less than (28/70) (which is exactly 40%). Since D is less than 40%, and A is 45%, A is the clear winner without needing a calculator. Common Pitfall: Students frequently confuse absolute expenditure with proportionate expenditure. Looking at the raw numbers, D spends the most (₹ 27). However, the question explicitly asks for the expenditure *proportionate to his resources*, meaning the percentage of their initial wallet size. Final Answer: Therefore, the correct answer is A.

Question 3

$x$% of 450 + 46% of 285 = 257.1

Accepted Answer

Concept & Formula: This problem requires evaluating a linear equation involving percentages. First, isolate the unknown percentage term by calculating the known percentage value and subtracting it from the total sum on the right side. ( (x/100) ) × A + ( (B/100) ) × C = D Step-by-Step Solution: • Given: Equation: x% of 450 + 46% of 285 = 257.1 • Calculation: Step 1: Simplify the known term (46% of 285). 46% of 285 = 0.46 × 285 = 131.1 Step 2: Substitute this value back into the original equation. x% of 450 + 131.1 = 257.1 Step 3: Isolate the term with the unknown x. x% of 450 = 257.1 - 131.1 x% of 450 = 126.0 Step 4: Solve for x. ( (x/100) ) × 450 = 126 4.5x = 126 x = (126/4.5) = (1260/45) = 28 Exam Strategy & Shortcut: To quickly find 46% of 285 mentally, split it into 50% minus 4%. 50% of 285 is half of 285, which is 142.5. 1% of 285 is 2.85, so 4% is 2.85 × 4 = 11.4 . Subtract the two: 142.5 - 11.4 = 131.1 . Then subtract 131.1 from 257.1 to get 126. Finally, to solve 4.5x = 126 , double both sides to remove the decimal: 9x = 252 . Then divide: 252 / 9 = 28 . Common Pitfall: Calculation errors during the decimal multiplication of 0.46 × 285 often lead students off track. Furthermore, forgetting to convert the percentage symbol into a denominator of 100 before isolating x will result in answers off by a factor of 100. Final Answer: Therefore, the correct answer is 28.

Question 4

35568 ÷ $x$% of 650 = 456

Accepted Answer

Concept & Formula: This problem tests the order of operations combined with percentage calculations. The term "x% of 650" acts as the entire denominator for the initial division operation. (A/(\frac{x){100} × B)} = C Step-by-Step Solution: • Given: Equation: 35568 \div (x% of 650) = 456 • Calculation: Step 1: Rearrange the equation to isolate the percentage term. If (35568/Term) = 456 , then Term = (35568/456) . x% of 650 = (35568/456) Step 2: Perform the division. (35568/456) = 78 So, x% of 650 = 78 Step 3: Solve for x. ((x/100)) × 650 = 78 6.5x = 78 x = (78/6.5) x = (780/65) x = 12 Exam Strategy & Shortcut: Instead of heavy division upfront, look at unit digits and relative scale. You know 456 × Something = 35568 . The unit digit of 456 is 6, and the target ends in 8. 6 × 3 = 18 or 6 × 8 = 48 . The multiplier likely ends in 8. Since 450 × 80 = 36000 , the multiplier must be exactly 78. Then solve 6.5 × x = 78 . 6 × 12 = 72 and 0.5 × 12 = 6 . 72 + 6 = 78 . Thus x = 12 . Common Pitfall: A major error is misinterpreting the order of operations as (35568 \div x%) × 650 . The word "of" binds the percentage and 650 together tightly as a single term before the division occurs. Final Answer: Therefore, the correct answer is 12.

Question 5

23% of 8040 + 42% of 545 = $x$% of 3000

Accepted Answer

Concept & Logic: This requires evaluating the sum of two separate percentage expressions and then determining what percentage that total sum represents out of a new base number (3000). Total Sum = ((x/100)) × Base Step-by-Step Solution: • Given: Equation: 23% of 8040 + 42% of 545 = x% of 3000 • Calculation: Step 1: Evaluate the first term (23% of 8040). 0.23 × 8040 = 1849.2 Step 2: Evaluate the second term (42% of 545). 0.42 × 545 = 228.9 Step 3: Add the two terms together to find the total sum. 1849.2 + 228.9 = 2078.1 Step 4: Set up the final equation and solve for x. 2078.1 = x% of 3000 2078.1 = ((x/100)) × 3000 2078.1 = 30x x = (2078.1/30) = 69.27 Exam Strategy & Shortcut: Break down complex percentages into easier chunks. For 23% of 8040: Find 20% (double 10%, so 804 × 2 = 1608 ). Then find 3% ( 80.4 × 3 = 241.2 ). Sum them: 1608 + 241.2 = 1849.2 . For 42% of 545: Find 40% ( 54.5 × 4 = 218 ). Then find 2% ( 5.45 × 2 = 10.9 ). Sum them: 218 + 10.9 = 228.9 . Add the two results: 1849.2 + 228.9 = 2078.1 . Divide by 30 to get x. The arithmetic is much safer and faster using this additive chunking method. Common Pitfall: Because the final equation is 2078.1 = 30x , students often mistakenly divide by 3000 instead of 30, forgetting that the percentage sign on x already canceled out two zeros from the 3000. Final Answer: Therefore, the correct answer is 69.27.

Question 6

40% of 60% of 32% of an amount is ₹ 432. What is the amount?

Accepted Answer

Concept & Formula: This requires solving a successive percentage chain. Converting percentages to simplified fractions before solving the algebraic equation will drastically reduce the risk of decimal multiplication errors. Final Amount = Base × ( (P_1/100) ) × ( (P_2/100) ) × ( (P_3/100) ) Step-by-Step Solution: • Given: Equation: 40% of 60% of 32% of Amount (A) = 432 • Calculation: Step 1: Convert all percentages into their simplest fractional forms. 40% = (40/100) = (2/5) 60% = (60/100) = (3/5) 32% = (32/100) = (8/25) Step 2: Set up the equation with the fractions. ( (2/5) ) × ( (3/5) ) × ( (8/25) ) × A = 432 Step 3: Multiply the numerators and denominators together. ( (2 × 3 × 8/5 × 5 × 25) ) × A = 432 ( (48/625) ) × A = 432 Step 4: Solve for A by isolating it. A = 432 × ( (625/48) ) A = 9 × 625 A = 5625 Since ₹ 5625 is not explicitly listed among options (a), (b), or (c), the correct choice must be "None of these". Exam Strategy & Shortcut: Look for clean multiples. Notice that the numerator of the combined fraction is 48 ( 4 × 6 × 2 dropping zeroes). You have to divide 432 by 48. 432 \div 48 . Notice that 48 × 10 = 480 . 432 is exactly 48 less than 480. So, 432 \div 48 is exactly 9. Then you just multiply 9 by the combined denominator ( 5 × 5 × 25 = 625 ). 9 × 625 = 5625 . This mental math shortcut avoids writing down a massive decimal chain like 0.4 × 0.6 × 0.32 = 0.0768 . Common Pitfall: The most common trap is attempting to multiply the decimals directly ( 0.4 × 0.6 × 0.32 = 0.0768 ) and then attempting long division ( 432 \div 0.0768 ) by hand. This wastes tremendous time and frequently leads to misplaced decimal points in the final answer. Final Answer: Therefore, the correct answer is None of these.

Question 7

If ₹ 2800 is $\frac{2}{7}$ percent of the value of a house, the worth of the house (in ₹) is

Accepted Answer

Concept & Formula: This problem requires finding the total base value when a fractional percentage of it is known. The critical step is remembering that the percentage sign (%) means dividing the fraction itself by an additional 100. Value = ( \frac{Fractional Percentage}{100} ) × Base Step-by-Step Solution: • Given: Part value = ₹ 2800 Percentage = (2/7)% • Calculation: Step 1: Set up the equation where V is the total worth of the house. ( (2/7)% ) of V = 2800 Step 2: Convert the percentage into a proper numerical fraction by dividing by 100. ( (\frac{2/7)}{100} ) × V = 2800 ( (2/700) ) × V = 2800 Step 3: Isolate V and solve. V = 2800 × ( (700/2) ) V = 1400 × 700 Step 4: Multiply out the final result. 14 × 7 = 98 Append the four zeros: 980000 V = 9,80,000 Exam Strategy & Shortcut: Think in terms of 1%. If (2/7)% = 2800 , you can find what (1/7)% is by halving both sides. (1/7)% = 1400 . Now find 1% by multiplying both sides by 7. 1% = 1400 × 7 = 9800 . If 1% of the house is 9800, then the total 100% value is simply 9800 with two more zeros added to the end. 100% = 9,80,000 . This completely avoids complex fraction-over-fraction algebra. Common Pitfall: The most frequent error on this specific question type is ignoring the word "percent". Students will set up the equation as (2/7) × V = 2800 (treating it as two-sevenths of the house, rather than two-sevenths of one percent). This will result in an incorrect answer of 9800, leaving them confused when looking at the options. Always divide by 100 when the "%" symbol is present. Final Answer: Therefore, the correct answer is 9,80,000.

Question 8

Nupur invests ₹ $89856$, which is $26\%$ of her annual income, in mutual funds. What is her monthly income?

Accepted Answer

Concept & Formula: This problem links an absolute investment amount to a percentage of an annual income, but requires the monthly income as the final answer. Converting between annual and monthly figures is the key algebraic step. Percentage × Base Income = Investment Amount Step-by-Step Solution: Given: • Investment amount = ₹ 89856 • Percentage of annual income = 26% Calculation: • Let the Annual Income be A. 26% of A = 89856 (26/100) × A = 89856 • Solve for Annual Income A: A = (89856 × 100/26) A = 3456 × 100 = 345600 • The question asks for the monthly income. Divide the annual income by 12: Monthly Income = (345600/12) = 28800 Exam Strategy & Shortcut: Calculate the monthly component directly in one equation to minimize large number division. Let Monthly Income be M. The Annual Income is 12M. 26% of (12M) = 89856 12M = (89856/0.26) M = (89856/12 × 0.26) = (7488/0.26) M = (748800/26) = 28800 By dividing by 12 early on (89856 \div 12 = 7488), the numbers stay much smaller and easier to manage during calculation. Common Pitfall: A common pitfall is successfully calculating the total annual income (₹ 345600) but forgetting to divide by 12 at the end because the word "annual" was prominent at the start of the problem statement. Final Answer: Therefore, the correct answer is ₹ 28800.

Question 9

If the average of a number, its $75\%$ and its $25\%$ is $240$, then the number is

Accepted Answer

Concept & Formula: The average of multiple terms is their total sum divided by the number of terms. By expressing all terms as percentages of a single unknown variable, you can establish a simple linear equation. Average = \frac{Sum of Terms}{Number of Terms} Step-by-Step Solution: Given: • The three terms are: a number, its 75%, and its 25%. • The average of these three terms is 240. Calculation: • Let the unknown base number be x. • The three terms expressed algebraically are x, 0.75x, and 0.25x. • Set up the mathematical average equation: (x + 0.75x + 0.25x/3) = 240 • Combine the like terms in the numerator: (2x/3) = 240 • Isolate x by multiplying both sides by 3: 2x = 720 • Divide by 2 to find the final number: x = 360 Exam Strategy & Shortcut: Recognize percentage complements to simplify the arithmetic immediately. 75% and 25% perfectly add up to 100%, which constitutes exactly 1 whole number. Thus, you have the original number (1) plus the two percentages which equal a second whole number (1), yielding a total sum of 2 times the number. If the average of 3 distinct items is 240, their total sum must be 240 × 3 = 720. Since 2 × Number = 720, the number is simply 360. Common Pitfall: A standard error is forgetting to include the original base number itself in the average calculation, effectively only averaging the 75% and 25% terms. This leads to incorrect denominator usage and a fundamentally flawed equation. Final Answer: Therefore, the correct answer is 360.

Question 10

The value of which of the following fractions is less than twenty percent?

Accepted Answer

Concept & Logic: To compare fractions with a percentage, it is easiest to convert the percentage into a baseline fraction or convert all fractions into approximate percentages. 20% = (20/100) = (1/5) Step-by-Step Solution: Calculation: • Establish the baseline threshold: 20% = (1/5) = 0.20. We are looking for a fraction smaller than this. • Evaluate each option: • (a) 5/6: Since 5 is close to 6, this is over 80%. (\approx 83.33%) • (b) 2/3: This is a standard fraction for 66.66%. • (c) 2/5: This is exactly double 1/5, so it equals 40%. • (d) 1/4: This is exactly 25%. • (e) 2/11: To compare 2/11 with 1/5, find a common numerator or convert to decimals. • 1/5 = 2/10 • Comparing 2/11 and 2/10: Since 11 is a larger denominator than 10, the fraction 2/11 must be smaller than 2/10. • Therefore, 2/11 < 1/5 (or 20%). Exam Strategy & Shortcut: Memorize standard fraction-to-percentage conversions for numbers 1 through 12. 1/11 \approx 9.09% Therefore, 2/11 \approx 18.18%. Since 18.18% < 20%, you can immediately identify (e) as the correct answer without manually evaluating the other simpler fractions. Common Pitfall: A common trap is attempting long division for every single fraction to find exact decimal values. This wastes valuable time. Use cross-multiplication or standard percentage equivalents to quickly eliminate options. Final Answer: Therefore, the correct answer is 2/11.

Question 11

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 12

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 13

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 14

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 15

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 16

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 17

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 18

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 19

Twenty-five percent of Reena's yearly income is equal to seventy-five percent of Anubhav's monthly income.  If Anubhav's yearly income is ₹ 240000, what is Reena's monthly income?

Accepted Answer

Concept & Strategy: This problem tests careful reading and unit consistency. The mathematical equations are simple, but the trap lies in the constant switching between "monthly" and "yearly" income terms. Always normalize all variables to the same time frame (either all monthly or all yearly) before equating them. Step-by-step Solution: Given: Anubhav's yearly income = ₹ 2,40,000 First, find Anubhav's monthly income: Anubhav's Monthly = (240000/12) = 20000 Set up the equation provided in the first sentence: 25% of Reena's Yearly = 75% of Anubhav's Monthly 0.25 × Reena's Yearly = 0.75 × 20000 Calculate the right side: 75% of 20000 = 15000 Substitute back to find Reena's yearly income: 0.25 × Reena's Yearly = 15000 (1/4) × Reena's Yearly = 15000 Reena's Yearly = 15000 × 4 = 60000 The question specifically asks for Reena's *monthly* income: Reena's Monthly = \frac{Reena's Yearly}{12} Reena's Monthly = (60000/12) = 5000 Reena's monthly income is ₹ 5000. Looking at the options (60000, 12000, 5200), ₹ 5000 is not listed. Exam Strategy & Shortcut: Simplify the percentages immediately. 25% is (1/4) and 75% is (3/4) . (1/4) Reena (Yr) = (3/4) Anubhav (Mo). Multiply both sides by 4: Reena (Yr) = 3 × Anubhav (Mo). Anubhav's monthly = 240000 \div 12 = 20000 . Reena's yearly = 3 × 20000 = 60000 . Reena's monthly = 60000 \div 12 = 5000 . This streamlined mental math reduces the chance of getting lost in decimals. Common Pitfall: The primary pitfall is calculating Reena's yearly income as ₹ 60,000 and immediately selecting Option A, completely missing that the question asks for her *monthly* income. Exam setters specifically place intermediate calculation results in the options to catch rushed students. Final Answer: Therefore, the correct answer is None of these.

Question 20

If a number is reduced by 40% it becomes two-thirds of another number.  What is the ratio of the first number to the second number?

Accepted Answer

Concept & Formula: This problem requires translating language into a simple linear algebraic equation to find a ratio between two unknown variables. Reducing a number by a percentage means subtracting that percentage from 100%. Reducing by 40% leaves exactly 60% of the original value. Step-by-step Solution: Let the first number be A and the second number be B. The problem states that if A is reduced by 40%, it equals two-thirds of B. First, express "A reduced by 40%": A - 40% of A = 60% of A = 0.6A Now, set up the equation according to the problem: 60% of A = (2/3) of B Convert the percentage to a fraction to make it easier to solve with the other fraction: (60/100)A = (2/3)B (3/5)A = (2/3)B To find the ratio of the first number to the second ( (A/B) ), isolate (A/B) on one side of the equation: (A/B) = (\frac{2/3)}{(3/5)} To divide fractions, multiply by the reciprocal: (A/B) = (2/3) × (5/3) (A/B) = (10/9) This means the ratio of A to B is 10 : 9. Exam Strategy & Shortcut: Memorize fraction-percentage equivalents to write the equation instantly. 40% reduction means 60% remains. 60% = (3/5) . You can write (3/5)A = (2/3)B in three seconds. Cross-multiply the denominators mentally: 3 × 3 = 9 (goes with A), 2 × 5 = 10 (goes with B). So 9A = 10B . Therefore, the ratio A : B is simply the reversed coefficients: 10 : 9 . Common Pitfall: The most frequent error is misreading "ratio of the first to the second" and accidentally giving the inverse ratio B : A , which is 9 : 10 (Option D). Always explicitly write out (A/B) to ensure your final terms are correctly oriented. Final Answer: Therefore, the correct answer is 10 : 9.

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