Averages for RRB NTPC: the total method, shortcuts and practice

Average questions look different on the surface: students and a teacher, a batsman's innings, a family's ages, a person replaced in a group. Underneath, nearly all of them are solved the same way. You turn averages into totals, add or subtract the totals, and turn the result back into an average.

This makes averages one of the most dependable scoring topics in NTPC Maths. Once the total method is a habit, most questions take three short lines.

The one formula

Average = total ÷ number of items. Turn it around:

Total = average × number.

Why this is the key: averages cannot be added or subtracted directly, but totals can. If five numbers average 20 and four of them average 18, you cannot subtract 18 from 20 to learn anything. You can subtract 72 from 100 and find the missing number is 28.

Shortcuts, and why each works

Equally spaced numbers. The average of numbers in an arithmetic progression (such as 12, 15, 18, 21, 24) is the middle term, or the mean of the two middle terms. It also equals (first + last) ÷ 2.
Why: the numbers pair off around the middle. 12 and 24 average 18, 15 and 21 average 18, and 18 is 18.

So the average of the first n natural numbers is (n + 1)/2, the first n even numbers is n + 1, and the first n odd numbers is n.

Adding the same amount to every item. If every item rises by k, the average rises by k. If every item is multiplied by k, so is the average.
Why: the total rises by n × k, and dividing by n gives k.

The deviation method. Pick a convenient number close to the data, add up how far each item is from it, and correct.
Why: this is the "add k to every item" rule in reverse. You are averaging the small deviations instead of the large numbers.

Ages over time. In n years, every member of a group is n years older, so the average age rises by n, as long as nobody joins or leaves.

Adding, removing and replacing members

Always count the group size carefully. When a teacher joins 30 students, the new total is for 31 people.

Weighted average

When two groups of different sizes combine, the overall average is (n₁ × a₁ + n₂ × a₂) ÷ (n₁ + n₂). It always lies between the two group averages and sits closer to the larger group's average.

Why: this is the total method again. Each group's total is its size times its average.

Worked questions at NTPC level

Q1. Find the average of 47, 52, 49, 55 and 57.

Take 50 as a base. Deviations: −3, +2, −1, +5, +7. Their sum is 10, and 10 ÷ 5 = 2.
Average = 50 + 2 = 52. Check: total = 260, and 260 ÷ 5 = 52.

Q2. The average age of 30 students is 14 years. When the teacher's age is included, the average becomes 15. Find the teacher's age.

Total of 31 = 31 × 15 = 465. Total of 30 = 30 × 14 = 420.
Teacher's age = 465 − 420 = 45 years.

A quicker view: the teacher lifts 31 people's average by 1, so the teacher is 31 years above the old average: 14 + 31 = 45.

Q3. A batsman's average after 10 innings is 32. How many runs must they score in the 11th innings to raise the average to 34?

Target total after 11 innings = 11 × 34 = 374. Current total = 10 × 32 = 320.
Runs needed = 54.

Q4. The average weight of 8 people rises by 2 kg when one person weighing 60 kg is replaced by a new person. Find the new person's weight.

The total rises by 8 × 2 = 16 kg. So the new person is 16 kg heavier than the person who left.
New person = 60 + 16 = 76 kg.

Q5. One section of 20 students averages 50 marks and another section of 30 students averages 60. Find the average of all 50 students.

(20 × 50 + 30 × 60) ÷ 50 = (1,000 + 1,800) ÷ 50 = 56.
It is closer to 60 because the second section is larger.

Q6. The average of 11 results is 50. The average of the first six is 49 and of the last six is 52. Find the sixth result.

The sixth result is counted in both groups of six.
First six + last six = 6 × 49 + 6 × 52 = 294 + 312 = 606. All eleven = 11 × 50 = 550.
Sixth result = 606 − 550 = 56.

Common mistakes

• Averaging two averages without weighting by group size.
• Miscounting the group after someone joins or leaves.
• Confusing change in average with change in total. A 2 kg rise in the average of 8 people is a 16 kg rise in the total.
• Using (x + y)/2 for average speed over equal distances.
• Forgetting that ages move with time. Five years later, every member is five years older.

Practice set

1. Find the average of 8, 12, 16 and 20.
2. The average of 6 numbers is 25. If one number is excluded, the average becomes 24. Find the excluded number.
3. The average weight of 20 people rises by 1 kg when one person weighing 45 kg is replaced. Find the new person's weight.
4. Find the average of the first 20 even numbers.
5. The average of three numbers is 40. Two of them are 30 and 42. Find the third.
6. Five years ago, the average age of a family of four was 24 years. A child has been born since then, and today the average age of the five members is 24 years. How old is the child?
7. The average of 11 results is 60. The first six average 58 and the last six average 63. Find the sixth result.
8. Find the average of the first 50 natural numbers.

Answers:

1. 14. Equally spaced, so the average of 8 and 20.
2. 30. 6 × 25 − 5 × 24 = 150 − 120.
3. 65 kg. Total rises by 20 × 1 = 20; 45 + 20 = 65.
4. 21. First n even numbers average n + 1. Check: (2 + 40) ÷ 2 = 21.
5. 48. 3 × 40 − (30 + 42) = 120 − 72.
6. 4 years. Today the four members total 4 × 24 + 4 × 5 = 116. The five total 5 × 24 = 120. Child = 4.
7. 66. 6 × 58 + 6 × 63 − 11 × 60 = 348 + 378 − 660.
8. 25.5. (50 + 1) ÷ 2.

What to do next

• For the next 20 average questions you solve, write the totals line first, before any other working.
• Practise the deviation method on five sets of numbers a day for a week.
• Revise ratio and proportion, then move on to profit and loss.