Mathematics Analysis

Step 1: Number of Male and Female Psychology Professors - Males: - Females: Step 2: Total Psychology Professors - Total = Males + Females = Step 3: Percentage of Female Psychology Professors - Percentage = (Females / Total) × 100 - Percentage = % Answer: C. 60%

To determine the discipline with the highest ratio of males to females, we calculate the male-to-female ratio for each discipline from the bar chart (Figure 1): Physics: Males: 32, Females: 8 Ratio = Mathematics: Males: 24, Females: 8 Ratio = Chemistry: Males: 12, Females: 20 Ratio = Economics: Males: 16, Females: 8 Ratio = Conclusion: The discipline with the highest male-to-female ratio is Physics (4:1).

To find the car that maintained the maximum average speed, we use the formula for average speed: Average Speed = Distance / Time Taken Now, let's calculate the average speed for each car: Car A: Average Speed = km / hours = km/h Car B: Average Speed = km / hours = km/h Car C: Average Speed = km / hours = km/h Car D: Average Speed = km / hours = km/h Conclusion: Car A and Car B both maintained the same maximum average speed of km/h. However, since the question asks for the car that maintained the maximum average speed, the answer is either Car A or Car B. If only one option must be chosen, the correct answer is: A. Car A

Let's denote: - x as the time taken to walk one way (in hours). - y as the time taken to drive one way (in hours). From the problem, we have the following two equations: 1. **Walking one way and driving back**: The total time taken for walking one way and driving back is 6 hours. Therefore: 2. **Walking both ways**: The total time taken for walking both ways is 10 hours. Therefore: Now, solve for x in the second equation: Substitute x = 5 into the first equation: Solving for y: Now, to find the time taken to drive both ways, we simply multiply y (the time taken to drive one way) by 2: Thus, the time taken to drive both ways is: 2 hours.

To solve this question, we follow these steps: 1. Determine the total number of Physics professors: From the bar chart (Figure 1): Total Physics professors = (males) + (females) = . 2. Find the percentage of Physics professors in the age group 25–34: From the pie chart (Figure 2): Percentage of Physics professors in the 25–34 age group = . 3. Calculate the total number of Physics professors in the 25–34 age group: . 4. Determine the number of female Physics professors in the 25–34 age group: It is given that females in this group make up 25% of all Physics professors in the group. . 5. Calculate the number of male Physics professors in the 25–34 age group: Total professors in this age group = . Males = . Final Answer: A. 9

Let the number of gold coins in the original collection be and the number of non-gold coins be . According to the problem, the ratio of gold coins to non-gold coins is 1:3, so: or . Next, 10 more gold coins are added to the collection, so the number of gold coins becomes , and the number of non-gold coins remains . The new ratio of gold coins to non-gold coins is given as 1:2, so: . Now, substitute into the equation: . To solve for , cross-multiply: . Expand the equation: . Subtract from both sides: . So, there are initially 20 gold coins. Using , the number of non-gold coins is: . Thus, the total number of coins before adding the 10 gold coins is: . After adding the 10 gold coins, the total number of coins is: . Therefore, the total number of coins in the collection now is: .

Let the value of the highest prize be x. The sequence of the seven prizes will be: . The sum of these prizes is given as Rs. 700, so: . Simplifying the equation: , . Now, solve for x: , . So, the highest prize is Rs. 160. The least prize will be: . Thus, the least value of the prize is: Rs. 40.

Let the original speed of the train be km/hr. The train travels 63 km at speed and 72 km at speed . Time taken for each part of the journey: Time_1 = Time_2 = Total time = 3 hours: Multiply the entire equation by : Simplify: Divide by 3: Solve using the quadratic formula: The two possible values for : or Since speed cannot be negative, the original speed is 42 km/hr.

Let's break down the problem step by step: - Initially, there is enough food for 1000 soldiers for 30 days. - After 10 days, 1000 soldiers join, so the total number of soldiers is now 2000. First, we calculate how much food is used in the first 10 days for 1000 soldiers: Food used in 10 days = soldier-days of food. Since the total food available is enough for 1000 soldiers for 30 days, the total amount of food available is: Total food = soldier-days of food. After the first 10 days, 10,000 soldier-days of food have been used, leaving: Remaining food = soldier-days of food. Now, with 2000 soldiers in the garrison, we calculate how many days the remaining food will last: Number of days = Remaining food / Number of soldiers = days. Therefore, the soldiers will be able to carry on with the remaining food for: 10 days.

Speed of policeman = 10 km/hr Speed of thief = 8 km/hr Relative speed = km/hr Convert relative speed to meters per second: 2 km/hr = m/s = m/s Distance between them = 100 meters Time = Distance / Relative speed = seconds = 3 minutes

| Step | Description | Calculation | Result | |------|------------------------------------------------------|----------------------|--------| | 1 | Total number of Physics professors (Male + Female) | | | | 2 | Percentage of professors in the 35–44 age group | | | | 3 | Number of professors in the 35–44 age group | | |

The gardener has 1000 plants and wants to plant them in such a way that the number of rows and columns is the same, meaning the number of rows and columns must form a perfect square. To find the nearest perfect square greater than or equal to 1000, we take the square root of 1000: . Rounding up, we get 32, meaning the gardener would need a square grid with 32 rows and 32 columns. The number of plants required for this is: . The gardener already has 1000 plants, so the number of extra plants needed is: . Therefore, the minimum number of extra plants the gardener needs is: 24.

Let X be the total number of professors in the university. Since 8 professors represent 2% of X, we set up the equation: Solving for X Rearranging the equation: Selecting the Correct Answer The correct answer from the given options is: A. 400

We know, as the no. of men ( ) increases, the no. of days ( ) to complete the job will decrease. Therefore, diagram IV is the correct answer.

Let's denote the following: - Total number of applicants: - Number of males: - Number of applicants with a driver's license: - Number of females: - Number of females with a driver's license: - (number of males with a driver's license) Now, we need to find the ratio of the minimum to maximum number of males who have a driver's license. Maximum number of males with a driver's license: For the maximum number of males with a driver's license, we assume that all the females do not have a driver's license. So, all 80 applicants with a driver's license must be male. Therefore, the maximum number of males with a driver's license is 70 (since there are only 70 males). Minimum number of males with a driver's license: For the minimum number of males with a driver's license, we assume that as many females as possible have a driver's license. Since there are 50 females, the maximum number of females with a driver's license is 50. Therefore, the remaining number of applicants with a driver's license must be male, which is: . Ratio of the minimum to maximum number of males with a driver's license: The minimum number of males with a driver's license is 30, and the maximum number is 70. The ratio is: . Thus, the ratio between the minimum to maximum number of males having a driver's license is: 3 to 7.

Combination Formula: Step 1: Select all three males. Since all three males are selected, there is only 1 way to choose them. Step 2: Select the fourth member from the remaining individuals. If there are N individuals in total, the number of remaining candidates is N - 3. The fourth member is chosen from these N - 3 individuals. Number of ways to select the fourth member = Example: If there are 5 individuals in total (3 males and 2 females): Remaining candidates = Number of ways to select the fourth member = Final Answer: Number of possible combinations = 2

Let: - x be the initial petrol consumption. - y be the number of days the petrol lasts initially. According to the question: - Initially, the petrol lasts for 10 days with a consumption of x. - After the consumption increases by 25% each day, the final consumption on the last day will be . This is a case of indirect variation, meaning that the product of the consumption and the number of days remains constant. So, we have the relation: Now, solving for y (the number of days the petrol will last after the increase in consumption): Dividing both sides by : Thus, the petrol will last for: 8 days.

To determine the total number of different routes from S to T, we multiply the number of possible paths at each stage. From S to U, there are 3 possible paths. From U to V, there are 2 possible paths. Since every path from S to U must continue through one of these, the total ways to reach V from S is: . From V to T, there are 3 possible paths. Extending the logic further, every route from S to V must continue through one of these paths, leading to: . Thus, the total number of different routes from S to T is 18.

Distance between A and B = km Speed of car 1 (from A) = km/h Speed of car 2 (from B) = km/h When two objects move towards each other, their relative speed is the sum of their individual speeds: Relative speed = km/h + km/h = km/h Now, to find the time taken to meet, we use the formula: Time = Distance / Relative speed Time = km / km/h = hours The cars start at 08:10 AM, so they will meet after 2 hours. 08:10 AM + 2 hours = 10:10 AM Therefore, the two cars will meet at 10:10 AM.

The time intervals at which the five persons fire the bullets are 6, 7, 8, 9, and 12 seconds. To find how often they will fire together in an hour, we need to find the least common multiple (LCM) of the time intervals. The LCM of 6, 7, 8, 9, and 12 gives the interval at which they will fire together. Find the prime factorization of each number: The LCM is obtained by taking the highest powers of all primes that appear: The highest power of 2: (from 8) The highest power of 3: (from 9) Highest power of 7: 7 (from 7) LCM = seconds. This means they will all fire together once every 504 seconds. To find how many times they fire together in an hour (3600 seconds), divide 3600 by 504: . Since they fire together once at the start, and the next time will be after 504 seconds, they will fire together 7 times in an hour. Thus, the correct answer is B. 7. NOTE: UPSC answer key considered both options B and C as correct.

Let the distance between the worker’s house and the factory be km. Case 1: Speed = km/hr Time taken at km/hr = hours. The worker is minutes late, so the actual time taken is minutes (where is the ideal time). Case 2: Speed = km/hr Time taken at km/hr = hours. The worker is minutes early, so the actual time taken is minutes. Set up the equation The difference in time between the two speeds is: hours Simplifying the left-hand side: Thus: Multiplying both sides by : km

We have a 36 cm line segment, and points are marked from both ends with increasing distances. Points from the left end: . Points from the right end: . Common point: . Total points: points from the left end. points from the right end (excluding common point). Total = points.

Total musicians = 120 Musicians who can play all three instruments = 6 Musicians who can play exactly two instruments = 30 Musicians who can play guitar alone = 40 We need to find musicians who can play either violin alone or flute alone. Total musicians who can play violin alone or flute alone = (Total musicians who can play violin or flute) - (Musicians who can play all three instruments) Final Answer = 44.

Let the total number of installments be x. The businessman has refunded 60% of his loan after paying 18 installments. This means that 18 installments account for 60% of the total loan amount. Each installment represents: % of the total loan amount. Now, to calculate the number of installments for the entire loan (100%): installments. Thus, the total number of installments in the agreement is 30.

The average profit of A in the year 1999 = 4000 The average profit of A in the year 2000 = 6000 Difference: Rs. 2000

We need to find the 5th day from the 21st of the month: , so we need to find the 26th day of the month. , meaning there are 23 days between the 3rd (Monday) and the 26th (required day). 23 days = 3 weeks and 2 extra (odd) days (since 3 weeks contribute 21 days, leaving 2 extra days). Starting from Monday, adding 2 days gives: Monday + 1 day = Tuesday Tuesday + 1 day = Wednesday Thus, the 26th day is Wednesday, so the correct answer is C. Wednesday.

The average profit of B in the year 1998 = The average profit of A in the year 1998 = , where is positive and not more than (by observation) Difference between the average profit of B and A in the year 1998 = = positive. Therefore, answer is C.

Option A is incorrect: To calculate the percentage growth in population from 1992 to 1993: Since the population increases by 1 lakh every year, the percentage growth will decline as the base (total population) increases. For 1993-1994: For 1994-1995: With a constant numerator and a growing denominator, the fraction's value decreases each year, meaning the percentage growth will always be lower than 5% going forward. Option B is incorrect: To calculate the income growth from 1992 to 1993: To calculate the income growth from 1993 to 1994: To calculate the income growth from 1994 to 1995: Option C is correct: To calculate the per capita income in 1992: To calculate the per capita income in 1993: To calculate the per capita income in 1994: To calculate the per capita income in 1995: Option D is incorrect: By calculation above, we know per capita income was highest in 1995.

To find when all three bells ring together again, we need to calculate the Least Common Multiple (LCM) of the given time intervals: The first bell rings every 18 minutes. The second bell rings every 24 minutes. The third bell rings every 32 minutes. Step 1: Prime Factorization Step 2: Find LCM The LCM is found by taking the highest powers of each prime factor: The highest power of 2 is (from 32). The highest power of 3 is (from 18). Thus, LCM = minutes. Step 3: Convert Minutes to Hours and Minutes 288 minutes = 4 hours 48 minutes. Step 4: Add to Initial Time The bells first ring together at 8:00 AM. Adding 4 hours 48 minutes to 8:00 AM gives 12:48 PM.

A is at a reference point. B is 5 km north of A. C is 12 km east of A. Using the Pythagorean theorem: Substituting the values: km Therefore, the shortest distance between B and C is 13 km.

Total tickets sold = . Half of these tickets were sold at Rs. each: Number of tickets sold at Rs. . Amount received = = Rs. . One-third of the tickets were sold at Rs. each: Number of tickets sold at Rs. . Amount received = = Rs. . The rest of the tickets were sold at Rs. each: Number of tickets sold at Rs. . Amount received = = Rs. . Total amount received = Rs. + Rs. + Rs. = Rs. .

Initially, all cards show '1' on the upper side. In each try, exactly four cards are turned upside down. Objective: All cards should show '2' on the upper side. Step-by-step: After the first try, 4 cards are flipped, showing '2' on the upper side, and 2 cards still show '1'. After the second try, the same 4 cards are flipped again, but 2 cards that were flipped earlier show '1' again, and the other 2 show '2'. After the third try, all 6 cards will show '2' on the upper side. Thus, the minimum number of tries required is 3.

Student A: Total marks obtained = Total maximum marks = Percentage of marks = % Student B: Total marks obtained = Total maximum marks = Percentage of marks = % Difference in Aggregate Percentage: Thus, the difference in aggregate percentage marks is zero.

Total students = 50 Students speaking both languages (English & Hindi) = 10 Students speaking English = 21 Students speaking only English = Students speaking Hindi (H): Students speaking only Hindi = Final Answer: Students speaking Hindi = 39 Students speaking only Hindi = 29 Students speaking only English = 11 Thus, the answer is 39, 29, 11.

Clearly, by observing the given graph we get the answer B. Non-decreasing

The death rate among army personnel is per , while the death rate among civilians is per . The comparison is made between two distinct groups: army personnel and civilians. These two groups are heterogeneous in nature, meaning they have different characteristics, environments, and lifestyles. Since the groups are different in many ways, directly comparing the death rates can be misleading. For example, army personnel may have better access to healthcare, are generally healthier, and have a more controlled environment compared to civilians. So, the comparison is not valid without considering these differences. Therefore, the correct inference is: D. The groups cannot be compared due to their heterogeneity.

Clearly, by observing the given graph we get the answer B. 1996

For the first goat: CP of first goat = SP / 1.10 For the second goat: CP of second goat = SP / 0.90 Total CP = CP of first goat + CP of second goat Total CP = (SP / 1.10) + (SP / 0.90) Total CP = SP * (1/1.10 + 1/0.90) Total CP = SP * (9/9.9 + 10/9) Total CP = SP * (99 + 100) / 99 Total CP = SP * 199 / 99 Total CP = SP * 1.909 Total SP = 2 * SP Profit or Loss = Total SP - Total CP Profit or Loss = 2 * SP - 1.909 * SP Profit or Loss = 0.091 * SP Percentage Profit or Loss = Percentage Profit or Loss = Percentage Profit or Loss % Thus, the correct answer is C. He suffers a loss of 1%.

A has Rs. 10, which is 80% of the bus fare for two persons. This means the full fare for two tickets is: Full fare = B has Rs. 3, which he hands over to A. Therefore, A now has: Total money A has = The full fare for two tickets is Rs. 12.5. After paying for the two tickets, A will be left with: Thus, after buying the two tickets, A will be left with 0.5 (Rs. 50 paise).

Let the original length and width of the garden be denoted by and . The original area = . The length of the garden is increased by 40%, so the new length = . The width of the garden is decreased by 20%, so the new width = . The new area of the garden = New length New width = . Simplifying, the new area = . The percentage increase in the area = . So, the percentage increase = . Therefore, the area of the new garden has increased by 12%.

Given conditions: - The time is between 8:00 and 9:00. - The hour and minute hands coincide (one above the other). - We need to find when they will coincide again. Step 1: Finding the first coincidence At 8:00, the hour hand is at . The minute hand starts at 0° and moves 360° per hour (6° per minute). The hour hand moves 30° per hour (0.5° per minute). Let be the time in minutes after 8:00 when they coincide: Step 2: Finding the next coincidence The hands coincide every . Rounding to the nearest integer: .

Given: Incomes ratio of Peter and Paul = 4:3 Expenses ratio of Peter and Paul = 3:2 Both save Rs. 6,000 at the end of the month Let: Peter's income = Paul's income = Peter's expense = Paul's expense = Savings Equation: Peter's savings = Income - Expense → Paul's savings = Income - Expense → Solve the system of equations: Multiply the second equation by 4 and the first equation by 3: → → Subtract the second equation from the first: Substitute y = 6,000 in the first equation: Incomes: Peter's income = Rs. 24,000 Paul's income = Rs. 18,000 Answer: Peter's income = Rs. 24,000 Paul's income = Rs. 18,000 Correct Option: A. Rs. 24,000 and Rs. 18,000

| Transaction | A's Amount | B's Amount | C's Amount | D's Amount | |-------------|------------|------------|------------|------------| | Initial | Rs. 100 | Rs. 100 | Rs. 100 | Rs. 100 | | Transaction 1 (A pays Rs. 20 to B) | Rs. 80 | Rs. 120 | Rs. 100 | Rs. 100 | | Transaction 2 (B pays Rs. 10 to C) | Rs. 80 | Rs. 110 | Rs. 110 | Rs. 100 | | Transaction 3 (C receives Rs. 30 from D) | Rs. 80 | Rs. 110 | Rs. 140 | Rs. 70 | Final Amounts: | Person | Final Amount | |-------------|--------------| | A | Rs. 80 | | B | Rs. 110 | | C | Rs. 140 | | D | Rs. 70 | Analysis of Statements: 1. C is the richest: True. C has Rs. 140, the highest amount. 2. D is the poorest: True. D has Rs. 70, the lowest amount. 3. C has more than what A and D together have: Not Correct. A and D together have Rs. 150, while C has Rs. 140. 4. B is richer than D: True. B has Rs. 110, which is more than D's Rs. 70. Conclusion: The statement "C has more than what A and D together have" is not correct.

Based on the chart: - Between the years 2000 and 2007, A's earnings were consistently higher than B's for all 7 years. - However, from 2007 to 2010, B's earnings exceeded A's for 3 consecutive years. - Despite this, A's average earnings over the entire period were still higher than B's average earnings. Therefore, A had a higher average income compared to B during the given timeframe.

Let's analyze the matrix: | 3 | 8 | 10 | 2 | ? | 1 | |----|----|----|----|----|----| | 6 | 56 | 90 | 2 | 20 | 0 | The first row is: 3, 8, 10, 2, ?, 1 The second row is: 6, 56, 90, 2, 20, 0 We need to find the relationship between the two rows to determine the value of '?'. Upon closer inspection, let's explore a potential relationship: 1. 2. 3. 4. 5. From the 5th point, we can solve for '?': Thus, the missing number at '?' is 5. The correct answer is A. 5.

Step 1: Selecting the Principal There are 2 eligible candidates for the Principal position, and we need to select 1 candidate for this post. So, the number of ways to select the Principal is: Ways to select Principal = Step 2: Selecting the Vice-Principal After selecting the Principal, 5 candidates remain for the Vice-Principal posts. We need to select 2 out of these 5 remaining candidates. The number of ways to select 2 candidates for the Vice-Principal position is: Ways to select Vice-Principal = Step 3: Total Combinations The total number of possible combinations is the product of the ways to select the Principal and the ways to select the Vice-Principal: Total combinations = Ways to select Principal Ways to select Vice-Principal = Conclusion The total number of possible combinations is 20. Therefore, the correct answer is: D. None of the above

Average Speed = Speed1 = 40 km/hr (from A to B) Speed2 = 60 km/hr (from B to A) Average Speed = = = 48 km/hr

1. Assume the volume of each glass is 1 liter for simplicity. 2. Calculate the amount of milk in each glass: - First glass: full of milk → liter of milk. - Second glass: full of milk → liter of milk. 3. Calculate the amount of water in each glass: - First glass: liter of water. - Second glass: liter of water. 4. Total amounts in the pot: - Total milk: liters. - Total water: liters. 5. Simplify the ratio of milk to water: - Ratio = . Therefore, the ratio of milk to water in the pot is 7:17, corresponding to option A.

Clearly, by observation answer is C.

Clearly, by observation answer is C.

Let's denote: - x as the number of pairs of brown socks in the original order. - p as the price of a pair of brown socks. Given that: - The price of a black pair is three times that of a brown pair, so the price of a black pair is . - The person ordered 5 pairs of black socks and x pairs of brown socks. - The clerk mistakenly interchanged the number of black and brown pairs, increasing the bill by 100%. Original Total Cost: The total cost of the original order is: Incorrect Total Cost: After the interchange, the total cost becomes: Condition Given: The bill increased by 100%, meaning the incorrect total cost is double the original total cost: Solving the Equation: 1. Expand both sides: 2. Simplify the equation: 3. Divide both sides by (assuming ): Conclusion: The number of pairs of brown socks in the original order was 25. Therefore, the correct answer is D. 25.

Let the son's age be x. The father's age is . The mother's age is . The sum of the father's and mother's age is 51: Simplifying: Solving for x: Therefore, the son's age is 3 years. Answer: D. 3 years

Step 1: Calculate the rate of each pipe. - Pipe A can fill the tank in 20 minutes, so its rate is of the tank per minute. - Pipe B can fill the tank in 30 minutes, so its rate is of the tank per minute. Step 2: Find the combined rate. Combined rate = The least common denominator of 20 and 30 is 60, so: Combined rate = Step 3: Calculate the total time. If the combined rate is of the tank per minute, it will take: Total time = minutes. Final Answer: B. 12 minutes

Let the total number of questions in the test be N. Each question carries equal marks, so the total marks in the test are proportional to N. Given: - The candidate attempted 8 questions and secured 50% marks in each. - He obtained a total of 40% in the test. - All questions in the test carried equal marks. Solution: 1. Marks per question: - Let the total marks for the test be M. - Since all questions carry equal marks, each question is worth marks. 2. Marks obtained by the candidate: - The candidate attempted 8 questions and secured 50% marks in each. - Therefore, the marks obtained in each attempted question are . - Total marks obtained by the candidate = . 3. Total marks obtained as a percentage: - The candidate obtained 40% of the total marks. - Therefore, . 4. Solving for N: - Simplifying the equation: . Therefore, the total number of questions in the test is .

Shahid and Rohit start from the same point in opposite directions. After each 1 km, Shahid always turns left, and Rohit always turns right. Let's analyze their movements: Assumptions: - Shahid starts by moving west. - Rohit starts by moving east. Shahid's Path: 1. Moves 1 km west. 2. Turns left (south) and moves 1 km. 3. Turns left (east) and moves 1 km. 4. Turns left (north) and moves 1 km. Rohit's Path: 1. Moves 1 km east. 2. Turns right (south) and moves 1 km. 3. Turns right (west) and moves 1 km. 4. Turns right (north) and moves 1 km.  - After both have traveled 2 km, Shahid is 1 km south of the starting point, and Rohit is 1 km north of the starting point. The distance between them is 2 km. - They meet after each has traveled 3 km. Conclusion: The correct statement is: - B. They meet after each has traveled 3 km.

Total Handshakes = Given: Handshakes = 24 Total Hugs = Tested pairs for m and f: For m = 6, f = 4: - Handshakes: (Correct) - Hugs among males: - Hugs among females: - Total Hugs: Therefore, the possible number of hugs is 21.

Given Data: - 45% read magazine A - 55% read magazine B - 40% read magazine C - 30% read both magazines A and B - 15% read both magazines B and C - 25% read both magazines A and C - 10% read all three magazines A, B, and C Using the Inclusion-Exclusion Principle: Formula: Substituting the given values: Calculations: Thus, 80% of the population reads at least one magazine. Conclusion: The percentage of the population that does not read any magazine is: Correct answer: C. 20%

Given: A cow costs more than 4 goats but less than 5 goats. The cost of a goat is between Rs. 600 and Rs. 800. Calculations: Minimum cost of a cow: If a goat costs Rs. 600: 4 goats cost Rs. 2,400. 5 goats cost Rs. 3,000. Therefore, the cow's cost is more than Rs. 2,400 but less than Rs. 3,000. Maximum cost of a cow: If a goat costs Rs. 800: 4 goats cost Rs. 3,200. 5 goats cost Rs. 4,000. Therefore, the cow's cost is more than Rs. 3,200 but less than Rs. 4,000. Conclusion: Combining both scenarios, the cow's cost is between Rs. 2,400 and Rs. 4,000. Most valid conclusion: D. A cow costs between Rs. 2,400 and Rs. 4,000.

Step 1: Determine the distances covered by A and B. A's distance: A covers the full 500 meters. B's distance: B starts 45 meters ahead, so B covers 500 meters minus 45 meters, which equals 455 meters. Step 2: Calculate the ratio of the distances covered. The ratio of the distances covered by A and B is: Distance Ratio = A's Distance / B's Distance = Simplifying this ratio: Step 3: Relate the distance ratio to the speed ratio. Since both A and B start at the same time and run for the same duration, their speeds are directly proportional to the distances they cover. Therefore, the ratio of their speeds is the same as the ratio of the distances: Speed Ratio = A's Speed / B's Speed = Step 4: Convert the ratio to a more familiar form. To express the ratio in whole numbers, we can multiply both the numerator and the denominator by 25: This simplifies to approximately 25:21. Conclusion: The ratio of the speeds of A to B is approximately 25:21. Therefore, the correct answer is A. 25:21.

To solve when the minute hand is 3 minutes ahead of the hour hand: At 6:00, the hour hand is at . The minute hand moves per minute, and the hour hand moves per minute. The difference between them should be (since 3 minutes = ). Equation: Simplify: So, at 6:36 PM, the minute hand is 3 minutes ahead of the hour hand. Answer: C. 6:36 PM

Let: = number of two-wheelers (scooters/motorbikes) = number of four-wheelers (cars) Given: - Total number of vehicles = - Total number of wheels = (since each two-wheeler has 2 wheels and each four-wheeler has 4 wheels) - Total number of wheels = Formulating the equation: Simplifying: Subtract from both sides: Subtract from both sides: Divide both sides by 2: Conclusion: The number of cars parked is 50. Answer: C. 50

Clearly, by the diagram below km east, then km north will bring X to his starting point.