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Question 1 / 20
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1
A company had reserves of 68,000, 72,000, and 67,000 for three consecutive months. What was the average monthly reserve?
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Solution: Step 1: Calculate the sum of reserves for the three months: 68,000 + 72,000 + 67,000 = 207,000 Step 2: Divide the sum by the number of months to find the average: 207,000 / 3 = 69,000 The average reserve per month is 69,000.
2
Calculate the mean value of the first 30 positive integers.
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Solution: Step 1: Sum of first n natural numbers = n(n+1)/2. Step 2: For n=30, sum = 30*31/2 = 465. Step 3: Calculate average: 465 / 30 = 15.5.
3
The average price of 10 books is Rs. 12. The average price of 8 of these books is Rs. 11.75. Of the two remaining books, the price of one is 60% higher than the price of the other. What is the price of each of these two books?
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Solution: Step 1: Calculate the total cost of all 10 books. * Total cost (10 books) = 10 books × Rs. 12/book = Rs. 120. Step 2: Calculate the total cost of the first 8 books. * Total cost (8 books) = 8 books × Rs. 11.75/book = Rs. 94. Step 3: Determine the combined cost of the two remaining books. * Combined cost of 2 books = Total cost (10 books) - Total cost (8 books) = 120 - 94 = Rs. 26. Step 4: Let the price of one of the remaining books be 'y' and the price of the other be 'x'. From the problem, x + y = 26 (Equation 1). Step 5: Formulate the relationship that one book's price is 60% more than the other. Let x be 60% more than y. * x = y + 0.60y = 1.6y (Equation 2). Step 6: Substitute Equation 2 into Equation 1. * 1.6y + y = 26 * 2.6y = 26 * y = 26 / 2.6 = 10. Step 7: Find the value of x using y = 10 in Equation 2. * x = 1.6 × 10 = 16. Step 8: Therefore, the prices of the two books are Rs. 16 and Rs. 10.
4
The average marks obtained by A and B are 15 less than the average marks obtained by B and C. If C scored 65 marks, what is the score obtained by A?
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Solution: Step 1: Express the given information as an equation. The average of B and C minus the average of A and B equals 15: * [(B + C) / 2] - [(A + B) / 2] = 15 Step 2: Combine the fractions (they have a common denominator): * (B + C - A - B) / 2 = 15 Step 3: Simplify the numerator: * (C - A) / 2 = 15 Step 4: Multiply both sides by 2: * C - A = 30 Step 5: Substitute the given value of C (65 marks) into the equation: * 65 - A = 30 Step 6: Solve for A: * A = 65 - 30 * A = 35. Therefore, A obtained 35 marks.
5
A man's income for three months covers his expenses for four months. If his monthly income is Rs. 1,000, calculate his total annual savings.
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Solution: Step 1: Understand the spending pattern: The man spends 3 months' income over a period of 4 months. Step 2: Calculate his income for 3 months: 3 * Rs. 1,000 = Rs. 3,000. Step 3: This means his expenditure for 4 months is Rs. 3,000. Step 4: Calculate his monthly expenditure: Rs. 3,000 / 4 = Rs. 750. Step 5: Calculate his total annual income: 12 months * Rs. 1,000/month = Rs. 12,000. Step 6: Calculate his total annual expenditure: 12 months * Rs. 750/month = Rs. 9,000. Step 7: Calculate his annual savings: Annual Income - Annual Expenditure = Rs. 12,000 - Rs. 9,000 = Rs. 3,000.
6
16 children are divided into two groups: Group A with 10 children and Group B with 6 children. If the average percentage marks for Group A is 75 and the overall average percentage marks for all 16 children is 76, what is the average percentage marks for the children in Group B?
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Solution: Step 1: Calculate the total marks for all 16 children. Total marks for all 16 children = Overall average × Total number of children = 76 × 16 = 1216 Step 2: Calculate the total marks for Group A (10 children). Total marks for Group A = Average marks of Group A × Number of children in Group A = 75 × 10 = 750 Step 3: Subtract the total marks of Group A from the total marks of all children to find the total marks of Group B. Total marks for Group B = Total marks for all children - Total marks for Group A = 1216 - 750 = 466 Step 4: Calculate the average marks for Group B by dividing its total marks by the number of children in Group B. Average marks for Group B = Total marks for Group B / Number of children in Group B = 466 / 6 Average marks for Group B = 233 / 3 = 77 and 2/3 %.
7
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.
8
A student's average mark in 4 subjects is 75. If the student scores 80 marks in a fifth subject, what will be their new average across all five subjects?
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Solution: Step 1: Calculate the total marks in the initial 4 subjects. * Total marks (4 subjects) = Average * Number of subjects = 75 * 4 = 300 marks. Step 2: Add the marks from the fifth subject to get the new total marks. * New total marks (5 subjects) = 300 + 80 = 380 marks. Step 3: Calculate the new average for 5 subjects. * New average = New total marks / Number of subjects = 380 / 5 = 76. Step 4: The new average is 76.
9
The average mathematics score for a group of 5 students was determined to be 50. It was later found that one student's marks of 48 were mistakenly recorded as 84. What is the correct average score?
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Solution: Step 1: Calculate the total sum of marks based on the incorrect average: Initial Sum = Average Marks × Number of Students = 50 × 5 = 250 marks. Step 2: Identify the incorrectly read mark (84) and the correct mark (48). Step 3: To correct the total sum, subtract the incorrectly recorded mark and add the correct mark: Corrected Sum = Initial Sum - Incorrect Mark + Correct Mark. Corrected Sum = 250 - 84 + 48. Step 4: Perform the calculation: Corrected Sum = 166 + 48 = 214 marks. Step 5: The number of students remains the same, which is 5. Step 6: Calculate the correct average: Correct Average = Corrected Sum / Number of Students = 214 / 5. Step 7: Perform the division: 214 / 5 = 42.8. Step 8: Therefore, the correct average mark is 42.8.
10
The average of three values is 87. The first value is 4 times the second and 5 times the third. What is the difference between the first and third values?
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Solution: Step 1: Let second value = x, third value = y Step 2: First value = 4x (given) Step 3: First value = 5y (given) Step 4: Since both expressions equal the first value, 4x = 5y Step 5: Average of three values = (4x + x + y) / 3 = 87 Step 6: Simplify: 5x + y = 261 Step 7: From Step 4: y = (4/5)x Step 8: Substitute y in Step 6: 5x + (4/5)x = 261 Step 9: Combine: (25x + 4x)/5 = 261 Step 10: Simplify: 29x/5 = 261 Step 11: Solve for x: 29x = 1305 => x = 45 Step 12: Find y: y = (4/5)*45 = 36 Step 13: First value = 4x = 4*45 = 180 Step 14: Difference = 180 - 36 = 144
11
Sachin Tendulkar has a particular batting average over 11 innings. In his 12th innings, he scores 120 runs, which causes his overall average to increase by 5 runs. What is his new average?
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Solution: Step 1: Let Sachin's average for the first 11 innings be 'x' runs. Step 2: Total runs scored in 11 innings = 11x runs. Step 3: After the 12th innings, his new average becomes (x + 5) runs. Step 4: Total runs scored in 12 innings = 12 × (x + 5) runs. Step 5: Set up the equation relating the total runs: (Total runs in 11 innings) + (Runs in 12th inning) = (Total runs in 12 innings). Step 6: So, 11x + 120 = 12(x + 5). Step 7: Expand the equation: 11x + 120 = 12x + 60. Step 8: Solve for x: 120 - 60 = 12x - 11x, which means x = 60. Step 9: His new average is x + 5 = 60 + 5 = 65 runs.
12
Five numbers have an average of 27. If one number is removed, the average of the remaining numbers becomes 25. What is the value of the excluded number?
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Solution: Step 1: Calculate the total sum of the five original numbers: Sum of 5 numbers = Average × Count = 27 × 5 = 135. Step 2: When one number is excluded, there are now 4 numbers left. Step 3: The average of these four numbers is 25. Calculate their total sum: Sum of 4 numbers = Average × Count = 25 × 4 = 100. Step 4: The excluded number is the difference between the total sum of the original five numbers and the total sum of the remaining four numbers: Excluded Number = Sum of 5 numbers - Sum of 4 numbers = 135 - 100 = 35. Step 5: The excluded number is 35.
13
The calculated mean of 10 numbers is 30. It was later discovered that two numbers, 15 and 23, were incorrectly recorded as 51 and 32, respectively. What is the accurate mean?
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Solution: Step 1: Calculate the initial sum based on the incorrect mean: Initial Sum = Mean × Number of items = 30 × 10 = 300. Step 2: Identify the incorrect values (51, 32) and the correct values (15, 23). Step 3: Calculate the sum of incorrect values: 51 + 32 = 83. Step 4: Calculate the sum of correct values: 15 + 23 = 38. Step 5: Determine the difference that needs to be adjusted in the sum: Difference = (Sum of Incorrect Values) - (Sum of Correct Values) = 83 - 38 = 45. Step 6: Calculate the actual correct sum: Actual Sum = Initial Sum - Difference = 300 - 45 = 255. Step 7: Calculate the correct mean: Correct Mean = Actual Sum / Number of items = 255 / 10 = 25.5.
14
A cricket player's average runs over 10 innings was 32. How many runs must he score in his next innings to increase his average runs by 4?
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Solution: Step 1: Calculate the total runs in the first 10 innings: Total runs = Average * Number of innings = 32 * 10 = 320 runs. Step 2: Determine the new target average after the 11th innings: New average = Original average + 4 = 32 + 4 = 36 runs. Step 3: Calculate the total runs needed after 11 innings to achieve the new average: New total runs = New average * New number of innings = 36 * 11 = 396 runs. Step 4: Find the runs required in the 11th innings: Runs in 11th innings = New total runs - Old total runs = 396 - 320 = 76 runs.
15
Calculate the total transportation cost for 30 employees over a month.
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Solution: Step 1: Number of employees = 30. Step 2: Daily transportation cost per employee = Rs 50. Step 3: Daily total transportation cost = 30 * 50 = Rs 1500. Step 4: Monthly total transportation cost (30 days) = 1500 * 30 = Rs 45000.
16
Three batches of students contain 55, 60, and 45 students, respectively. Their average marks are 50, 55, and 60. What is the combined average marks of all the students?
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Solution: Step 1: Calculate the total marks for each batch. - Batch 1: 55 students * 50 marks/student = 2750 marks. - Batch 2: 60 students * 55 marks/student = 3300 marks. - Batch 3: 45 students * 60 marks/student = 2700 marks. Step 2: Calculate the total marks of all students. - Total marks = 2750 + 3300 + 2700 = 8750 marks. Step 3: Calculate the total number of students. - Total students = 55 + 60 + 45 = 160 students. Step 4: Calculate the combined average marks. - Combined average = Total marks / Total students = 8750 / 160 = 54.6875. Step 5: Round the average marks: 54.68.
17
A car owner purchases petrol at Rs. 17 per liter in the first year, Rs. 19 per liter in the second year, and Rs. 20 per liter in the third year. If he spends Rs. 6460 on petrol each year, calculate the average cost per liter over these three years.
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Solution: Step 1: Calculate the quantity of petrol purchased in each of the three years. Quantity in Year 1 = Total spent / Price per litre = 6460 / 17 = 380 litres Quantity in Year 2 = Total spent / Price per litre = 6460 / 19 = 340 litres Quantity in Year 3 = Total spent / Price per litre = 6460 / 20 = 323 litres Step 2: Calculate the total quantity of petrol consumed over three years. Total quantity = 380 + 340 + 323 = 1043 litres Step 3: Calculate the total amount of money spent on petrol over three years. Total amount spent = 3 × Rs. 6460 = Rs. 19380 Step 4: Calculate the average cost per liter by dividing the total amount spent by the total quantity consumed. Average cost per litre = Total amount spent / Total quantity = 19380 / 1043 ≈ Rs. 18.58.
18
The average of 'n' numbers, x1, x2, ..., xn, is x_bar. What is the value of the summation from i=1 to n of (xi - x_bar)?
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Solution: Step 1: Understand the definition of the arithmetic mean. The arithmetic mean (average) x_bar of 'n' numbers (x1, x2, ..., xn) is defined as: x_bar = (x1 + x2 + ... + xn) / n Step 2: Rewrite the sum of the numbers in terms of the average. From the definition, Sum of numbers = x1 + x2 + ... + xn = n × x_bar. Step 3: Expand the given summation expression. Summation (xi - x_bar) from i=1 to n = (x1 - x_bar) + (x2 - x_bar) + ... + (xn - x_bar). Step 4: Rearrange the terms in the expanded summation. This can be rewritten as: (x1 + x2 + ... + xn) - (x_bar + x_bar + ... + x_bar) [n times] Step 5: Substitute the sum of numbers and simplify. Summation = (n × x_bar) - (n × x_bar) = 0. Therefore, the sum of deviations of individual data points from their mean is always 0.
19
A person buys 1 kg of tomatoes from each of four different locations. The rates are 1 kg/rupee, 2 kg/rupee, 3 kg/rupee, and 4 kg/rupee respectively. What is the average quantity of tomatoes purchased per rupee (x kg/rupee)?
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Solution: Step 1: Determine the cost to purchase 1 kg of tomatoes from each place based on the given rates. - Place 1 (1 kg/rupee): Cost for 1 kg = Rs. 1. - Place 2 (2 kg/rupee): Cost for 1 kg = Rs. 1/2 = Rs. 0.50. - Place 3 (3 kg/rupee): Cost for 1 kg = Rs. 1/3. - Place 4 (4 kg/rupee): Cost for 1 kg = Rs. 1/4 = Rs. 0.25. Step 2: Calculate the total quantity of tomatoes purchased. - Total quantity = 1 kg + 1 kg + 1 kg + 1 kg = 4 kg. Step 3: Calculate the total money spent for the 4 kg of tomatoes. - Total money spent = 1 + 1/2 + 1/3 + 1/4 - Find a common denominator (12): (12/12) + (6/12) + (4/12) + (3/12) = 25/12 rupees. Step 4: Calculate the average kg of tomatoes purchased per rupee (x). - x = Total quantity purchased / Total money spent - x = 4 kg / (25/12) rupees - x = 4 * (12/25) = 48/25 kg/rupee. Step 5: Convert the fraction to a decimal. - x = 1.92 kg/rupee.
20
Four consecutive integers sum to 1290. What is the largest integer?
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Solution: Step 1: Let the four consecutive integers be n, n+1, n+2, n+3 Step 2: Sum of integers = n + (n+1) + (n+2) + (n+3) = 1290 Step 3: Combine like terms: 4n + 6 = 1290 Step 4: Subtract 6 from both sides: 4n = 1284 Step 5: Divide by 4: n = 321 Step 6: Largest integer = n + 3 = 321 + 3 = 324
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