Percentages

The average pass rate is calculated by dividing the total number of students who passed in both years by the total number of students who appeared in both years. (75% of 100) + (60% of 75) = 75 + 45 = 120 passed out of 175 total students, so the average pass rate = (120/175)*100 = 68 4/7%.

Let the original fraction be x/y. Increasing numerator by 140% means numerator becomes x + 1.4x = 2.4x. Increasing denominator by 150% means denominator becomes y + 1.5y = 2.5y. The new fraction is (2.4x)/(2.5y) = 4/15. Simplifying, (2.4x)/(2.5y) = 4/15 => (2.4/2.5)(x/y) = 4/15 => (0.96)(x/y) = 4/15 => x/y = (4/15)/0.96 = (4/15)*(25/24) = 100/360 = 5/18, which is not among options. Rechecking calculation: 2.4x/2.5y = 4/15 => (2.4/2.5)(x/y) = 4/15 => 0.96(x/y) = 4/15 => x/y = (4/15)/0.96 = (4/15)*(25/24) = (100/360) = 5/18. Since 5/18 is not an option, check options for closest fraction: 2/9 = 4/18, which is close to 5/18. Testing 2/9: numerator increased by 140%: 2 + 2*1.4 = 2 + 2.8 = 4.8; denominator increased by 150%: 9 + 9*1.5 = 9 + 13.5 = 22.5; new fraction = 4.8/22.5 = 0.2133, original fraction 2/9 = 0.2222. New fraction should be 4/15 = 0.2667, so 2/9 is not exact. Testing 3/5: numerator increased: 3 + 4.2 = 7.2; denominator increased: 5 + 7.5 = 12.5; new fraction = 7.2/12.5 = 0.576, not 4/15. Testing 5/16: numerator increased: 5 + 7 = 12; denominator increased: 16 + 24 = 40; new fraction = 12/40 = 3/10 = 0.3, not 4/15. Since none matches exactly, the original fraction is 5/18, which is not listed, so correct answer is D (None of these).

When a number is increased by 10%, it becomes 110% (or 1.10 times) of the original number. Then, reducing this new number by 10% means multiplying by 90% (or 0.90). The final number is therefore 1.10 × 0.90 = 0.99 times the original number, which is 99% of the original. This ratio is 99:100. Among the given options, 4:5 (which is 0.8) is not a correct approximation. However, since none of the options exactly match 99:100, the closest correct ratio is 99:100, and the answer should be corrected to reflect this ratio. The original answer (B: 4:5) is incorrect. The correct ratio is 99:100, which is not listed in the options. Therefore, the question options should be updated to include 99:100 as the correct answer.

The total area of the paper is 30 cm × 15 cm = 450 cm². The margins reduce the usable area: top and bottom margins total 2 × 2.5 cm = 5 cm, so the height available for typing is 15 - 5 = 10 cm. The side margins total 2 × 1.25 cm = 2.5 cm, so the width available for typing is 30 - 2.5 = 27.5 cm. Therefore, the typing area is 27.5 cm × 10 cm = 275 cm². The percentage of the paper area available for typing is (275 / 450) × 100 ≈ 61.11%, which is closest to 60%. Hence, option D (60%) is correct.

If the price increases by 15%, the new price is 115% of the original. To keep expenditure the same, consumption must be reduced so that (new consumption) × (new price) = (original consumption) × (original price). Thus, new consumption = original consumption × (100/115) = original consumption × (20/23). The reduction in consumption is (1 - 20/23) × 100% = (3/23) × 100% ≈ 13.04%. However, the options are given as mixed fractions, and option A is '10 20/23 %' which equals approximately 10.87%, option B is '13 1/23 %' which equals approximately 13.04%, option C is '16 2/3 %' (16.67%), and option D is 15%. The correct reduction is approximately 13.04%, matching option B. Therefore, the original correct_answer 'B' is correct. The explanation was missing and has been added. The difficulty is medium due to the calculation complexity. Tags added for clarity.

Given that X is 20% less than Z, X = 0.8Z. Given that Y is 28% less than Z, Y = 0.72Z. To find by what percentage Y is less than X, calculate ((X - Y) / X) × 100 = ((0.8Z - 0.72Z) / 0.8Z) × 100 = (0.08Z / 0.8Z) × 100 = 10%. Therefore, Y is 10% less than X. The correct answer is option C (10%).

When the side of a square is increased by 25%, the new side length is 1.25 times the original. The area increases by (1.25)^2 - 1 = 1.5625 - 1 = 0.5625, or 56.25%.

Let the two numbers be x and y. Given x + y = 2490 and 6.5% of x = 8.5% of y, which means 0.065x = 0.085y. From this, x/y = 0.085/0.065 = 17/13. Using x + y = 2490 and x = (17/13)y, solving gives y = 1479 and x = 1011. Hence, the numbers are 1011 and 1479, matching option B.

When a number is increased by 10%, it becomes 1.1 times the original. Then reducing this new number by 10% multiplies it by 0.9, resulting in 1.1 × 0.9 = 0.99 times the original number, which is a 1% decrease.

If the price increases by 25%, the new price is 125% of the original. To keep expenditure constant, consumption must be reduced so that (new price) × (new consumption) = (original price) × (original consumption). Let the reduction be x%. Then, (1 + 0.25) × (1 - x) = 1, so 1.25 × (1 - x) = 1, which gives 1 - x = 1/1.25 = 0.8, so x = 0.2 or 20%. However, this is a common misconception. The correct approach is to find the percentage decrease in consumption as (increase%)/(1 + increase%) = 25/125 = 0.2 or 20%. So the correct answer is 20%. The original answer D (20%) is correct, but the explanation was missing. Therefore, the correct answer is D, not B.

Given: 5% of A's income = 15% of B's income, so A = 3B. Also, 10% of B's income = 20% of C's income, so B = 2C. Given C = Rs. 2000, B = 2 × 2000 = Rs. 4000, and A = 3 × 4000 = Rs. 12000. Total income = A + B + C = 12000 + 4000 + 2000 = Rs. 18000. However, this contradicts the calculation; rechecking: 5% A = 15% B → A = (15/5)B = 3B; 10% B = 20% C → B = (20/10)C = 2C; with C=2000, B=4000, A=12000; total = 12000+4000+2000=18000. The correct total is Rs. 18000, which corresponds to option C. Therefore, the original correct_answer 'C' is correct. The initial confusion was in the explanation step. The explanation now clearly supports option C.

Let A be the set of students who failed in English and B be the set of students who failed in Mathematics. Given n(A) = 50, n(B) = 30, and n(A ∩ B) = 12. The number of students failing in at least one subject is n(A ∪ B) = n(A) + n(B) - n(A ∩ B) = 50 + 30 - 12 = 68. Therefore, the number of students passing both subjects is the total number of students minus those failing at least one subject: 100 - 68 = 32. Hence, the correct answer is 32 (option D).

The total votes cast are 1136 + 7636 + 11628 = 20400. The winning candidate received 11628 votes. The percentage of total votes received by the winning candidate is (11628 / 20400) × 100 ≈ 57%. Since 57% is not among the given options (11.6%, 60%, 76.4%), the correct choice is 'None of these'.

Given 60% employees are men, and 40% of men earn more than Rs. 50,000, so men earning >50,000 = 0.6 * 0.4 = 24%. Total employees earning >50,000 is 36%, so women earning >50,000 = 36% - 24% = 12%. Since women are 40% of employees, women earning less than Rs. 50,000 = 40% - 12% = 28%, which is 70% of women (28%/40% = 0.7 or 70%). Therefore, 70% of women earn less than Rs. 50,000 per year.

When the length of a rectangle is increased by 40% and the width is decreased by 20%, the new area is calculated by multiplying the new length factor and the new width factor: (1 + 0.40) × (1 - 0.20) = 1.40 × 0.80 = 1.12. This means the new area is 1.12 times the original area, which corresponds to a 12% increase in area. Therefore, the correct answer is that the area has increased by 12%.

Given AB = 5 cm, AC = 3 cm, so CB = 2 cm. Increasing AC by 6% means AC_new = 3 × 1.06 = 3.18 cm. Since AB is fixed, CB_new = 5 - 3.18 = 1.82 cm. The decrease in CB is 2 - 1.82 = 0.18 cm. The percentage decrease in CB is (0.18 / 2) × 100% = 9%. Therefore, the correct answer is 9%, which corresponds to option D.

Calculating the number of men and women who cleared the final test shows more men succeeded overall. Although women's success rates at each stage are higher, the larger initial number of men results in more men clearing the examination.

Let the total number of voters be N. 10% did not vote, so votes cast = 0.9N. Of these, 10% were invalid, so valid votes = 0.9N × 0.9 = 0.81N. The winning candidate got 54% of valid votes = 0.54 × 0.81N = 0.4374N. The other candidate got 46% of valid votes = 0.46 × 0.81N = 0.3726N. The majority is the difference: 0.4374N - 0.3726N = 0.0648N = 1620. Solving for N gives N = 1620 / 0.0648 = 25000. However, this contradicts the options. Rechecking calculations: 10% did not vote → votes cast = 0.9N; 10% of votes polled invalid → valid votes = 0.9N × 0.9 = 0.81N; majority = (54% - 46%) × 0.81N = 8% × 0.81N = 0.0648N = 1620; N = 1620 / 0.0648 = 25000. Since 25000 is option A, correct answer is A, not B. Therefore, correct answer is A (25000).

Let the number of children be x. Each child receives 20% of x sweets, i.e., 0.2x sweets. Total sweets distributed = number of children × sweets per child = x × 0.2x = 0.2x². Given total sweets = 605, so 0.2x² = 605 ⇒ x² = 3025 ⇒ x = 55. Each child receives 0.2 × 55 = 11 sweets. This matches option A. Therefore, the correct answer is 11 sweets per child.

Ram spends 65% of Rs. 720, which is 0.65 × 720 = Rs. 468. Therefore, he saves Rs. 720 - Rs. 468 = Rs. 252.

Given the total population is 300,000 with 180,000 males, the number of females is 120,000. Since 50% of the population is illiterate, the literate population is 50% of 300,000, which is 150,000. Among males, 70% are literate, so literate males = 70% of 180,000 = 126,000. Therefore, literate females = total literate - literate males = 150,000 - 126,000 = 24,000. This matches option A.

If the price increases by 20%, to keep expenditure constant, consumption must decrease so that (1 + price increase) × (1 - consumption decrease) = 1. Solving, consumption decrease = 20/120 = 1/6 = 16 2/3%.

Given that 8% of the voters did not cast their votes, 92% of the total voters actually voted. The winner obtained 48% of the total votes, so the loser got 44% (92% - 48%) of the total votes. The difference between the winner and the loser is 4% of the total voters, which equals 1100 votes. Therefore, the total number of voters is 1100 divided by 0.04, which equals 27,500. This matches option D, making it the correct answer.

Using the formula for union of two sets: P(Car or Mobile) = P(Car) + P(Mobile) - P(Both) = 80% + 60% - 55% = 85%. Therefore, 85% of those surveyed owned a car or a mobile phone or both.

Let the total percentage of students be 100%. Given 18% play neither football nor cricket, so 82% play football or cricket or both. Using the formula for union of two sets: P(F ∪ C) = P(F) + P(C) - P(F ∩ C), we have 82% = 40% + 50% - P(F ∩ C). Solving gives P(F ∩ C) = 8%.