Averages for SSC CHSL

An average is a total shared out equally. That one idea, total = average × count, solves nearly every average question in SSC CHSL faster than adding long lists. The questions look varied (cricket scores, ages, weights, marks), but the method underneath is the same every time: turn the average into a total, make the change, and turn it back.

Averages also feed other topics. Weighted averages are the same arithmetic as mixtures and alligation, and average speed is a special case you will meet in speed and distance.

The total method

Average = sum ÷ count, so sum = average × count. Whenever something is added, removed, replaced or corrected, work on the sum.

Why this is faster: the question almost never gives you the individual values, but it always lets you find the total. Changing a total is one addition or subtraction.

The deviation shortcut

To average numbers that sit close together, assume a round value, add up how far each number is from it, and share that out.

For 67, 72, 75, 69 and 77, assume 70. The deviations are −3, +2, +5, −1 and +7, which total +10. Shared among 5 numbers, that is +2 each, so the average is 72.

Why: every number is (70 + its deviation). Adding them gives 5 × 70 + (sum of deviations), and dividing by 5 gives 70 + (average deviation).

Consecutive and evenly spaced numbers

For any evenly spaced list (consecutive numbers, consecutive odds, multiples of 5), the average is (first + last)/2, which is also the middle term.

• Average of the first n natural numbers = (n + 1)/2.
• Average of the first n odd numbers = n. The first 25 odd numbers average 25.
• Average of the first n even numbers = n + 1.

Why: in an evenly spaced list, the first and last terms pair up, then the second and second-last, and every pair has the same sum.

Joining, leaving and replacement

• Someone joins: new member = new total − old total.
• Someone leaves: leaving member = old total − new total.
• Replacement (count unchanged): new member = old member + (change in average × count).

Why the replacement rule works: the count stays the same, so if the average rises by 2 in a group of 10, the total must have risen by 20. All of that rise came from the swap.

Weighted averages

When groups of different sizes are combined, the overall average is (n₁a₁ + n₂a₂)/(n₁ + n₂). It is not the plain average of the two group averages, unless the groups are the same size. The combined average always lies between the two group averages, closer to the larger group.

Seven worked questions

Q1. The average of 8 numbers is 25. If the number 32 is removed, what is the new average?
Total = 200. New total = 168. New average = 168 ÷ 7 = 24.

Q2. The average weight of 20 students is 45 kg. When the teacher's weight is included, the average rises by 1 kg. Find the teacher's weight.
New total = 21 × 46 = 966. Old total = 900. Teacher = 66 kg.
Quick view: the teacher brings 46 for themselves plus 1 for each of the 20 students: 46 + 20 = 66.

Q3. The average age of 10 workers rises by 2 years when a worker aged 30 is replaced by a new worker. Find the new worker's age.
The total rises by 10 × 2 = 20. New worker = 30 + 20 = 50.

Q4. A batter scores 87 runs in the 17th innings and so raises their average by 3. Find the new average.
Let the old average be a. Then 16a + 87 = 17(a + 3), so a = 87 − 51 = 36. The new average is 39.
Quick view: the 87 runs must cover the new average once and lift each of the 16 earlier innings by 3, so 87 = new average + 48, giving 39.

Q5. A class has 30 boys with an average score of 60 and 20 girls with an average of 70. Find the class average.
(30 × 60 + 20 × 70) ÷ 50 = 3,200 ÷ 50 = 64. Note it sits closer to 60, because there are more boys.

Q6. The average of 25 observations was 36. Later, one observation of 48 was found to have been recorded as 23. Find the correct average.
Correct total = 900 + (48 − 23) = 925. Correct average = 37.

Q7. The average of 11 results is 50. The average of the first 6 is 49 and of the last 6 is 52. Find the 6th result.
The 6th result is counted in both groups. 6 × 49 + 6 × 52 − 11 × 50 = 294 + 312 − 550 = 56.

Common mistakes

Practice

1. The average of 6 numbers is 18. If one number is excluded, the average becomes 16. Find the excluded number.
2. Find the average of all even numbers from 2 to 40.
3. The average age of 15 students is 14. When their teacher's age is included, the average becomes 15. Find the teacher's age.
4. A batter's average after 10 innings is 32. How many runs in the 11th innings will raise it to 34?
5. The average of 20 numbers was 50, but 65 was wrongly taken as 45. Find the correct average.
6. The average of five consecutive odd numbers is 27. Find the largest.
7. The average of A, B and C is 40. The average of A and B is 35, and of B and C is 42. Find B.
8. In a class, boys average 54 marks, girls average 44, and the class average is 50. Find the ratio of boys to girls.

Answers:

1. 28. 6 × 18 − 5 × 16 = 108 − 80.
2. 21. These are the first 20 even numbers, so the average is 20 + 1.
3. 30. 16 × 15 − 15 × 14 = 240 − 210.
4. 54. 11 × 34 − 10 × 32 = 374 − 320.
5. 51. The total rises by 20, so the average rises by 20 ÷ 20 = 1.
6. 31. The middle number is 27, so the list is 23, 25, 27, 29, 31.
7. 34. A + B + C = 120 and A + B = 70, so C = 50. Then B + C = 84 gives B = 34.
8. 3 : 2. By alligation, boys : girls = (50 − 44) : (54 − 50) = 6 : 4. Check: (3 × 54 + 2 × 44) ÷ 5 = 250 ÷ 5 = 50.

What to do next

• Redo every example above using only totals, without looking at the solutions.
• Practise the deviation method on five sets of marks or temperatures a day for a week.
• Solve 20 mixed average questions from previous CHSL papers at under 40 seconds each.
• Carry the weighted-average idea into mixtures and alligation, where it becomes a two-second ratio.