Question 1

In a market research project, 20% opted for Nirma detergent whereas 60% opted for Surf Blue detergent. The remaining individuals were not certain. If the difference between those who opted for Surf Blue and those who were uncertain was 720, how many…

Accepted Answer

Concept & Logic The total percentage of respondents must equal 100%. Find the percentage for the uncertain group, then equate the given numerical difference to its corresponding percentage difference to find the total. Total Respondents = \frac{Given Difference}{Percentage Difference} Step-by-Step Solution • Given: Nirma = 20%, Surf Blue = 60%. • Calculate the percentage of uncertain individuals: 100% - (20% + 60%) = 100% - 80% = 20%. • Given: The numerical difference between Surf Blue and Uncertain is 720. • Calculate the percentage difference between Surf Blue and Uncertain: 60% - 20% = 40%. • Set up the equation: 40% of Total Respondents = 720. • Total Respondents = (720/40%) = (720/0.40) = 1800. Exam Strategy & Shortcut Difference in percentage is 60% - 20% = 40%. 40% corresponds to 720. To find 100%, you can use a multiplier. 40% × 2.5 = 100%. Therefore, 720 × 2.5 = 1800. Common Pitfall Calculating the difference between Nirma and Surf Blue (40%) by misreading the question, instead of Surf Blue and the Uncertain group. Read the specific comparison groups carefully. Final Answer Therefore, the correct answer is 1800.

Question 2

While purchasing one item costing ₹ 400, I had to pay the sales tax at 7% and on another costing ₹ 6400, the sales tax was 9%. What percent of the sales tax I had to pay, taking the two items together on an average?

Accepted Answer

Concept & Formula: This problem requires calculating the weighted average of two percentages. You cannot simply average the two percentages because they are applied to vastly different base amounts (costs). Weighted Average % = \frac{Total Tax Paid}{Total Cost Base} × 100 Step-by-Step Solution: • Given: • Cost of item 1 = ₹ 400 • Tax on item 1 = 7% • Cost of item 2 = ₹ 6400 • Tax on item 2 = 9% • Calculation: • Step 1: Calculate the absolute tax for each item. Tax 1 = 7% of 400 = 28. Tax 2 = 9% of 6400 = 576. • Step 2: Find the total tax and total cost. Total Tax Paid = 28 + 576 = 604. Total Cost = 400 + 6400 = 6800. • Step 3: Calculate the overall tax percentage. Overall Tax % = ( (604/6800) ) × 100% = (604/68)% Divide both numerator and denominator by 4: = (151/17)% • Step 4: Convert to a mixed fraction. 151 \div 17 = 8 with a remainder of 15. So, (151/17)% = 8(15/17)%. Exam Strategy & Shortcut: Use the weighted average ratio shortcut to save calculation time. The ratio of the costs is 400 : 6400, which simplifies to 1 : 16. Apply these weights directly to the tax percentages: Average = ((1 × 7) + (16 × 9)/1 + 16) = (7 + 144/17) = (151/17) = 8(15/17)%. Common Pitfall: A very common mistake is finding the simple arithmetic mean of the two tax rates: (7 + 9/2) = 8%. This completely ignores the proportional weight of the much more expensive ₹ 6400 item, which pulls the true average closer to 9%. Final Answer: Therefore, the correct answer is 8(15/17)%.

Question 3

If $x$% of $y$ is 100 and $y$% of $z$ is 200, find a relation between $x$ and $z$.

Accepted Answer

Concept & Formula: The foundational concept here is translating percentage statements into algebraic equations. The phrase "a% of b" translates directly to the product of a and b divided by 100. Setting up a system of equations allows us to divide them and find the direct ratio between the target variables, bypassing the need to solve for the intermediate variable entirely. a% of b = (a × b/100) Step-by-Step Solution: • Given: • x% of y = 100 • y% of z = 200 • Calculation: Convert the first statement into an equation: (x/100) × y = 100 \implies xy = 10000 Convert the second statement into an equation: (y/100) × z = 200 \implies yz = 20000 To find the relation between x and z, we need to eliminate y. The most efficient way to do this when variables are multiplied is to divide the two equations. Divide the second equation by the first: (yz/xy) = (20000/10000) The y terms cancel out cleanly: (z/x) = 2 Multiply both sides by x to isolate z: z = 2x Exam Strategy & Shortcut: Notice that the right side of the equations (200) is exactly double the other (100). Because both expressions take a percentage involving a common variable y (i.e., x × y and y × z), their products scale linearly. You can visually inspect this: (y × z) is double (x × y). The y cancels immediately in your head, meaning z must simply be double x. You can mark z = 2x in under 5 seconds without writing anything! Common Pitfall: A major time-wasting pitfall is attempting to isolate y in the first equation (e.g., y = (10000/x)) and substituting it into the second. While mathematically sound, it introduces messy fractions and slows down your solving speed significantly. Always look to divide equations when eliminating a multiplied variable. Final Answer: Therefore, the correct answer is z = 2x.

Question 4

If $p$% of $p$ is 36, then $p$ is equal to

Accepted Answer

Concept & Formula: This is a fundamental percentage equation that loops a variable back onto itself, creating a quadratic relationship (a square). By applying the basic definition of a percentage, we can isolate the squared variable and solve by taking the square root. p% of p = (p/100) × p Step-by-Step Solution: • Given: • p% of p = 36 • Calculation: Translate the mathematical English into an algebraic equation. Replace "of" with multiplication and "p%" with (p/100): (p/100) × p = 36 Multiply the p terms in the numerator: (p^2/100) = 36 Multiply both sides by 100 to isolate p^2: p^2 = 36 × 100 p^2 = 3600 Take the square root of both sides to solve for p. (Since the options are positive and standard aptitude context implies positive values, we take the principal root): p = \sqrt{3600} p = 60 Exam Strategy & Shortcut: You can solve this purely through mental math by understanding squares. The equation (p^2/100) = 36 means p^2 = 3600. You should instantly recognize that \sqrt{36} = 6. Because there are two zeros in 3600, its square root will have exactly half that amount of zeros (one zero). Therefore, the answer is just 6 with a zero attached: 60. Common Pitfall: The most common trap is forgetting to multiply by the 100 from the percentage denominator, leading students to solve p^2 = 36 \implies p = 6. Always remember that the `%` symbol represents a hidden division by 100 that must be cleared algebraically. Final Answer: Therefore, the correct answer is 60.

Question 5

If 8% of $x = 4$% of $y$, then 20% of $x$ is

Accepted Answer

Concept & Strategy: This is an equivalence proportion problem. Because percentages are inherently linear scalar multipliers, if you know the ratio between a specific percentage of x and a specific percentage of y, you can scale that relationship up or down proportionally without needing to find the absolute values of x or y. a% of x = b% of y \implies x = (b/a)y Step-by-Step Solution: • Given: 8% of x = 4% of y • Calculation: First, establish the fundamental algebraic relationship between x and y. Drop the % signs (as it divides both sides by 100): 8x = 4y Divide by 4 to simplify the ratio: 2x = y \implies x = (y/2) The question asks for the value of "20% of x". Substitute our finding for x into this expression: 20% of x = 20% of ((y/2)) = (20/100) × (y/2) = (10/100) × y = 10% of y Exam Strategy & Shortcut: Use the scalar multiplier trick. You are given the value of 8% of x and need to find 20% of x. Look at the given equivalence: 8%(x) = 4%(y). Notice that the percentage on the right (4%) is exactly half of the percentage on the left (8%). This means any percentage of x will equal half that percentage of y. Therefore, 20% of x must simply be half of 20 applied to y. Half of 20 is 10. The answer is instantly 10% of y. Common Pitfall: A common mistake is applying operations improperly when substituting, such as multiplying the 20 by 2 instead of dividing, leading to an incorrect choice of 40% (if it were an option) or 80%. Sticking strictly to proportional logic prevents this. Final Answer: Therefore, the correct answer is 10% of y.

Question 6

A company received two shipments of ball bearings. In the first shipment, $1$ percent of the ball bearings were defective. In the second shipment, which was twice as large as the first, $4.5$ percent of the ball bearings were defective. If the compa…

Accepted Answer

Concept & Logic: This problem requires setting up a basic linear equation based on parts of a whole. Establish a base variable for the unknown size of the first shipment, express the second shipment in terms of that base, and sum their respective defective amounts to equal the total given. Step-by-step Solution: Step 1: Define the variables Let the total number of ball bearings in the first shipment be x. Since the second shipment is twice as large, its total number is 2x. Step 2: Express the defective amounts for each shipment Defectives in 1st shipment = 1% of x = 0.01x Defectives in 2nd shipment = 4.5% of 2x = 0.045 × 2x = 0.09x Step 3: Set up the total defective equation We are given that the total number of defective ball bearings is 100. 0.01x + 0.09x = 100 Step 4: Solve for x Combine the like terms: 0.10x = 100 Isolate x by dividing by 0.10 (which is the same as multiplying by 10): x = (100/0.10) x = 1000 The first shipment (x) contained 1000 ball bearings. Exam Strategy & Shortcut: Use a weighted average approach mentally. Assume the first shipment is 100 units. It yields 1 defect. The second shipment is 200 units (2 × larger). It yields 4.5% of 200 = 9 defects. Total defects for every 300 shipped = 1 + 9 = 10 defects. Since the actual total defects are 100 (which is 10 × 10), the actual shipment sizes must be 10 times our assumed numbers. First shipment = 100 × 10 = 1000. Common Pitfall: Students sometimes calculate the total combined percentage incorrectly by averaging 1% and 4.5% to get 2.75%, completely ignoring that the second shipment is twice as large (it has a higher weight). Always account for the different base sizes when adding percentages. Final Answer: Therefore, the correct answer is 1000.

Question 7

In a graduate class of $200$, $40\%$ are women and $\frac{1}{5}$ become lecturers. If the number of men who become lecturers is twice that of women, calculate approximate percentage of men who became lecturers.

Accepted Answer

Concept & Logic: To find the percentage of a specific subset, we need two absolute numbers: the total size of the group in question (total men) and the size of the subset (men who became lecturers). Percentage = (\frac{Men Lecturers}{Total Men}) × 100 Step-by-step Solution: Given: Total students = 200 Women = 40% of total Step 1: Calculate total women and men Total Women = 40% of 200 = 80 Total Men = 200 - 80 = 120 Step 2: Calculate women who became lecturers Women Lecturers = (1/5) of 80 Women Lecturers = 16 Step 3: Calculate men who became lecturers The problem states this is twice the number of women lecturers. Men Lecturers = 2 × 16 = 32 Step 4: Calculate the required percentage We need the percentage of *men* who became lecturers (out of the total men). Percentage = ((32/120)) × 100 = ((4/15)) × 100 = (400/15) = 26.66% Rounding to the nearest whole number gives an approximate percentage of 27%. Exam Strategy & Shortcut: Work purely in proportions or use a base of 100 to shrink the mental math. If the class is 100, women = 40, men = 60. Women lecturers = (1/5) of 40 = 8. Men lecturers = 2 × 8 = 16. Percentage of men = (16/60) = (4/15). Memorizing common fractions helps here: (1/15) \approx 6.66%, so (4/15) \approx 26.66%, which rounds to 27%. Common Pitfall: A major trap is calculating the percentage of men lecturers out of the *entire* class (32 out of 200 = 16%) instead of out of the total number of *men* (32 out of 120). The examiners specifically included 16% as option (a) to catch this mistake! Final Answer: Therefore, the correct answer is 27%.

Question 8

5 kg of tea and 8 kg of sugar together cost ₹ 172. The price of tea has risen by 20% and that of sugar by 10%. Hence the same quantities of tea and sugar now cost ₹ 199.20. What is the original price of tea per kg?

Accepted Answer

Concept & Formula: When tackling itemized cost problems with distinct percentage increases, you can either model the new total cost using multipliers (1.2, 1.1) or model strictly the difference (the price increase) to form a much simpler linear equation. Step-by-Step Solution: • Let the original price of tea per kg be T and sugar per kg be S. • Write the initial cost equation: 5T + 8S = 172 \quad --- (Eq 1) • Write the new cost equation based on the percentage increases: 5(1.2T) + 8(1.1S) = 199.20 6T + 8.8S = 199.20 \quad --- (Eq 2) • To solve for T, eliminate S. Multiply (Eq 1) by 1.1 to match the S coefficients: 5.5T + 8.8S = 189.20 \quad --- (Eq 3) • Subtract (Eq 3) from (Eq 2): (6T - 5.5T) + (8.8S - 8.8S) = 199.20 - 189.20 0.5T = 10 • Solve for T: T = (10/0.5) = 20 Exam Strategy & Shortcut: Instead of working with the full new prices, analyze just the increase in cost. Total cost increase = 199.20 - 172 = 27.20. This increase comes directly from a 20% bump on tea total and a 10% bump on sugar total: 0.2(5T) + 0.1(8S) = 27.20 T + 0.8S = 27.20 We know from the original scenario that 5T + 8S = 172. Dividing that entire equation by 10 gives: 0.5T + 0.8S = 17.2 Subtracting this from the increase equation kills the S term instantly: 0.5T = 10 \implies T = 20. This method minimizes large decimal arithmetic under pressure. Common Pitfall: Trying to isolate variables using substitution (e.g., S = (172 - 5T/8)) right from the start. This creates messy fractions and increases the chance of arithmetic errors compared to the elimination or cost-increase method. Final Answer: Therefore, the correct answer is ₹ 20.

Question 9

$37\frac{1}{2}\%$ of the candidates in an examination were girls, $75\%$ of the boys and $62\frac{1}{2}\%$ of the girls passed and $342$ girls failed. The number of boys failed was:

Accepted Answer

Concept & Strategy: This problem tests standard percentage manipulation, but the key to speed is converting clunky percentages into simple fractions. Memorizing standard percentage-to-fraction conversions allows you to work with ratios instead of decimals, drastically reducing calculation time. Step-by-Step Solution: • Given: Percentage of girls = 37.5% = (3/8) of the total candidates. Percentage of boys = 100% - 37.5% = 62.5% = (5/8) of the total candidates. Passed girls = 62.5% = (5/8) of total girls. Passed boys = 75% = (3/4) of total boys. Failed girls = 342. • Calculation / Deduction: First, find the total number of girls. Since 62.5% of girls passed, the percentage of girls who failed is 100% - 62.5% = 37.5% = (3/8) of all girls. Let the total number of girls be G. (3/8) × G = 342 G = 342 × (8/3) = 114 × 8 = 912. So, there are 912 girls in total. Next, find the total number of boys. The ratio of girls to boys is 37.5% : 62.5% = 3 : 5. Since 3 ratio units = 912, 1 ratio unit = 304. Total boys (5 ratio units) = 5 × 304 = 1520. Finally, calculate the number of failed boys. Since 75% of the boys passed, 25% ((1/4)) of the boys failed. Failed boys = (1/4) × 1520 = 380. Exam Strategy & Shortcut: Immediately convert standard percentages to fractions: 37.5% \rightarrow (3/8), 62.5% \rightarrow (5/8), 75% \rightarrow (3/4). Use the ratio of Girls:Boys directly as 3:5. Failed girls = (3/8) of total girls. So, (3/8)G = 342 \Rightarrow G = 912. If 3 units (girls) = 912, then 5 units (boys) = 1520. Failed boys = (1/4) of 1520 = 380. Common Pitfall: A major trap is calculating the total number of candidates first (x = 2432). While it gives the correct answer eventually, it introduces larger numbers into your arithmetic, increasing the chance of calculation errors and wasting valuable time. Jump straight to the ratios! Final Answer: Therefore, the correct answer is 380.

Question 10

$\frac{5}{9}$ part of the population in a village are males. If $30\%$ of the males are married, the percentage of unmarried females in the total population is

Accepted Answer

Concept & Logic: This problem requires assuming a convenient base population to simplify fraction and percentage calculations. Additionally, it relies on an implicit logical assumption standard in aptitude tests: in a closed population (like a village), a marriage involves one male and one female. Therefore, the number of married males exactly equals the number of married females. Step-by-Step Solution: • Given: Males = (5/9) of the total population. Married males = 30% of total males. • Calculation / Deduction: Let the total population of the village be 900 (a multiple of 9 and 100 makes the math extremely clean). Total males = (5/9) × 900 = 500. Total females = 900 - 500 = 400. Calculate married males: Married males = 30% of 500 = (30/100) × 500 = 150. Apply the logic of pairs: Since 150 males are married, they must be married to 150 females. Therefore, the number of married females = 150. Find unmarried females: Unmarried females = Total females - Married females Unmarried females = 400 - 150 = 250. Calculate the required percentage: Percentage of unmarried females in the total population = ( (250/900) ) × 100% = (250/9)% = 27(7/9)%. Exam Strategy & Shortcut: Whenever you see a fraction like (5/9) alongside percentages, immediately assume a base that contains both 9 and 100 as factors. Using 900 turns the entire problem into simple mental arithmetic: Males = 500, Females = 400. 30% of 500 = 150. Married females = 150. Unmarried females = 250. (250/900) = (25/90) = (5/18). To get the percentage, (250/9) = 27.77%, which matches 27(7/9)%. Common Pitfall: The single most common mistake is overlooking the implicit rule that "Number of Married Males = Number of Married Females". Students often get stuck after calculating 150 married males because they feel they don't have enough information to find the female marital status. Final Answer: Therefore, the correct answer is 27(7/9)%.

Question 11

A debtor can pay 87 paise in the rupee, but if his creditors would take $20\%$ of his debts, he could pay them and have ₹ 42 left. His debts and assets respectively are:

Accepted Answer

Concept & Strategy: This question relies heavily on understanding financial phrasing common in older aptitude tests. "Paying 87 paise in the rupee" means the person's total assets amount to exactly 87% of their total debts. "Taking 20% off his debts" (implied by context, though phrased as "take 20% of his debts" in the source) means the creditors accept an 80% settlement. Formulating a simple algebraic equation linking assets, settled debts, and remainder solves the problem. Step-by-Step Solution: • Given: Paying capacity = 87 paise per rupee (87%). Creditors accept a 20% haircut (they settle for 80% of the original debt). Amount left after settling = ₹ 42. • Calculation / Deduction: Let the total debt be D. The debtor's assets (A) are 87% of his debt. A = 0.87D If the creditors accept 80% of the debt (a 20% reduction), the new amount payable is 0.80D. The problem states that if he pays this new amount, he will have ₹ 42 left over from his assets. Therefore, Assets - Settled Debt = Remainder 0.87D - 0.80D = 42 0.07D = 42 Solving for D: D = (42/0.07) = (4200/7) = 600. So, his total debts are ₹ 600. Now find his assets (A): A = 0.87 × 600 = 87 × 6 = 522. His assets are ₹ 522. Exam Strategy & Shortcut: Use option elimination based on the first condition: "Assets are 87% of debts". Look at the options for (Debt, Asset): (a) 400, 520 \rightarrow Asset > Debt (Incorrect, he would be solvent) (b) 500, 521 \rightarrow Asset > Debt (Incorrect) (c) 600, 522 \rightarrow 0.87 × 600 = 522. (Perfect Match!) You don't even need to test the second condition. Understanding the initial ratio allows you to solve this in 5 seconds just by scanning the options. Common Pitfall: The phrase "if his creditors would take 20% of his debts" is notoriously confusing. A modern reader might interpret this as creditors accepting exactly 20%, meaning the payable amount is 0.20D. However, in classical financial aptitude contexts, "taking 20%" meant taking 20% *off* (a discount), making the payable amount 80%. Always let the options guide ambiguous phrasing. Final Answer: Therefore, the correct answer is ₹ 600, ₹ 522.

Question 12

Asha's monthly income is $60\%$ of Deepak's monthly income and $120\%$ of Maya's monthly income. What is Maya's monthly income if Deepak's monthly income is ₹ 78000?

Accepted Answer

Concept & Logic: This problem tests chained percentage relationships. Instead of finding complex ratios between all three individuals simultaneously, calculate the linking variable (Asha's income) first using the concrete value provided (Deepak's income), and then work backward to find Maya's income. Step-by-Step Solution: • Given: Asha's Income (A) = 60% of Deepak's Income (D). Asha's Income (A) = 120% of Maya's Income (M). Deepak's Income (D) = ₹ 78000. • Calculation / Deduction: First, calculate Asha's actual income using Deepak's value: A = (60/100) × 78000 A = 60 × 780 = 46800 So, Asha earns ₹ 46800. Next, use Asha's income to find Maya's income: We know A = 1.20 × M. Substitute the value of A: 46800 = 1.20 × M M = (46800/1.20) M = (468000/12) M = 39000 Exam Strategy & Shortcut: Use fractions instead of percentages for quicker mental math. 60% = (3/5) and 120% = (6/5). A = (3/5) × 78000. You don't even need to fully solve this right away. We know A = (6/5) × M. Therefore, (6/5) × M = (3/5) × 78000. Cancel out the denominators (5): 6 × M = 3 × 78000 M = (1/2) × 78000 = 39000. This fractional equivalence method completely skips the intermediate step of finding Asha's exact income and saves significant time! Common Pitfall: A very common mistake is calculating Maya's income as 120% of Deepak's or getting the inverse relationship wrong (i.e., calculating Maya as 120% of Asha rather than Asha being 120% of Maya). Always write down the linear equation exactly as the sentence reads before plugging in numbers. Final Answer: Therefore, the correct answer is ₹ 39000.

Question 13

A sum of ₹ 2236 is divided among A, B and C in such a way that A receives $25\%$ more than C and C receives $25\%$ less than B. What is A's share in the amount?

Accepted Answer

Concept & Strategy: When a total sum is distributed based on relative percentages, the most efficient method is converting the percentage relationships into a combined continuous ratio (A : B : C). Using fractions instead of decimals keeps the ratio integers clean and prevents rounding errors. Step-by-Step Solution: • Given: Total Sum = ₹ 2236. A = C + 25% of C = 1.25 × C = (5/4) C. C = B - 25% of B = 0.75 × B = (3/4) B. • Calculation / Deduction: First, establish the ratios: From A = (5/4) C, we get the ratio A : C = 5 : 4. From C = (3/4) B, we get the ratio C : B = 3 : 4. To combine them into A : C : B, make the common term (C) equal. Multiply A : C by 3 \Rightarrow 15 : 12. Multiply C : B by 4 \Rightarrow 12 : 16. Combined ratio A : C : B = 15 : 12 : 16. So, A : B : C = 15 : 16 : 12. Now, calculate the share values. Let the shares be 15x, 16x, and 12x. Total units = 15 + 16 + 12 = 43 units. 43 units = 2236 1 unit = (2236/43) = 52 We need to find A's share: A's share = 15 units = 15 × 52 = 780. Exam Strategy & Shortcut: Use the Ratio Merging Method rapidly. A:C = 5:4 and C:B = 3:4. Write it like a block: A \quad C \quad B 5 \quad 4 \quad \quad 3 \quad 4 Fill the empty spaces with the adjacent numbers: 5 \quad 4 \quad 4 3 \quad 3 \quad 4 Multiply vertically: (5 × 3) : (4 × 3) : (4 × 4) = 15 : 12 : 16. This grid method is much faster than algebraically balancing fractions under time pressure. Common Pitfall: A common trap is assuming that because A is 25% *more* than C, and C is 25% *less* than B, that A and B are equal. Percentages are relative to their base. A 25% increase on a smaller base (C) does not equal a 25% decrease from a larger base (B). The ratio method completely bypasses this logical trap. Final Answer: Therefore, the correct answer is ₹ 780.

Question 14

A district has 64000 inhabitants. If the population increases at the rate of $2 \frac{1}{2}\%$ per annum, then the number of inhabitants at the end of 3 years will be

Accepted Answer

Concept & Formula: Population growth over multiple years is calculated using the compound interest formula, because the growth rate applies to an expanding base population each year. A = P (1 + (R/100))^T Where P is the initial population, R is the annual growth rate, and T is the number of years. Step-by-Step Solution: 1. Given Data: • Initial population (P) = 64000 • Rate (R) = 2 (1/2)% = 2.5% = (5/2)% • Time (T) = 3 years 2. Find the Fractional Multiplier: • First, express the rate as a fraction of 100: (2.5/100) = (1/40). • The multiplier for each year is (1 + (1/40)) = (41/40). 3. Apply the Compound Growth Formula: • Population after 3 years = 64000 × ( (41/40) )^3 • = 64000 × ( (41/40) ) × ( (41/40) ) × ( (41/40) ) • = 64000 × (41 × 41 × 41/40 × 40 × 40) • = 64000 × (68921/64000) 4. Simplify: • The 64000 in the numerator and denominator perfectly cancel each other out. • = 68921 Exam Strategy & Shortcut: Memorize standard fractional equivalents and their cubes. A 2.5% rate is exactly (1/40). Over 3 years, the denominator will be 40^3 = 64000. Since the initial population is exactly 64000, it will completely cancel out the denominator. Thus, the final answer is simply the numerator cubed: 41^3. If you know 40^3 = 64000, 41^3 must end in a 1 (since 1 × 1 × 1 = 1). Looking at the options, only 68911 and 68921 end in 1. 41^3 calculation reveals 68921. Common Pitfall: Students often waste immense amounts of time calculating the decimal values sequentially (e.g., finding 2.5% of 64000, adding it, then finding 2.5% of the new number, etc.). Always convert unusual percentage rates like 2.5%, 12.5%, or 16.66% into simple fractions before applying the power. Final Answer: Therefore, the correct answer is 68921.

Question 15

The population of a town 2 years ago was 62,500. Due to migration to big cities, it decreases every year at the rate of 4%. The present population of the town is

Accepted Answer

Concept & Formula: When a value undergoes a constant percentage decrease successively, it behaves like compound interest in reverse (compound depreciation). The formula for successive decrease is: P_{final} = P_{initial} × (1 - (R/100))^n To calculate faster, we convert the percentage decrease directly into a fraction multiplier. Step-by-Step Solution: • Given: Initial population = 62,500. Rate of decrease = 4% per year. Time = 2 years. • Fractional Multiplier: A decrease of 4% means the population becomes 96% of its previous value each year. Fraction equivalent: 4% = (4/100) = (1/25). The multiplying factor for a decrease is 1 - (1/25) = (24/25). • Apply the multiplier twice (for 2 years): P_{present} = 62500 × (24/25) × (24/25) • Simplify: P_{present} = 62500 × (576/625) • Notice that 62500 \div 625 = 100. • Final Calculation: P_{present} = 100 × 576 = 57600 Exam Strategy & Shortcut: Divisibility Rule of 9: The multiplying factor is (24/25). Since 24 = 8 × 3, multiplying by 24 twice means we are multiplying by 3 × 3 = 9. Therefore, the final answer must be a multiple of 9. Check the sum of digits of the options: (a) 5+6+7+0+0 = 18 (Possible) (b) 5+7+6+0+0 = 18 (Possible) (c) 5+8+8+0+0 = 21 (Not a multiple of 9) (d) 6+0+0+0+0 = 6 (Not a multiple of 9) Since two options are multiples of 9, the fractional method above is the most reliable and takes barely 10 seconds to execute because 625 cancels out perfectly. Common Pitfall: A frequent mistake is calculating 4% of 62,500 (which is 2,500), doubling it for two years (5,000), and subtracting it to get 57,500. This calculates simple decrease instead of successive (compound) decrease. Final Answer: Therefore, the correct answer is 57,600.

Question 16

Depreciation applicable to an equipment is 20%. The value of the equipment 3 years from now will be less by

Accepted Answer

Concept & Formula: This problem asks for the total effective percentage decrease over 3 years. This requires applying the successive percentage change concept on an assumed base value. For an annual depreciation rate R, the remaining value after n years is calculated as: Final Value = Initial Value × (1 - (R/100))^n Step-by-Step Solution: • Assume Initial Value: Let the current value of the equipment be 100. • Depreciation Rate: 20% per year. • Fractional Multiplier: 100% - 20% = 80%. As a fraction, this is (80/100) = (4/5) or 0.8. • Calculate Final Value after 3 years: Final Value = 100 × 0.8 × 0.8 × 0.8 Final Value = 100 × 0.512 = 51.2 • Calculate the Total Depreciation: The question asks how much the value will be *less by*. Total Reduction = Initial Value - Final Value Total Reduction = 100 - 51.2 = 48.8 • Since our base was 100, this reduction is directly 48.8%. Exam Strategy & Shortcut: Effective Percentage Formula (for 3 successive changes): Alternatively, remember the cube of the multiplier. A 20% decrease means the value is multiplied by 0.8 each year. 0.8^3 = 0.512. This means 51.2% of the value remains. The depreciation is simply 100% - 51.2% = 48.8%. Knowing your cubes (like 8^3 = 512) makes this a 5-second mental math problem. Common Pitfall: The single most common mistake is choosing 51.2%. Students correctly calculate the final remaining value (51.2%) but fail to read the exact wording of the question: "will be less by". You must subtract the remaining value from the original 100% to find the actual depreciation amount. Final Answer: Therefore, the correct answer is 48.8%.

Question 17

Ashish started a business with an initial investment of ₹ 500000. In the first year, he incurred a loss of 4%. However, during the second year, he earned a profit of 5% which in the third year rose to 10%. Calculate his net profit for the entire per…

Accepted Answer

Concept & Formula: This requires calculating the final amount after three successive percentage changes (one loss, two profits), and then finding the difference between the final amount and the initial investment to determine the net profit. Final Amount = Initial × (1 - (L_1/100)) × (1 + (P_1/100)) × (1 + (P_2/100)) Net Profit = Final Amount - Initial Amount Step-by-Step Solution: • Given: Initial investment = ₹ 500000. Year 1: 4% loss (multiplier 0.96). Year 2: 5% profit (multiplier 1.05). Year 3: 10% profit (multiplier 1.10). • Set up the final amount equation: Final Amount = 500000 × (96/100) × (105/100) × (110/100) • Simplify by cancelling zeros: There are 5 zeros in 500000 and 1 zero in 110. That's 6 zeros in the numerator. The denominator has 100 × 100 × 100 = 1000000 (6 zeros). They cancel out perfectly! • Calculate the product: Final Amount = 5 × 96 × 105 × 11 Final Amount = 480 × 105 × 11 Final Amount = 50400 × 11 = 554400 • Calculate Net Profit: Net Profit = 554400 - 500000 = 54400 Exam Strategy & Shortcut: Effective Percentage Method: You can combine the percentage changes successively using the formula x + y + (xy/100). 1. Combine Year 1 (-4%) and Year 2 (+5%): -4 + 5 + ((-4 × 5)/100) = 1 - 0.20 = 0.8% (Net 0.8% profit after 2 years) 2. Combine this result (+0.8%) with Year 3 (+10%): 0.8 + 10 + ((0.8 × 10)/100) = 10.8 + 0.08 = 10.88% The total net profit is exactly 10.88% of the initial investment. 10.88% of 500000 = 10.88 × 5000 = 54400 This eliminates heavy multiplication and delivers the answer directly. Common Pitfall: A frequent mistake is applying the 10% profit in the third year to the base 5% profit of the second year (meaning only a 15% total profit), rather than applying a 10% compound growth onto the actual final accrued capital of the second year. Always use successive multipliers. Final Answer: Therefore, the correct answer is ₹ 54400.

Question 18

A building worth ₹ 1,33,100 is constructed on land worth ₹ 72,900. After how many years will the value of both be the same if land appreciates at 10% p.a. and building depreciates at 10% p.a.?

Accepted Answer

Concept & Equation: This problem combines Compound Appreciation and Compound Depreciation. We need to track the future value of two separate assets changing at different rates (one positive, one negative) and find the point in time (n years) when their values are equal. • Formula for Appreciation (Land): A_1 = P_1(1 + (R_1/100))^n • Formula for Depreciation (Building): A_2 = P_2(1 - (R_2/100))^n We set A_1 = A_2 and solve for n. Step-by-step Solution: • Given: • Initial Land Value (P_1) = ₹ 72,900 • Land Appreciation Rate (R_1) = +10% p.a. • Initial Building Value (P_2) = ₹ 1,33,100 • Building Depreciation Rate (R_2) = -10% p.a. • Setting up the Equation: • Future value of Land = 72900 × (1 + (10/100))^n = 72900 × (1.1)^n • Future value of Building = 133100 × (1 - (10/100))^n = 133100 × (0.9)^n • Since their values become equal: 72900 × (1.1)^n = 133100 × (0.9)^n • Calculation: • Rearrange to group the exponential terms on one side and constants on the other: ((1.1)^n/(0.9)^n) = (133100/72900) • Simplify both sides: ((1.1/0.9))^n = (1331/729) • Multiply the numerator and denominator of the fraction inside the parenthesis by 10: ((11/9))^n = (1331/729) • Recognize the cubes of the numbers. You should know that 11^3 = 1331 and 9^3 = 729: ((11/9))^n = ((11/9))^3 • Equating the exponents: n = 3 Exam Strategy & Shortcut: In competitive exams, numbers like 1331 and 729 are massive hints. They are highly recognizable perfect cubes (11^3 and 9^3). When you see +10% and -10%, you are dealing with multipliers of (11/10) and (9/10). The ratio of their starting values is (133100/72900), which instantly reduces to (1331/729). Because this ratio perfectly matches the cubes of the rate multipliers ((11^3/9^3)), you can instantly conclude that n = 3 without writing out the full algebraic equation. Common Pitfall: A common trap is accidentally swapping the rates (e.g., applying depreciation to the land and appreciation to the building). Always double-check which asset is increasing and which is decreasing. Buildings generally depreciate due to wear and tear, while land appreciates due to scarcity, which aligns with logical real-world principles. Final Answer: Therefore, the correct answer is 3.

Question 19

The population of a town increases $4\%$ annually but is decreased by emigration annually to the extent of $(1/2)\%$. What will be the increase percent in 3 years?

Accepted Answer

Concept & Formula: This problem is about finding the Net Compound Growth Rate. When there are simultaneous increases and decreases happening constantly per period, we first find the net rate of change per period, and then apply the standard compound interest formula over the given time. The net effective annual growth rate is the difference between the increase rate and the decrease rate. Step-by-step Solution: • Given: • Annual increase rate = 4% • Annual decrease (emigration) rate = (1/2)% = 0.5% • Time period (n) = 3 years • Calculation: • First, calculate the net annual growth rate: Net Rate (R) = 4% - 0.5% = 3.5% • The population after 3 years will be: Final Population = P(1 + (3.5/100))^3 • Let the initial population P be 100 to make percentage calculations easier. Final Population = 100 × (1 + (7/200))^3 = 100 × ((207/200))^3 • Calculating this explicitly takes time, so let's use the successive percentage change formula or approximation. The formula for successive changes of x%, y%, and z% is complex. Instead, let's approximate using simple interest for a rough estimate, then add a bit for compounding. • Simple interest over 3 years: 3.5% × 3 = 10.5% • Because it's compound growth, the actual increase will be slightly higher than 10.5%. • Let's look at the options: (a) 9.8, (b) 10, (c) 10.5, (d) 10.8. • Since it must be greater than 10.5%, the only logical answer is 10.8. Exam Strategy & Shortcut: Use the Effective Percentage Change Approximation. For a small rate R over n years, the effective change is roughly nR + a small compounding factor. Here, R = 3.5% and n = 3. 3 × 3.5% = 10.5%. Because it compounds, the actual value must be strictly greater than 10.5%. Looking at the options, 10.8% is the only viable choice. This avoids calculating 1.035^3 entirely. Common Pitfall: The most common mistake is simply multiplying the net rate by the number of years (3.5% × 3 = 10.5%) and selecting option (c), forgetting that population growth is a compounding process, not a simple linear process. Final Answer: Therefore, the correct answer is 10.8.

Question 20

The current birth rate per thousand is 32, whereas the corresponding death rate is 11 per thousand. The net growth rate in terms of population increase in percent is given by

Accepted Answer

Concept & Logic: This question involves converting Rates per Thousand into Percentage Rates (per hundred). The net growth is the difference between births and deaths. The crucial step is understanding the base unit. "Per thousand" means out of every 1000 people. "Percent" means out of every 100 people. Step-by-step Solution: • Given: • Birth rate = 32 per thousand • Death rate = 11 per thousand • Calculation: • First, find the net growth rate per thousand: Net Growth per thousand = Birth rate - Death rate Net Growth per thousand = 32 - 11 = 21 • This means the population grows by 21 people for every 1000 people. • To find the percentage (per 100), we divide this rate by 10: Net Growth Percent = (21/1000) × 100 Net Growth Percent = (21/10) = 2.1% Exam Strategy & Shortcut: Understand the direct relationship between "per mille" (per thousand) and "per cent" (per hundred). Since 1000 is ten times 100, to convert a rate per thousand to a percent, simply divide by 10. Net rate = 32 - 11 = 21. Percent = (21/10) = 2.1%. Common Pitfall: The trap here is the decimal placement. Students might see the net difference of 21 and hastily select option (d) 21% without converting "per thousand" to "percent". Always pay close attention to the units requested in the final question. Final Answer: Therefore, the correct answer is 2.1%.

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