Question 1

The ratio of the students in schools $A, B$ and $C$ is 5 : 4 : 7. If the number of students in the schools are increased by 20%, 25% and 20% respectively, what would be the new ratio of the students in schools $A, B$ and $C$?  (Bank P.O., 2010)

Accepted Answer

Concept & Percentage Increase in Ratios: To find the new ratio after each part of a ratio undergoes a percentage increase, multiply the original ratio parts by their respective growth factors (1 + (r/100)). Step-by-Step Solution: • Assume Initial Values: Let the number of students in schools A, B, and C be 5x, 4x, and 7x. To make percentage calculations easy, assume x = 10. So, the initial students are 50, 40, and 70. • Calculate Increases: • School A new students = 50 × (1 + (20/100)) = 50 × 1.20 = 60. • School B new students = 40 × (1 + (25/100)) = 40 × 1.25 = 50. • School C new students = 70 × (1 + (20/100)) = 70 × 1.20 = 84. • Find the New Ratio: New Ratio = 60 : 50 : 84. • Simplify: Divide all terms by 2. New Ratio = 30 : 25 : 42. Exam Strategy & Shortcut: Skip the variables and multiply the ratio numbers directly by their percentage multipliers: A = 5 × 1.2 = 6 B = 4 × 1.25 = 5 C = 7 × 1.2 = 8.4 Ratio = 6 : 5 : 8.4. Multiply by 10 to remove the decimal \implies 60 : 50 : 84. Simplify by dividing by 2 \implies 30 : 25 : 42. Common Pitfall: Trying to add the percentages directly to the ratio values (e.g., 5 + 0.20 = 5.20) instead of applying the percentage as a multiplier of the base ratio unit. Final Answer: Therefore, the correct answer is 30 : 25 : 42.

Question 2

The expenses on rice, fish and oil of a family are in the ratio 12 : 17 : 3. The prices of these articles are increased by 20%, 30% and 50% respectively. The total expenses of family on these articles are increased by  (S.S.C., 2007)

Accepted Answer

Concept & Overall Percentage Change: The overall percentage increase of a total is the sum of the absolute increases divided by the original total. Overall % Increase = \frac{Total Increase}{Original Total} × 100 Step-by-Step Solution: • Set Initial Values: Let the expenses be 120, 170, and 30 (scaling the ratio 12:17:3 by 10 for easy percentage calculations). • Find Initial Total: 120 + 170 + 30 = 320. • Calculate Individual Increases: • Rice increase: 20% of 120 = (20/100) × 120 = 24. • Fish increase: 30% of 170 = (30/100) × 170 = 51. • Oil increase: 50% of 30 = (50/100) × 30 = 15. • Calculate Total Increase: 24 + 51 + 15 = 90. • Calculate Percentage Increase: Percentage Increase = (90/320) × 100 = (9/32) × 100 = (900/32) (900/32) = (225/8) = 28.125% • Convert to a mixed fraction: 28 (1/8)%. Exam Strategy & Shortcut: Use weighted averages. The overall percentage increase is the weighted sum of the individual percentage increases: Overall Change = ((12 × 20) + (17 × 30) + (3 × 50)/12 + 17 + 3) Overall Change = (240 + 510 + 150/32) = (900/32) = (225/8) = 28 (1/8)% Common Pitfall: Averaging the three percentage values directly (20 + 30 + 50) / 3 without taking into account the unequal weights of the initial expenses. Final Answer: Therefore, the correct answer is 28 (1/8)%.

Question 3

A sum of money is divided among 160 males and some females in the ratio 16 : 21. Individually each male gets ₹ 4 and a female ₹ 3. The number of females is

Accepted Answer

Concept & Ratio of Totals vs Individuals: The total amount received by a group is the product of the number of individuals in that group and the amount each individual receives. Total Amount for Group = Number of People × Amount per Person Step-by-Step Solution: 1. Given: - Number of males = 160 - Amount per male = ₹ 4 - Amount per female = ₹ 3 - Ratio of total male money to total female money = 16 : 21 2. Calculate Total Male Amount: Total money received by all males = 160 × 4 = 640. 3. Use Ratio to Find Total Female Amount: Let total male money be 16x and total female money be 21x. 16x = 640 \implies x = 40. Total money received by all females = 21x = 21 × 40 = 840. 4. Calculate Number of Females: Since each female gets ₹ 3, the number of females is: Number of Females = \frac{Total Female Money}{Amount per Female} = (840/3) = 280 Exam Strategy & Shortcut: Set up a single proportion equation: \frac{Total Male Money}{Total Female Money} = (160 × 4/F × 3) = (16/21). Solve for F: (640/3F) = (16/21) \implies 16 × 3F = 640 × 21 \implies 48F = 13440 \implies F = 280. Common Pitfall: Confusing the ratio of total money with the ratio of individuals. Always differentiate between total pool ratios and per-head ratios. Final Answer: Therefore, the correct answer is 280.

Question 4

Prices of foodgrains have risen by 10% and of other items of consumption by 15%. If the ratio of an employee's expenditure on foodgrains and other items is 2 : 5, by how much should his salary be increased so that he may maintain the same level of c…

Accepted Answer

Concept & Proportional Distribution: When an overall total is divided in a given ratio, you can find the absolute value of each part. The net increase required is the sum of the absolute increases of each individual component. Net Increase = Increase in Part 1 + Increase in Part 2 Step-by-Step Solution: 1. Given: Total salary = ₹ 3500. Expenditure ratio (Foodgrains : Other items) = 2 : 5. 2. Calculate Absolute Expenditures: Total ratio parts = 2 + 5 = 7 parts. Value of 1 part = (3500/7) = 500. Expenditure on foodgrains = 2 parts × 500 = 1000. Expenditure on other items = 5 parts × 500 = 2500. 3. Calculate the Increase for Each Category: Price of foodgrains rises by 10%. Increase = 10% of 1000 = (10/100) × 1000 = 100. Price of other items rises by 15%. Increase = 15% of 2500 = (15/100) × 2500 = 375. 4. Calculate Total Required Salary Increase: Total increase needed = 100 + 375 = 475. Exam Strategy & Shortcut: Calculate the net percentage increase directly using the weighted average of the percentages based on the ratio parts: Net % increase = ((2 × 10) + (5 × 15)/2 + 5) = (20 + 75/7) = (95/7)%. Required increase amount = (95/700) × 3500 = 95 × 5 = 475. Common Pitfall: A common mistake is averaging the two percentages directly (getting 12.5%) and applying it to the total salary, completely ignoring the 2:5 weighting ratio. Final Answer: Therefore, the correct answer is ₹ 475.

Question 5

A student took five papers in an examination, where full marks were same on each paper. Her marks in these papers were in the proportion of 6 : 7 : 8 : 9 : 10. In all these papers together, the candidate obtained 60% of the total marks. Then the num…

Accepted Answer

Concept & Proportional Thresholds: To compare proportional marks to a percentage threshold, we must establish a relationship between the "ratio parts" of the marks obtained and the "maximum marks" per paper. Threshold = Percentage × Maximum Marks per Paper Step-by-Step Solution: 1. Define Obtained Marks: Let the marks obtained in the five papers be 6x, 7x, 8x, 9x, and 10x. 2. Calculate Total Obtained Marks: Total marks obtained = 6x + 7x + 8x + 9x + 10x = 40x. 3. Define Maximum Marks: Let the maximum marks for each single paper be M. Total maximum marks for all 5 papers = 5M. 4. Equate Using Given Percentage: The total marks obtained (40x) is 60% of the total maximum marks (5M). 40x = 0.60 × 5M 40x = 3M M = (40x/3) \approx 13.33x. 5. Determine the 50% Threshold: We need to find papers where marks are greater than 50% of M. 50% of M = 0.5 × (40x/3) = (20x/3) \approx 6.67x. 6. Compare Marks to Threshold: We check which papers have marks > 6.67x: - Paper 1: 6x (Less) - Paper 2: 7x (More) - Paper 3: 8x (More) - Paper 4: 9x (More) - Paper 5: 10x (More) She scored more than 50% in 4 papers. Exam Strategy & Shortcut: Ignore variables. Assume the sum of the ratio parts (40) is exactly the total marks obtained. Since 40 represents 60% of the grand total, the grand total is (40/0.6) = 66.67. Since there are 5 identical papers, the maximum marks per paper is (66.67/5) = 13.33. 50% passing threshold per paper = (13.33/2) = 6.67. Looking at the ratio numbers (6, 7, 8, 9, 10), the numbers 7, 8, 9, 10 are greater than 6.67. Thus, 4 papers. Common Pitfall: Attempting to calculate the average of the ratio numbers directly without tying them to the maximum possible marks, leading to arbitrary and incorrect threshold comparisons. Final Answer: Therefore, the correct answer is 4.

Question 6

Smita works in her office for 6 hours and Kajal works in her office for 7 hours 30 minutes. Find the ratio of their working hours. [ESIC—UDC Exam, 2016]

Accepted Answer

Concept & Ratio Calculation: To find the ratio between two quantities, they must be expressed in the same unit. Ratio = \frac{Value A}{Value B} Step-by-Step Solution: • Given: Smita's working time = 6 hours. Kajal's working time = 7 hours 30 minutes. • Convert both times to minutes to ensure consistency. • Smita's time in minutes = 6 × 60 = 360 minutes. • Kajal's time in minutes = (7 × 60) + 30 = 420 + 30 = 450 minutes. • Now, find the ratio: 360 : 450 • Divide both sides by 10: 36 : 45 • Divide both sides by their greatest common divisor (9): 4 : 5 Exam Strategy & Shortcut: Instead of converting to minutes, you can keep them in hours using decimals: 6 hours to 7.5 hours. Multiply both by 2 to remove the decimal, getting 12 to 15. Divide by 3 to get the ratio 4:5. Common Pitfall: A common mistake is trying to write the ratio directly as 6 : 7.3, which incorrectly assumes 30 minutes is 0.3 hours rather than 0.5 hours. Final Answer: Therefore, the correct answer is 4 : 5.

Question 7

Directions: Study the following information and answer the questions that follow :  The graph given below represents the production (in tonnes) and sales (in tonnes) of Company A from 2006 to 2011.  [Graph Data of Company A] | Year | Production (in …

Accepted Answer

Concept & Ratio of Aggregates: Calculate the sum of all values in the first set and the sum of all values in the second set independently, then express their relation in simplest fractional form. Ratio = \frac{\sum Production_A}{\sum Sales_A} Step-by-Step Solution: • Calculate the total production of Company A across all years: • Total Production = 750 + 800 + 600 + 550 + 700 + 650 = 4050 tonnes. • Calculate the total sales of Company A across all years: • Total Sales = 300 + 550 + 450 + 400 + 500 + 550 = 2750 tonnes. • Form the ratio of Total Production to Total Sales: • Ratio = 4050 : 2750. • Simplify by dividing both sides by 10: 405 : 275. • Simplify further by dividing both sides by 5: • 405 \div 5 = 81 • 275 \div 5 = 55 • The final simplified ratio is 81 : 55. Exam Strategy & Shortcut: End digits ending in zero make for quick addition. Once you establish the initial ratio of 4050 : 2750, dropping the zero leaves 405 : 275. Since both end in 5, dividing by 5 immediately gives 81 : 55, circumventing more complex simplification tests. Common Pitfall: A common pitfall is reversing the order requested in the prompt (finding Sales to Production instead of Production to Sales), or making arithmetic errors when summing the six data points. Final Answer: Therefore, the correct answer is 81 : 55.

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