Mean, weighted average, replacing members and average speed.
45 solved questions · 15 easy · 20 medium · 10 hard
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What is the average of the first 10 natural numbers?
Answer: A. 5.5
Solution
Sum of first 10 natural numbers = 10 × 11 ÷ 2 = 55 Average = 55 ÷ 10 = 5.5 Answer: 5.5
Shortcut: Average of first n natural numbers = (n + 1)/2 = 11/2.
Find the average of 12, 15, 18, 20 and 25.
Answer: D. 18
Solution
Sum = 12 + 15 + 18 + 20 + 25 = 90 Average = 90 ÷ 5 = 18 Answer: 18
The average of 3 consecutive odd numbers is 15. What is the largest of them?
Answer: C. 17
Solution
Consecutive odd numbers with average 15 have 15 as the middle number. The numbers are 13, 15 and 17. Largest = 17 Answer: 17
Shortcut: For an equally spaced list, the average is the middle term.
What is the average of the first 5 multiples of 6?
Answer: B. 18
Solution
The multiples are 6, 12, 18, 24, 30. Sum = 90, average = 90 ÷ 5 = 18 Answer: 18
Shortcut: Middle term of 6, 12, 18, 24, 30 is 18.
Meena's average mark in 4 subjects is 72. How many marks must she score in the fifth subject to raise her average to 75?
Answer: B. 87
Solution
Total needed for 5 subjects = 5 × 75 = 375 Total of first 4 = 4 × 72 = 288 Fifth subject = 375 − 288 = 87 Answer: 87
The average age of 20 students is 14 years. When the teacher's age is included, the average becomes 15 years. What is the teacher's age?
Answer: A. 35 years
Solution
Total age of 20 students = 20 × 14 = 280 Total age of 21 persons = 21 × 15 = 315 Teacher = 315 − 280 = 35 years Answer: 35 years
Find the average of all even numbers from 2 to 40.
Answer: D. 21
Solution
Even numbers 2, 4, ..., 40 are 20 terms. Average = (first + last)/2 = (2 + 40)/2 = 21 Answer: 21
The average of 25 and a number x is 40. Find x.
Answer: C. 55
Solution
(25 + x)/2 = 40 25 + x = 80 x = 55 Answer: 55
The average of 7 numbers is 30. If each number is increased by 5, what is the new average?
Answer: C. 35
Solution
When every number increases by 5, the average also increases by 5. New average = 30 + 5 = 35 Answer: 35
The average of 50 numbers is 38. Two numbers, 45 and 55, are removed. What is the average of the remaining numbers?
Answer: B. 37.5
Solution
Sum of 50 numbers = 50 × 38 = 1,900 Sum after removal = 1,900 − 45 − 55 = 1,800 Average = 1,800 ÷ 48 = 37.5 Answer: 37.5
The average of 8 numbers is 15 and the average of another 12 numbers is 20. What is the average of all 20 numbers?
Answer: A. 18
Solution
Sum of first group = 8 × 15 = 120 Sum of second group = 12 × 20 = 240 Total = 360, average = 360 ÷ 20 = 18 Answer: 18
A batsman has an average of 45 runs in 4 innings. He scores 70 in the fifth innings. What is his new average?
Answer: D. 50
Solution
Total in 4 innings = 4 × 45 = 180 Total in 5 innings = 180 + 70 = 250 New average = 250 ÷ 5 = 50 Answer: 50
What is the average of the first 20 odd numbers?
Answer: D. 20
Solution
Sum of first n odd numbers = n² = 400 Average = 400 ÷ 20 = 20 Answer: 20
Shortcut: Average of first n odd numbers is simply n.
The average of 6 numbers is 25. If one number, 40, is removed, what is the sum of the remaining 5 numbers?
Answer: C. 110
Solution
Total of 6 numbers = 6 × 25 = 150 Sum of remaining 5 = 150 − 40 = 110 Answer: 110
The average of five numbers is 12. The average of three of them is 10. What is the average of the other two?
Answer: B. 15
Solution
Sum of 5 numbers = 5 × 12 = 60 Sum of 3 numbers = 3 × 10 = 30 Sum of other 2 = 30, so average = 30 ÷ 2 = 15 Answer: 15
The average weight of 8 men increases by 2.5 kg when a man weighing 65 kg is replaced by a new man. What is the weight of the new man?
Answer: A. 85 kg
Solution
Increase in total weight = 8 × 2.5 = 20 kg New man = 65 + 20 = 85 kg Answer: 85 kg
Shortcut: New − old = (number of men) × (change in average).
A car travels from Pune to Mumbai at 60 km/h and returns at 40 km/h along the same road. What is its average speed for the whole journey?
Answer: A. 48 km/h
Solution
Let the distance be 120 km (LCM of 60 and 40). Time going = 120 ÷ 60 = 2 h, time returning = 120 ÷ 40 = 3 h Average speed = 240 ÷ 5 = 48 km/h Answer: 48 km/h
Shortcut: Average speed = 2xy/(x + y) = 2 × 60 × 40/100 = 48.
A cyclist covers three equal distances at 30 km/h, 40 km/h and 60 km/h. What is the average speed for the whole journey?
Answer: D. 40 km/h
Solution
Take each distance = 120 km. Times = 120/30 + 120/40 + 120/60 = 4 + 3 + 2 = 9 hours Total distance = 360 km Average speed = 360 ÷ 9 = 40 km/h Answer: 40 km/h
Section A has 30 students with an average mark of 60 and Section B has 20 students with an average mark of 70. What is the average mark of all 50 students?
Answer: C. 64
Solution
Total marks of A = 30 × 60 = 1,800 Total marks of B = 20 × 70 = 1,400 Total = 3,200, average = 3,200 ÷ 50 = 64 Answer: 64
The average of 11 numbers is 50. The average of the first six is 49 and of the last six is 52. What is the sixth number?
Answer: B. 56
Solution
Sum of first six = 6 × 49 = 294 Sum of last six = 6 × 52 = 312 The sixth number is counted twice: 294 + 312 = 606 = total + sixth Total = 11 × 50 = 550 Sixth = 606 − 550 = 56 Answer: 56
A batsman's average rises by 3 to 48 after his 17th innings. How many runs did he score in the 17th innings?
Answer: B. 96
Solution
Average after 17 innings = 48, so total = 48 × 17 = 816 Average after 16 innings = 45, so total = 45 × 16 = 720 17th innings = 816 − 720 = 96 Answer: 96
Shortcut: Score = new average + (innings × rise) = 48 + 17 × 3 − 3 = 96. Equivalently 45 + 17 × 3.
Five years ago, the average age of a husband and wife was 35 years. Now, with their child, the average age of the family of three is 28 years. What is the child's present age?
Answer: A. 4 years
Solution
Husband + wife 5 years ago = 2 × 35 = 70 Husband + wife now = 70 + 2 × 5 = 80 Total of three now = 3 × 28 = 84 Child = 84 − 80 = 4 years Answer: 4 years
The average of five consecutive integers is 12. If the next two integers are also included, what is the new average?
Answer: D. 13
Solution
The five integers are 10, 11, 12, 13, 14. Next two are 15 and 16. Sum of all seven = 10 + 11 + 12 + 13 + 14 + 15 + 16 = 91 Average = 91 ÷ 7 = 13 Answer: 13
Shortcut: Seven consecutive integers 10..16 have the middle term 13 as average.
The average of three numbers is 30. They are in the ratio 2 : 3 : 4. What is the largest number?
Answer: C. 40
Solution
Sum = 3 × 30 = 90 Total parts = 2 + 3 + 4 = 9, one part = 10 Largest = 4 × 10 = 40 Answer: 40
A shopkeeper mixes 4 kg of rice at ₹50 per kg with 6 kg of rice at ₹70 per kg. What is the average price per kg of the mixture?
Answer: C. ₹62
Solution
Total cost = 4 × 50 + 6 × 70 = 200 + 420 = ₹620 Total weight = 10 kg Average = 620 ÷ 10 = ₹62 per kg Answer: ₹62
The average of 25 numbers is 30. It is later found that 75 was wrongly copied as 25. What is the correct average?
Answer: B. 32
Solution
Correct sum = wrong sum + (75 − 25) = wrong sum + 50 Average increases by 50 ÷ 25 = 2 Correct average = 30 + 2 = 32 Answer: 32
The average weight of A, B and C is 40 kg. The average of A and B is 38 kg and the average of B and C is 43 kg. What is B's weight?
Answer: A. 42 kg
Solution
A + B + C = 3 × 40 = 120 A + B = 2 × 38 = 76, so C = 120 − 76 = 44 B + C = 2 × 43 = 86, so A = 120 − 86 = 34 B = 120 − 34 − 44 = 42 kg Answer: 42 kg
Shortcut: B = (A+B) + (B+C) − (A+B+C) = 76 + 86 − 120 = 42.
The average of 10 numbers is 18. Each number is multiplied by 3 and then decreased by 2. What is the new average?
Answer: D. 52
Solution
New average = 3 × 18 − 2 = 54 − 2 = 52 Answer: 52
The average of 12 results is 14. The average of the first seven is 12 and of the last seven is 17. What is the seventh result?
Answer: D. 35
Solution
Sum of first seven = 7 × 12 = 84 Sum of last seven = 7 × 17 = 119 Sum of all 12 = 12 × 14 = 168 The seventh result is counted twice: 84 + 119 − 168 = 35 Answer: 35
A cricketer has an average of 42 runs in 10 matches. What average must he score in the next 5 matches to make his overall average 45?
Answer: C. 51
Solution
Required total for 15 matches = 15 × 45 = 675 Runs in first 10 = 10 × 42 = 420 Runs needed in next 5 = 255, so average = 255 ÷ 5 = 51 Answer: 51
A company has 60 employees with an average salary of ₹20,000. The 10 managers earn an average of ₹38,000. What is the average salary of the other employees?
Answer: B. ₹16,400
Solution
Total salary = 60 × 20,000 = ₹12,00,000 Managers' total = 10 × 38,000 = ₹3,80,000 Others' total = 12,00,000 − 3,80,000 = ₹8,20,000 Average = 8,20,000 ÷ 50 = ₹16,400 Answer: ₹16,400
The average of the first n natural numbers is 25. Find n.
Answer: A. 49
Solution
Average = (n + 1)/2 = 25 n + 1 = 50 n = 49 Answer: 49
The average of four consecutive even numbers is 21. What is the largest of them?
Answer: A. 24
Solution
The four numbers are symmetric about 21: 18, 20, 22, 24. (Check: sum = 84, average = 21.) Largest = 24 Answer: 24
The average of five numbers is 27. If one number is excluded, the average becomes 25. What is the excluded number?
Answer: D. 35
Solution
Sum of five = 5 × 27 = 135 Sum of four = 4 × 25 = 100 Excluded number = 135 − 100 = 35 Answer: 35
The average age of 24 students in a class is 15 years. When 6 new students with an average age of 20 years join, what is the average age of the whole class?
Answer: C. 16 years
Solution
Total age of 24 students = 24 × 15 = 360 Total age of 6 new students = 6 × 20 = 120 Total = 480 for 30 students Average = 480 ÷ 30 = 16 years Answer: 16 years
A bus travels the first 90 km at 30 km/h and the next 150 km at 75 km/h. What is its average speed for the whole journey?
Answer: B. 48 km/h
Solution
Time for first part = 90 ÷ 30 = 3 h Time for second part = 150 ÷ 75 = 2 h Total distance = 240 km, total time = 5 h Average speed = 240 ÷ 5 = 48 km/h (The simple mean of 30 and 75 = 52.5 is wrong because the times differ.) Answer: 48 km/h
The average age of 8 persons is 32 years. Two persons aged 35 and 40 leave and two new persons join, after which the average age of the 8 persons is 30 years. What is the average age of the two new persons?
Answer: B. 29.5 years
Solution
Original total = 8 × 32 = 256 After the two leave: 256 − 35 − 40 = 181 New total = 8 × 30 = 240 Ages of two newcomers = 240 − 181 = 59 Average = 59 ÷ 2 = 29.5 years Answer: 29.5 years
The average age of five friends was 30 years five years ago. Today one of them leaves and a new friend joins, and the average age of the group is now 33 years. How many years younger is the newcomer than the person who left?
Answer: A. 10 years
Solution
Total age five years ago = 5 × 30 = 150 Total age now (same five) = 150 + 5 × 5 = 175 New total = 5 × 33 = 165 Newcomer − leaver = 165 − 175 = −10 So the newcomer is 10 years younger. Answer: 10 years
In an examination, the average mark of the students who passed is 39 and of those who failed is 15. If the average of all 100 students is 33, how many students passed?
Answer: D. 75
Solution
Let x students pass. 39x + 15(100 − x) = 33 × 100 39x + 1,500 − 15x = 3,300 24x = 1,800, x = 75 Answer: 75
Shortcut: Alligation: (33 − 15) : (39 − 33) = 18 : 6 = 3 : 1, so 3/4 of 100 = 75 passed.
The mean of 25 observations is 50. It was later found that two observations, 84 and 12, were wrongly recorded; the correct values were 48 and 23. What is the correct mean?
Answer: C. 49
Solution
Wrong sum = 25 × 50 = 1,250 Wrong values total = 84 + 12 = 96, correct values total = 48 + 23 = 71 Correct sum = 1,250 − 96 + 71 = 1,225 Correct mean = 1,225 ÷ 25 = 49 Answer: 49
A batsman scores 90 runs in his 20th innings, which raises his average by 2. What is his average after the 20th innings?
Answer: C. 52
Solution
Let the average after 19 innings be x. 19x + 90 = 20(x + 2) 19x + 90 = 20x + 40 x = 50 New average = 52 Answer: 52
Shortcut: New average = 90 − 20 × 2 + ... : score − (innings × rise) = 90 − 40 = 50 is the old average, so new = 52.
Three classes A, B and C have 25, 35 and 40 students with average marks of 40, 60 and x respectively. If the average mark of all 100 students is 59, find x.
Answer: B. 70
Solution
Total marks of all = 100 × 59 = 5,900 Class A = 25 × 40 = 1,000, class B = 35 × 60 = 2,100 Class C = 5,900 − 3,100 = 2,800 x = 2,800 ÷ 40 = 70 Answer: 70
A man goes from his house to the market at 30 km/h and returns at a different speed. If his average speed for the round trip is 36 km/h, at what speed did he return?
Answer: A. 45 km/h
Solution
Average speed = 2xy/(x + y) = 36 with x = 30. 60y = 36(30 + y) 60y = 1,080 + 36y 24y = 1,080, y = 45 km/h Answer: 45 km/h
Shortcut: Using 1/36 = (1/2)(1/30 + 1/y): 1/y = 1/18 − 1/30 = 1/45.
The average of six numbers is 30. The average of the first four is 28. The fifth number is 4 more than the sixth. What is the fifth number?
Answer: D. 36
Solution
Sum of six = 6 × 30 = 180 Sum of first four = 4 × 28 = 112 Fifth + sixth = 68 Let the sixth = s, fifth = s + 4: 2s + 4 = 68, so s = 32 Fifth = 36 Answer: 36
Ravi's average score in 6 tests is 72. What average must he score in the next 2 tests to raise his overall average to 75 after 8 tests?
Answer: D. 84
Solution
Required total for 8 tests = 8 × 75 = 600 Total of first 6 = 6 × 72 = 432 Needed in next 2 = 168, so average = 84 Answer: 84
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