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Question 1 / 20
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1
An MBA graduating class has an overall mean monthly salary of Rs. 16,000. Students with prior work experience earn a mean monthly salary of Rs. 18,000, while those without experience earn Rs. 12,000. Calculate the percentage breakdown of students with and without work experience in the class.
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Solution: **Method 1: Using Equations** Step 1: Let 'W' be the number of students with work experience and 'N' be the number of students without work experience. Step 2: The total salary earned by students with work experience = 18000W. Step 3: The total salary earned by students without work experience = 12000N. Step 4: The overall total salary for the class = 16000 * (W + N). Step 5: Set up the equation: 18000W + 12000N = 16000(W + N). Step 6: Divide all terms by 1000 to simplify: 18W + 12N = 16W + 16N. Step 7: Rearrange terms to solve for the ratio of W to N: 18W - 16W = 16N - 12N. 2W = 4N. W/N = 4/2 = 2/1. Step 8: The ratio of students with work experience to those without is 2:1. Step 9: Total parts in the ratio = 2 + 1 = 3 parts. Step 10: Percentage of students with work experience = (2 / 3) * 100% = 66.67%. Step 11: Percentage of students without work experience = (1 / 3) * 100% = 33.33%. **Method 2: Using Alligation (Mixture and Alligation Rule)** Step 1: Write down the average salaries of the two groups and the overall average: Work Experience Average: 18000 No Work Experience Average: 12000 Overall Average: 16000 Step 2: Apply the alligation rule to find the ratio of students (No Work Exp : Work Exp): (Work Exp Avg - Overall Avg) : (Overall Avg - No Work Exp Avg) (18000 - 16000) : (16000 - 12000) 2000 : 4000 Step 3: Simplify the ratio: 1 : 2. This ratio represents the proportion of students Without work experience : With work experience. Step 4: So, the ratio of students With work experience : Without work experience = 2 : 1. Step 5: Total parts in the ratio = 2 + 1 = 3 parts. Step 6: Percentage of students with work experience = (2 / 3) * 100% = 66.67%. Step 7: Percentage of students without work experience = (1 / 3) * 100% = 33.33%.
2
A business records sales of 5420, 5660, 6200, 6350, and 6500 units over five consecutive periods. What sales target is needed in the sixth period to achieve an overall average of 6000 units?
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Solution: Step 1: Calculate total sales for 5 periods: 5420 + 5660 + 6200 + 6350 + 6500 = 30,130 Step 2: Determine required total for 6 periods: 6000 * 6 = 36,000 Step 3: Find needed sales for 6th period: 36,000 - 30,130 = 5,870
3
Given the frequency distribution table showing the number of working hours per day for employees in a small-scale industry, calculate the average number of working hours per employee.
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Solution: Step 1: Determine the midpoint (class mark) for each working hour range. | Working Hours | No. of Employees (f) | Midpoint (x) | f × x | |---|---|---|---| | 3 - 5 | 7 | (3+5)/2 = 4 | 7 × 4 = 28 | | 5 - 7 | 10 | (5+7)/2 = 6 | 10 × 6 = 60 | | 7 - 9 | 18 | (7+9)/2 = 8 | 18 × 8 = 144 | | 9 - 11 | 57 | (9+11)/2 = 10 | 57 × 10 = 570 | | 11 - 13 | 14 | (11+13)/2 = 12 | 14 × 12 = 168 | | 13 - 15 | 8 | (13+15)/2 = 14 | 8 × 14 = 112 | Step 2: Sum all (f × x) values to get the total sum of working hours. Total (f × x) = 28 + 60 + 144 + 570 + 168 + 112 = 1082 Step 3: Sum all frequencies (number of employees) to get the total number of employees. Total employees (Σf) = 7 + 10 + 18 + 57 + 14 + 8 = 114 Step 4: Calculate the average by dividing the total sum of working hours by the total number of employees. Average working hours = (Σf × x) / Σf = 1082 / 114 ≈ 9.491 Step 5: Round the average to the nearest option. Average working hours ≈ 9.5.
4
The average marks of 14 students were 71. It was later found that one student's marks were wrongly entered as 42 instead of 56, and another's as 74 instead of 32. What is the correct average?
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Solution: Step 1: Calculate the initial total sum of marks for 14 students: 14 * 71 = 994. Step 2: Determine the net adjustment needed for the total sum: First error: Correct 56, Wrong 42. Difference = +14 (sum was less by 14). Second error: Correct 32, Wrong 74. Difference = -42 (sum was more by 42). Step 3: Calculate the overall net adjustment: +14 + (-42) = -28. Step 4: Apply the net adjustment to the initial sum to get the actual total sum: 994 - 28 = 966. Step 5: Calculate the correct average: 966 / 14 = 69.
5
The average height of 25 boys is 1.4 meters. If 5 boys leave the group, the average height of the remaining boys increases by 0.15 meters. What is the average height of the 5 boys who left?
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Solution: Step 1: Calculate the total height of the original 25 boys: Total Height = 1.4 m × 25 = 35 m. Step 2: After 5 boys leave, the remaining number of boys is 25 - 5 = 20 boys. Step 3: The new average height of the remaining 20 boys is 1.4 m + 0.15 m = 1.55 m. Step 4: Calculate the total height of the remaining 20 boys: New Total Height = 1.55 m × 20 = 31 m. Step 5: The total height of the 5 boys who left is the difference between the original total height and the new total height: Height of 5 boys = 35 m - 31 m = 4 m. Step 6: Calculate the average height of the 5 boys who left: Average Height = 4 m / 5 = 0.8 m.
6
The average cost of 5 apples and 4 mangoes is Rs. 36. The average cost of 7 apples and 8 mangoes is Rs. 48. Determine the total cost of 24 apples and 24 mangoes.
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Solution: Step 1: Calculate the total cost of the first group of fruits (5 apples + 4 mangoes = 9 items): Total cost1 = Average cost * Number of items = 36 * 9 = 324 Rs. Step 2: Calculate the total cost of the second group of fruits (7 apples + 8 mangoes = 15 items): Total cost2 = Average cost * Number of items = 48 * 15 = 720 Rs. Step 3: Sum the quantities and costs from both groups to find the total cost of (5+7=12) apples and (4+8=12) mangoes: Total cost of 12 apples and 12 mangoes = Total cost1 + Total cost2 = 324 + 720 = 1044 Rs. Step 4: The problem asks for the total cost of 24 apples and 24 mangoes. Notice that 24 apples and 24 mangoes is exactly double the quantity of 12 apples and 12 mangoes. Step 5: Calculate the final total cost: 2 * (Total cost of 12 apples and 12 mangoes) = 2 * 1044 = 2088 Rs.
7
The average monthly income of P and Q is Rs. 5050. The average monthly income of Q and R is Rs. 6250. The average monthly income of P and R is Rs. 5200. Calculate P's monthly income.
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Solution: Step 1: Convert average incomes into sum of incomes equations. (P + Q) / 2 = 5050 => P + Q = 10100 (Equation 1) (Q + R) / 2 = 6250 => Q + R = 12500 (Equation 2) (P + R) / 2 = 5200 => P + R = 10400 (Equation 3) Step 2: Add all three equations together. (P + Q) + (Q + R) + (P + R) = 10100 + 12500 + 10400 2P + 2Q + 2R = 33000 2(P + Q + R) = 33000 P + Q + R = 16500 (Equation 4) Step 3: Find P's income by subtracting Equation 2 from Equation 4. (P + Q + R) - (Q + R) = 16500 - 12500 P = 4000. Step 4: State P's monthly income. P's monthly income is Rs. 4000.
8
Determine the average of all integers between 6 and 34 that are divisible by 5.
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Solution: Step 1: Identify the numbers between 6 and 34 that are divisible by 5. These numbers are 10, 15, 20, 25, 30. Step 2: Count the total number of identified values. There are 5 numbers. Step 3: Calculate the sum of these numbers: 10 + 15 + 20 + 25 + 30 = 100. Step 4: Calculate the average by dividing the sum by the count. Average = Sum / Number of values = 100 / 5 = 20. (Alternatively, since these numbers form an arithmetic progression, the average is the mean of the first and last term: (10 + 30) / 2 = 40 / 2 = 20.)
9
The average weight of A, B, and C is 65 kg. If the average weight of C and B is 61.5 kg, and the average weight of A and C is 68.5 kg, determine the weight of C in kilograms.
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Solution: Step 1: Convert the given averages into sums: * Sum of A, B, and C's weights: A + B + C = 65 kg × 3 = 195 kg (Equation 1) * Sum of B and C's weights: B + C = 61.5 kg × 2 = 123 kg (Equation 2) * Sum of A and C's weights: A + C = 68.5 kg × 2 = 137 kg (Equation 3) Step 2: Substitute Equation 2 into Equation 1 to find A's weight: * A + (B + C) = 195 * A + 123 = 195 * A = 195 - 123 = 72 kg. Step 3: Substitute the value of A into Equation 3 to find C's weight: * A + C = 137 * 72 + C = 137 * C = 137 - 72 = 65 kg. (Alternative for Step 2 and 3: Add Equation 2 and Equation 3: (B + C) + (A + C) = 123 + 137 => A + B + 2C = 260. Subtract Equation 1 from this result: (A + B + 2C) - (A + B + C) = 260 - 195 => C = 65 kg.)
10
A class of 50 students has an average mark of 70%. The first 25 students have an average of 60%, and the next 24 students have an average of 80%. What percentage mark did the last (50th) student obtain?
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Solution: Step 1: Calculate the total marks for all 50 students. Total students = 50. Overall average mark = 70%. Total marks = 50 * 70 = 3500. Step 2: Calculate the total marks for the first 25 students. Number of students = 25. Average mark = 60%. Total marks for first 25 students = 25 * 60 = 1500. Step 3: Calculate the total marks for the next 24 students. Number of students = 24. Average mark = 80%. Total marks for next 24 students = 24 * 80 = 1920. Step 4: Calculate the marks obtained by the 50th student. Marks of last student = Total marks of 50 students - (Total marks of first 25 students + Total marks of next 24 students) Marks of last student = 3500 - (1500 + 1920) Marks of last student = 3500 - 3420 = 80. Since marks are expressed as percentages, the mark obtained by the last student is 80%.
11
Among three numbers, the second number is double the first and also triple the third. If the average of these three numbers is 44, determine the largest number.
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Solution: Step 1: Let the three numbers be N1, N2, and N3. Step 2: According to the problem, N2 = 2 × N1 and N2 = 3 × N3. Step 3: Express all numbers in terms of a common variable. Let N1 = 3x. Then N2 = 2 × (3x) = 6x. Step 4: Since N2 = 3 × N3, 6x = 3 × N3, which implies N3 = 2x. Step 5: The three numbers are 3x, 6x, and 2x. Step 6: The average of the three numbers is 44. So, (N1 + N2 + N3) / 3 = 44. Step 7: Substitute the expressions: (3x + 6x + 2x) / 3 = 44. Step 8: Simplify: 11x / 3 = 44. Step 9: Solve for x: 11x = 132, so x = 12. Step 10: Find the largest number. The numbers are 3x=36, 6x=72, and 2x=24. Step 11: The largest number is 6x = 6 × 12 = 72.
12
Gautam travels to his office at a speed of 12 kmph and returns home at 10 kmph. What is his average speed for the entire journey?
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Solution: Step 1: Identify the two speeds for the equal distances traveled (to office and back home). Speed_1 (v1) = 12 km/hr. Speed_2 (v2) = 10 km/hr. Step 2: Use the formula for average speed when two equal distances are covered at different speeds: Average Speed = (2 × v1 × v2) / (v1 + v2). Step 3: Substitute the values into the formula. Average Speed = (2 × 12 × 10) / (12 + 10) Average Speed = (2 × 120) / 22 Average Speed = 240 / 22 Average Speed = 120 / 11 km/hr. Step 4: Calculate the decimal value. Average Speed ≈ 10.909... km/hr, which is approximately 10.9 km/hr. Step 5: Gautam's average speed is approximately 10.9 km/hr.
13
Calculate the average of all natural numbers from 21 to 39, inclusive.
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Solution: Step 1: The numbers from 21 to 39 form an arithmetic progression (AP) with a common difference of 1. Step 2: For an arithmetic progression, the average is (First Term + Last Term) / 2. Step 3: Here, the First Term = 21 and the Last Term = 39. Step 4: Calculate the average: Average = (21 + 39) / 2 = 60 / 2 = 30. Step 5: The average of all natural numbers from 21 to 39 is 30.
14
For five consecutive years, the revenues of a company are $21,37,100, $19,28,700, $26,34,500, $22,85,400, and $24,14,300. Find by what percentage the highest revenue exceeds the average revenue.
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Solution: Step 1: Calculate the total revenue = $21,37,100 + $19,28,700 + $26,34,500 + $22,85,400 + $24,14,300 = $114,00,000 Step 2: Calculate the average revenue = Total revenue / Number of years = $114,00,000 / 5 = $22,80,000 Step 3: Identify the highest revenue = $26,34,500 Step 4: Calculate the difference between the highest and average revenue = $26,34,500 - $22,80,000 = $3,54,500 Step 5: Calculate the percentage by which the highest revenue exceeds the average revenue = ($3,54,500 / $22,80,000) * 100 = 15.54%
15
What is the arithmetic mean of the first 11 natural numbers?
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Solution: Step 1: Identify the first 11 natural numbers. These are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. Step 2: Calculate the sum of these 11 numbers using the sum of an arithmetic progression formula. Sum of first 'n' natural numbers = n × (n + 1) / 2 Here, n = 11. Sum = 11 × (11 + 1) / 2 = 11 × 12 / 2 = 11 × 6 = 66. Step 3: Divide the sum by the count of numbers (11) to find the mean. Mean = Sum / Number of terms = 66 / 11 = 6.0.
16
The average of ten positive numbers is 'x'. If each of these numbers is increased by 10%, how is 'x' affected?
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Solution: Step 1: Let the ten positive numbers be n1, n2, ..., n10. Step 2: The original average 'x' is given by: x = (n1 + n2 + ... + n10) / 10. Step 3: If each number is increased by 10%, each new number becomes 1.1 times its original value (e.g., 1.1 * n1, 1.1 * n2, etc.). Step 4: The new average, let's call it x', will be: x' = (1.1*n1 + 1.1*n2 + ... + 1.1*n10) / 10. Step 5: Factor out 1.1 from the numerator: x' = 1.1 * (n1 + n2 + ... + n10) / 10. Step 6: Recognize that (n1 + n2 + ... + n10) / 10 is the original average 'x'. Step 7: Substitute 'x' back into the equation: x' = 1.1 * x. Step 8: This shows that the new average is 1.1 times the original average, which means the average 'x' is increased by 10%.
17
The average age of a group of 4 individuals is 36 years. The youngest individual is 6 years old. What was the average age of the group at the time of the youngest individual's birth?
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Solution: Step 1: Current total age of group = 4 * 36 = 144 years Step 2: Youngest individual's current age = 6 years Step 3: At birth, youngest individual's age = 0 years Step 4: Time elapsed since birth = 6 years Step 5: Total age of other 3 individuals 6 years ago = 144 - 6 - 3 * 6 = 144 - 6 - 18 = 120 years Step 6: Average age 6 years ago = 120 / 3 = 40 years
18
Five distinct positive numbers have an average of 25. If the smallest number among them is replaced by 0, the average decreases by 'x'. What statement is true about 'x'?
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Solution: Step 1: Let the five distinct positive numbers be n1, n2, n3, n4, n5, where n1 is the smallest. Since they are distinct positive numbers and their average is 25, the smallest number (n1) must be positive and less than 25. * 0 < n1 < 25 Step 2: The average of the five numbers is 25. So, their sum is: * Sum (S) = n1 + n2 + n3 + n4 + n5 = 5 * 25 = 125. Step 3: When the smallest number (n1) is replaced by 0, the new sum (S') becomes: * S' = 0 + n2 + n3 + n4 + n5 = (n1 + n2 + n3 + n4 + n5) - n1 = S - n1 = 125 - n1. Step 4: The new average (Avg') is the new sum divided by 5. * Avg' = S' / 5 = (125 - n1) / 5 = 25 - (n1 / 5). Step 5: The decrease in average, 'x', is the original average minus the new average. * x = 25 - Avg' = 25 - (25 - n1/5) = n1/5. Step 6: From Step 1, we established that 0 < n1 < 25. Divide this inequality by 5: * 0/5 < n1/5 < 25/5 * 0 < n1/5 < 5. Step 7: Since x = n1/5, it follows that 0 < x < 5. Therefore, 'x is less than 5'.
19
A boy cycles 10 km at an average speed of 12 km/hr and then travels another 12 km at an average speed of 10 km/hr. What is his approximate average speed for the entire trip?
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Solution: Step 1: Calculate the total distance traveled: Total Distance = 10 km + 12 km = 22 km. Step 2: Calculate the time taken for the first part of the journey: Time1 = Distance1 / Speed1 = 10 km / 12 km/hr = 5/6 hours. Step 3: Calculate the time taken for the second part of the journey: Time2 = Distance2 / Speed2 = 12 km / 10 km/hr = 6/5 hours. Step 4: Calculate the total time taken for the entire journey: Total Time = Time1 + Time2 = (5/6) + (6/5) hours. Step 5: Find a common denominator (30) to add the fractions: Total Time = (25/30) + (36/30) = 61/30 hours. Step 6: Calculate the average speed for the entire trip: Average Speed = Total Distance / Total Time. Step 7: Average Speed = 22 km / (61/30) hours = (22 * 30) / 61 km/hr = 660 / 61 km/hr. Step 8: Perform the division: 660 / 61 ≈ 10.819 km/hr. Step 9: Round to one decimal place, the average speed is approximately 10.8 km/hr.
20
The average temperature for the first four days of a week is 40.2°C, and for the last four days, it is 41.3°C. If the average temperature for the entire week (7 days) is 40.6°C, what was the temperature on the fourth day?
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Solution: Step 1: Calculate the total temperature for the first four days. Total Temp (Day 1-4) = 40.2°C/day × 4 days = 160.8°C. Step 2: Calculate the total temperature for the last four days. Total Temp (Day 4-7) = 41.3°C/day × 4 days = 165.2°C. Step 3: Calculate the total temperature for the entire week (7 days). Total Temp (Day 1-7) = 40.6°C/day × 7 days = 284.2°C. Step 4: Sum the totals from Step 1 and Step 2. This sum includes the temperature of the fourth day twice because it's part of both periods. Sum of (Day 1-4) + (Day 4-7) = 160.8 + 165.2 = 326°C. Step 5: The temperature on the fourth day is found by subtracting the total temperature for the whole week from the sum calculated in Step 4. Temperature on 4th day = 326°C - 284.2°C = 41.8°C.
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