Question 1

Which one of the following shows the best percentage?

Accepted Answer

Concept & Strategy: To compare fractions and determine which represents the highest percentage, convert them into decimals or percentages. The fraction with the highest numerical value represents the "best" percentage. Percentage = (\frac{Numerator}{Denominator}) × 100 Step-by-Step Solution: • Calculation: Let us evaluate each option systematically. Option (a): (384/540) Simplify the fraction: (384/540) = (38.4/54) . Calculate percentage: \approx 71.11% Option (b): (425/500) Simplify the fraction: Multiply numerator and denominator by 2 to get a base of 1000. (850/1000) = (85/100) Calculate percentage: 85% Option (c): (570/700) Simplify the fraction: (57/70) . 7 × 8 = 56 , so it is slightly more than (56/70) (which is 80%). Calculate percentage: \approx 81.42% Option (d): (480/660) Simplify the fraction: (48/66) = (8/11) . Calculate percentage: Since (1/11) = 9.09% , (8/11) = 8 × 9.09% = 72.72% . Comparing all values: 71.11%, 85%, 81.42%, 72.72%. The highest value is 85%. Exam Strategy & Shortcut: Instead of full division, use estimation and baseline comparisons. Look at (b): 425 out of 500 is clearly 85%. Now look at (c): 570 out of 700. 80% of 700 is 560, so 570 is roughly 81-82%. Look at (d): 480 out of 660. 70% of 660 is 462, so 480 is around 72-73%. Look at (a): 384 out of 540. 70% of 540 is 378, so it is just over 71%. It is immediately obvious that (b) is the highest without doing exact decimal calculations. Common Pitfall: A common mistake is trying to find a common denominator for all four fractions. Given the large and diverse denominators (540, 500, 700, 660), this approach will consume an immense amount of time and is highly prone to arithmetic errors. Final Answer: Therefore, the correct answer is (425/500).

Question 2

In a class of 65 students and 4 teachers, each student got sweets that are 20% of the total number of students and each teacher got sweets that are 40% of the total number of students. How many sweets are there?

Accepted Answer

Concept & Logic: This problem requires understanding dynamic distributions. The number of sweets given to each individual is dependent on a fixed variable—the total number of students—rather than a fixed number. Total Sweets = (Students × Sweets per Student) + (Teachers × Sweets per Teacher) Step-by-Step Solution: • Given: Total number of students = 65 Total number of teachers = 4 Sweets per student = 20% of total students Sweets per teacher = 40% of total students • Calculation: Step 1: Calculate the number of sweets each student received. Sweets per student = 20% of 65 0.20 × 65 = 13 sweets per student. Step 2: Calculate the number of sweets each teacher received. Sweets per teacher = 40% of 65 0.40 × 65 = 26 sweets per teacher. Step 3: Calculate the total number of sweets distributed. Total sweets for students = 65 × 13 = 845 Total sweets for teachers = 4 × 26 = 104 Step 4: Sum them up. Total Sweets = 845 + 104 = 949 Exam Strategy & Shortcut: Notice that the percentage for teachers (40%) is exactly double the percentage for students (20%). Since 20% of 65 is 13, you immediately know that 40% is 26 without having to calculate the percentage again. Furthermore, use unit digits to quickly verify addition: The last digit of 65 × 13 is 5 (since 5 × 3 = 15 ). The last digit of 4 × 26 is 4 (since 4 × 6 = 24 ). 5 + 4 = 9 . The answer must end in 9. The only option ending in 9 is 949. Common Pitfall: A very common reading error is calculating the percentage against the total number of *people* in the class (students + teachers = 69) instead of strictly the total number of *students* (65) as explicitly stated in the problem text. Final Answer: Therefore, the correct answer is 949.

Question 3

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 4

125% of 3060 - 85% of $x$ = 408

Accepted Answer

Concept & Logic: This is a linear equation problem involving percentages. The core strategy is to simplify the known numerical percentage term first, isolate the algebraic term containing the variable x, and then solve for x. Step-by-Step Solution: • Given: Equation: 125% × 3060 - 85% × x = 408 • Calculation: Step 1: Simplify the first term. Convert 125% to a fraction for easier multiplication. 125% = (125/100) = (5/4) Value = (5/4) × 3060 = 5 × 765 = 3825 Step 2: Substitute this value back into the equation and isolate the term with x. 3825 - 0.85x = 408 3825 - 408 = 0.85x 3417 = 0.85x Step 3: Solve for x. Convert the decimal to a fraction to avoid decimal division errors. 0.85 = (85/100) = (17/20) 3417 = ((17/20)) × x x = (3417 × 20/17) x = 201 × 20 x = 4020 Exam Strategy & Shortcut: Use digital root (sum of digits) to quickly check options, or use unit digit analysis. The equation is 3417 = 0.85x . This means x = (3417/0.85) . Since 3417 ends in 7, and we are dividing by an expression that has a 5 in the denominator structure ( (17/20) means we divide by 17 and multiply by 20). Multiplying 3417 by 20 gives a number ending in 40. Dividing by 17, the unit digit of the quotient must be 0, and the digit before it must be determined by what times 7 ends in 4. 2 × 7 = 14 . Thus, the answer must end in 20. Option (d) 4020 is the only plausible choice ending in 20. Common Pitfall: Students often waste time trying to multiply 1.25 × 3060 directly using decimals. Recognizing that 125% is exactly (5/4) turns a complex decimal multiplication into a very simple division by 4 followed by multiplication by 5. Final Answer: Therefore, the correct answer is 4020.

Question 5

$x$% 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 6

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 7

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 8

3.2% of 500 × 2.4% of $x$ = 288

Accepted Answer

Concept & Formula: This problem features the multiplication of two percentage terms. The goal is to simplify the known term first, treat it as a constant multiplier, and isolate x. (Value_1) × ( \frac{Percent_2}{100} × x ) = Result Step-by-Step Solution: • Given: Equation: 3.2% of 500 × 2.4% of x = 288 • Calculation: Step 1: Simplify the first term completely. 3.2% of 500 = ((3.2/100)) × 500 = 3.2 × 5 = 16 Step 2: Substitute this value into the equation. 16 × (2.4% of x) = 288 Step 3: Isolate the term containing x. 2.4% of x = (288/16) 2.4% of x = 18 Step 4: Solve for x. ((2.4/100)) × x = 18 x = (18 × 100/2.4) x = (1800/2.4) = (18000/24) = 750 Exam Strategy & Shortcut: Recognize that "Percent of a number" is commutative. 3.2% of 500 is exactly the same as 500% of 3.2. Since 500% means multiplying by 5, 3.2 × 5 = 16 . Then, immediately divide the result side (288) by 16 to get 18. Now you just need to solve 2.4% of x = 18 . Since 2.4% is close to 2.5% (which is (1/40) ), x will be slightly larger than 18 × 40 = 720 . 750 fits perfectly and is the only logical option. Common Pitfall: A common trap is attempting to multiply the two percentages together first ( 3.2% × 2.4% ). This leads to extremely small, error-prone decimals ( 0.000768 ). Always simplify known constants against whole numbers first before combining terms. Final Answer: Therefore, the correct answer is 750.

Question 9

85% of 485.5 = 50% of $x$

Accepted Answer

Concept & Strategy: When one side of an equation equals 50% (or exactly half) of an unknown variable, the unknown variable must be exactly double the value of the other side. Value = 0.5 × x \implies x = 2 × Value Step-by-Step Solution: • Given: Equation: 85% of 485.5 = 50% of x • Calculation: Step 1: Simplify the equation by removing the 50% fraction. 0.85 × 485.5 = 0.5 × x Multiply both sides by 2 to isolate x: x = 2 × (0.85 × 485.5) Step 2: Reorder the multiplication to make it easier. x = (2 × 0.85) × 485.5 x = 1.7 × 485.5 Step 3: Perform the final multiplication. x = 1.7 × 485.5 = 825.35 Exam Strategy & Shortcut: You can entirely avoid decimal multiplication by using fractions. 85% is (17/20) and 50% is (1/2) . (17/20) × 485.5 = (1/2) × x Multiply both sides by 2: x = (17/10) × 485.5 Dividing by 10 shifts the decimal point: x = 17 × 48.55 17 × 48 = 816 , and 17 × 0.55 = 9.35 . 816 + 9.35 = 825.35 . Common Pitfall: The most common mistake is calculating 85% of 485.5 (which is 412.675) and forgetting that this number is only *half* of the answer. Always ensure you complete the final isolation step for x. Final Answer: Therefore, the correct answer is 825.35.

Question 10

40% of 4.5 + $x$% of $\frac{2}{3}$ = 20% of 10

Accepted Answer

Concept & Formula: This problem requires evaluating a linear equation containing percentages and fractions. We must simplify the known terms on both sides of the equation to isolate the unknown percentage variable. Value_1 + ( (x/100) ) × Base = Value_2 Step-by-Step Solution: • Given: Equation: 40% of 4.5 + x% of (2/3) = 20% of 10 • Calculation: Step 1: Simplify the first term (40% of 4.5). 40% = (4/10) = 0.4 0.4 × 4.5 = 1.8 Step 2: Simplify the term on the right side of the equation (20% of 10). 20% of 10 = 2 Step 3: Substitute the simplified values back into the equation. 1.8 + ( (x/100) ) × ( (2/3) ) = 2 Step 4: Isolate the term containing x. ( (2x/300) ) = 2 - 1.8 ( (2x/300) ) = 0.2 Step 5: Solve for x. 2x = 0.2 × 300 2x = 60 x = 30 Exam Strategy & Shortcut: Use mental math to speed up the process. For 40% of 4.5: double 4.5 to get 9 (which is 20%), then double again to get 1.8 (40%). For 20% of 10: easily 2. The difference is 2 - 1.8 = 0.2 . So, what percentage of (2/3) gives 0.2? Convert 0.2 to a fraction: (1/5) . x% × ((2/3)) = (1/5) \implies x% = ((1/5)) × ((3/2)) = (3/10) = 30% . Common Pitfall: Students often struggle with mixing decimals (4.5) and fractions ( (2/3) ). A common error is converting (2/3) to 0.666..., which introduces rounding errors and makes isolating x extremely difficult. Always keep fractions as fractions until the final step. Final Answer: Therefore, the correct answer is 30.

Question 11

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 12

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 13

If $35\%$ of a number is $175$, then what percent of $175$ is that number?

Accepted Answer

Concept & Formula: This is a two-step percentage problem. First, find the unknown base number using the given percentage. Then, calculate what percentage that new base number is relative to the original given value. The percentage formula is: Percentage = ( \frac{Part}{Base} ) × 100 Step-by-Step Solution: Given: • 35% of a number = 175 Calculation: • Let the unknown number be x. (35/100) × x = 175 • Solve for x: x = (175 × 100/35) • Simplify (since 35 × 5 = 175): x = 5 × 100 = 500 • The question asks: What percent of 175 is 500? Here, 175 is the new base. Required Percentage = ( (500/175) ) × 100 • Simplify the fraction by dividing by 25: Required Percentage = ( (20/7) ) × 100 = (2000/7)% • Convert to a mixed number or decimal: (2000/7)% \approx 285.71% Exam Strategy & Shortcut: You can skip finding the absolute number entirely by using ratios! Let the number be N. We know 0.35N = 175. The question asks for the ratio of N to 175, expressed as a percentage. Since 175 = 0.35N, substituting this gives: (N/175) = (N/0.35N) = (1/0.35) (1/0.35) = (100/35) = (20/7) Convert the fraction (20/7) to a percentage: (20/7) × 100 \approx 285.7%. Common Pitfall: Students often misread "what percent of 175 is that number" as "what percent of that number is 175". If misread, you would just circle 35% (Option a). Always pay close attention to what follows the word "of", as it identifies the base of the percentage. Final Answer: Therefore, the correct answer is None of these.

Question 14

In an examination X secures $58\%$ marks and Y secures $105$ marks less than X. If the maximum marks were $700$, then what percent of marks did Y secure?

Accepted Answer

Concept & Logic: When working with percentages of a common base (the maximum marks), you can convert absolute differences directly into percentage differences to save calculation time. Step-by-Step Solution: Given: • Maximum marks = 700 • X's score = 58% of maximum marks • Y's score = X's score - 105 marks Calculation: • First, find X's absolute marks: X = (58/100) × 700 X = 58 × 7 = 406 • Next, calculate Y's absolute marks: Y = 406 - 105 = 301 • Finally, convert Y's marks back to a percentage: Y_{%} = (301/700) × 100 Y_{%} = (301/7) = 43% Exam Strategy & Shortcut: Instead of calculating the absolute marks for both students, convert the difference in marks into a percentage difference! • Difference in marks = 105 • Percentage equivalent of this difference = (105/700) × 100 = 15% • Y's percentage = X's percentage - 15% • Y's percentage = 58% - 15% = 43% This method completely bypasses the need to multiply 58 by 7. Common Pitfall: A common mistake is calculating X's marks (406) and stopping there, or incorrectly subtracting the 105 marks directly from the 58% figure without converting it. Always ensure you are adding or subtracting like units (marks with marks, percentages with percentages). Final Answer: Therefore, the correct answer is 43 percent.

Question 15

Ms. Pooja invests $13\%$ of her monthly salary, i.e. ₹ $8554$ in Mediclaim Policies. Later she invests $23\%$ of her monthly salary on Child Education Policies ; also she invests another $8\%$ of her monthly salary on Mutual Funds. What is the total…

Accepted Answer

Concept & Formula: This problem requires mapping a known percentage to a given absolute value to find the base salary, and then calculating a new cumulative percentage from that base. The timeframe (monthly vs. annual) is a crucial component of the calculation. Total Percentage × Base Value = Target Value Step-by-Step Solution: Given: • Investment in Mediclaim = 13% of Monthly Salary (MS) = ₹ 8554 • Investment in Child Education = 23% of MS • Investment in Mutual Funds = 8% of MS Calculation: • First, find the total percentage of the monthly salary that is invested: Total Invested % = 13% + 23% + 8% = 44% • Next, use the known 13% valuation to find the total monthly salary: 13% of MS = 8554 (13/100) × MS = 8554 MS = (8554 × 100/13) = 65800 • Calculate the total monthly investment (44% of MS): Monthly Investment = (44/100) × 65800 = 28952 • Calculate the total annual investment by multiplying by 12: Annual Investment = 28952 × 12 = 347424 Exam Strategy & Shortcut: To save time, bypass finding the absolute monthly salary entirely. You know 13% equals 8554. You need to find 44%, but annualized (multiplied by 12). Let I be the monthly investment. I = ( (44/13) ) × 8554 Since 8554 \div 13 = 658, the monthly investment is 44 × 658. Multiply by 12 for the annual amount: 44 × 658 × 12 = 347424. Focusing on the unit digit can quickly verify the result: 4 × 8 × 2 = 64. The unit digit must be 4. Looking at the options, only (d) ends in 4. Common Pitfall: The most common mistake is stopping after calculating the monthly investment (₹ 28952) and selecting option (a). The question explicitly asks for the total annual amount, requiring multiplication by 12. Final Answer: Therefore, the correct answer is ₹ 347424.

Question 16

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 17

David and his wife each receives an $8$ percent annual rise. If David receives a raise of ₹ $800$ and his wife receives a raise of ₹ $840$, what is the difference between their annual incomes after their raises?

Accepted Answer

Concept & Logic: When two distinct values are increased by the exact same percentage, the difference between the absolute increases is directly proportional to the difference between the original base values. Step-by-Step Solution: Given: • David's raise = ₹ 800 (which is 8% of his income) • Wife's raise = ₹ 840 (which is 8% of her income) Calculation: • Find David's original annual income (D): 8% of D = 800 D = (800/0.08) = 10000 • Find his wife's original annual income (W): 8% of W = 840 W = (840/0.08) = 10500 • Calculate their incomes after the respective raises: David's New Income = 10000 + 800 = 10800 Wife's New Income = 10500 + 840 = 11340 • Find the absolute difference: Difference = 11340 - 10800 = 540 Exam Strategy & Shortcut: Work strictly with the differences to bypass full income calculations entirely. The difference in raise amounts = 840 - 800 = 40. This ₹ 40 represents 8% of the difference in their base incomes. 8% of Base Difference = 40 Base Difference = (40/0.08) = 500 The final difference after the raise will simply be the Base Difference plus the Difference in Raises: Final Difference = 500 + 40 = 540 Common Pitfall: Students frequently calculate the difference in the base incomes (₹ 500) and hastily select option (c). However, the question specifically requests the difference after their raises, requiring the addition of the ₹ 40 raise discrepancy to the base discrepancy. Final Answer: Therefore, the correct answer is ₹ 540.

Question 18

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 19

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 20

$76\%$ of the students in a school are boys. If the number of girls is $204$, then the total number of students is

Accepted Answer

Concept & Logic: In percentage problems involving a binary split (like boys and girls), the percentages of both groups must add up to 100%. Total Percentage = % Boys + % Girls = 100% Step-by-Step Solution: Given: • Percentage of boys = 76% • Number of girls = 204 Calculation: • Find the percentage of girls in the school: % of Girls = 100% - 76% = 24% • Let the total number of students be x. • We know that 24% of the total students are girls, which equals 204. 24% of x = 204 (24/100) × x = 204 • Isolate x to find the total number of students: x = (204 × 100/24) • Simplify the fraction (divide by 12): x = 17 × 50 = 850 Exam Strategy & Shortcut: If 24% = 204, then 1% = (204/24) = 8.5. The total number of students is 100%, so multiply by 100: 8.5 × 100 = 850 This unitary method bypasses setting up a full algebraic equation and is much faster for mental math. Common Pitfall: A frequent mistake is calculating 76% of 204, confusing the given part (girls) with the whole (total students). Always align the percentage with its corresponding absolute value. Final Answer: Therefore, the correct answer is 850.

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