Averages for RRB Group D

Average = total ÷ number of items. Turn it around and you get the formula that solves almost every average question:

• Total = average × number.

Whatever the story (a new worker joins, a batsman plays another innings, a mark was written wrongly), convert every average into a total, do the change on the totals, and divide at the end. Candidates who try to work with averages directly get tangled. Candidates who work with totals rarely do.

Quick facts worth memorising

The main question types

1. Someone joins or leaves

Find the old total and the new total. The difference is the person who joined or left.

2. A wrong entry

The total changes by (correct value − wrong value). So the average changes by that difference ÷ number of items.

3. Replacement

One person leaves and another takes their place, so the group size stays the same. If the average of n people changes by d:

• New person = old person + (n × d)

Use a minus sign when the average falls.

4. Combining two groups

You cannot just average two averages unless the groups are the same size. Combine the totals:

• Combined average = (n₁ × a₁ + n₂ × a₂) ÷ (n₁ + n₂)

The answer always lies between the two averages, closer to the larger group.

Worked examples

Example 1: The average of 4 numbers is 25. When a fifth number is added, the average becomes 28. Find the fifth number.

• Old total = 4 × 25 = 100. New total = 5 × 28 = 140.
• Fifth number = 140 − 100 = 40.

Example 2: The average age of 20 workers is 30 years. When the supervisor is included, the average becomes 31. How old is the supervisor?

• Total of 21 = 21 × 31 = 651. Total of 20 = 600.
• Supervisor = 51 years.
• Shortcut: the supervisor is 30 + 21 × 1 = 51. They bring the group's 30 plus one extra year for each of the 21 people.

Example 3: A student's average in 5 subjects is 72. One mark was wrongly written as 58 instead of 85. What is the correct average?

• The total rises by 85 − 58 = 27.
• The average rises by 27 ÷ 5 = 5.4.
• Correct average = 77.4.

Example 4: The average weight of 8 people rises by 2.5 kg when a person weighing 65 kg is replaced by a new person. What does the new person weigh?

• New person = 65 + 8 × 2.5 = 65 + 20 = 85 kg.

Example 5: Section A has 30 students with an average of 60 marks. Section B has 20 students with an average of 70. Find the average of all 50 students.

• Totals: 30 × 60 = 1,800 and 20 × 70 = 1,400.
• Combined = 3,200 ÷ 50 = 64.
• Not 65. The bigger section pulls the average towards 60.

Example 6: A batsman has an average of 40 runs after 10 innings. How many runs must they score in the 11th innings to raise the average to 42?

• Needed total after 11 innings = 11 × 42 = 462. Current total = 400.
• Runs needed = 62.

Common mistakes

• Averaging two averages when the groups are of different sizes.
• Forgetting that the group size changes when someone joins or leaves.
• In replacement questions, adding d instead of n × d.
• Dividing a wrong-entry correction by the wrong count.
• Using the average of "first n natural numbers" formula for numbers that do not start at 1.

Practice set

1. Find the average of 10, 20, 30 and 40.
2. The average of 5 numbers is 40. After one is removed, the average is 38. Find the removed number.
3. Find the average of the first 9 natural numbers.
4. The average of 3 numbers is 50. Two of them are 40 and 65. Find the third.
5. The average weight of 10 people rises by 2 kg when a person weighing 50 kg is replaced. How much does the new person weigh?
6. The average of 5 consecutive odd numbers is 27. Find the largest.
7. The average of 11 results is 50. The average of the first 6 is 49 and of the last 6 is 52. Find the sixth result.
8. Find the average of the first 10 even numbers.

Answers:

1. 25. Equally spaced: (10 + 40) ÷ 2.
2. 48. 5 × 40 = 200; 4 × 38 = 152; 200 − 152 = 48.
3. 5. (9 + 1) ÷ 2.
4. 45. Total 150; 150 − 40 − 65 = 45.
5. 70 kg. 50 + 10 × 2 = 70.
6. 31. The middle number is 27, so the numbers are 23, 25, 27, 29, 31.
7. 56. The sixth result is counted in both groups: 6 × 49 + 6 × 52 − 11 × 50 = 294 + 312 − 550 = 56.
8. 11. 2 to 20 has an average of (2 + 20) ÷ 2 = 11, which is n + 1.

What to do next

• Write "total = average × number" at the top of your averages page, and use it in every question.
• Practise five replacement and five combined-group questions.
• Try the "overlapping groups" type (practice question 7) until you can set it up without help.
• Move on to problems on ages, where average questions often reappear, and revise ratio.