Averages

Prerequisites

• Percentages — you will meet the “averaging percentages” trap again here.
• Ratio and Proportion — a weighted average is a ratio-weighted mean.

The Hook: The 11th Innings

Why it matters for CGPSC: average (औसत) questions are a dependable source of marks in Prelims Paper II, and they sit inside Data Interpretation sets (“average production over five years”). The reasoning-and-mathematics part of Mains Paper IV (GS-II) uses the same ideas with steps shown.

The One Formula Behind Everything

$$\boxed{\text{Average} = \frac{\text{Sum of observations}}{\text{Number of observations}}} \qquad\qquad \boxed{\text{Sum} = \text{Average} \times \text{Number}}$$

Nearly every average question is solved by converting averages to sums, doing the arithmetic on sums, and converting back.

The deviation (assumed-mean) method

Pick a convenient base value, average the deviations from it, and add back.

Averages of Number Patterns

For numbers in arithmetic progression (equally spaced), the average is the middle term, or the mean of the first and last:

$$\boxed{\text{Average of an AP} = \frac{\text{First term} + \text{Last term}}{2}}$$

Set	Average
First \( n \) natural numbers \( 1, 2, \dots, n \)	\( \frac{n+1}{2} \)
First \( n \) even numbers \( 2, 4, \dots, 2n \)	\( n + 1 \)
First \( n \) odd numbers \( 1, 3, \dots, 2n-1 \)	\( n \)
Squares of first \( n \) natural numbers	\( \frac{(n+1)(2n+1)}{6} \)
Consecutive numbers, odd count	The middle number

Weighted (Combined) Average

When groups of different sizes are combined, weight each average by its group size:

$$\boxed{\bar{x} = \frac{n_1 a_1 + n_2 a_2 + \cdots}{n_1 + n_2 + \cdots}}$$

The combined average always lies between the group averages, closer to the bigger group.

Adding, Removing and Replacing a Member

Let a group of \( n \) have average \( A \).

Situation	Shortcut
New member joins; average rises by \( d \)	New member \( = A + (n+1)d \) (or new average \( + nd \))
Member leaves; average rises by \( d \)	Leaving member \( = A - (n-1)d \)
One member replaced; average rises by \( d \)	New member \( = \) old member \( + nd \)

Correcting a Wrong Entry

If one value was recorded wrongly, the sum is off by (correct − wrong), so the average is off by that difference divided by \( n \):

$$\boxed{\text{Correct average} = \text{Old average} + \frac{\text{Correct value} - \text{Wrong value}}{n}}$$

Overlapping Groups

When two groups share a member (the “sixth result” or “Wednesday” questions), add the two group sums and subtract the overall sum: the overlap is counted twice.

Average Speed

Average speed is total distance ÷ total time — not the average of the speeds.

For two equal distances covered at speeds \( x \) and \( y \):

$$\boxed{\text{Average speed} = \frac{2xy}{x + y}}$$

For equal times at speeds \( x \) and \( y \), the average speed is simply \( \frac{x+y}{2} \).

Memory Tricks

Don’t Confuse

This	vs	That
Equal distances at 40 and 60 km/h → 48 km/h		Equal times at 40 and 60 km/h → 50 km/h
Combined average of unequal groups (weighted)		Simple mean of the two averages
Average of first \( n \) odd numbers \( = n \)		Average of first \( n \) even numbers \( = n+1 \)

Common Mistakes

Practice

Level 1: Foundation

1. Find the average of 12, 15, 18, 21 and 24.
2. Find the average of the first 10 natural numbers.
3. The average of 8 numbers is 45. Find their sum.
4. Find the average of the first 20 even numbers.

Answers

1. 18 — equally spaced, so the middle term.
2. \( \frac{10+1}{2} = \) 5.5.
3. \( 45 \times 8 = \) 360.
4. \( 20 + 1 = \) 21.

Level 2: Prelims MCQs

Q1. The average of the first 25 odd natural numbers is:

(a) 24 (b) 25 (c) 26 (d) 50

Answer: (b) 25

The average of the first \( n \) odd numbers is \( n \). Check: the numbers run from 1 to 49, and \( \frac{1 + 49}{2} = 25 \). Option (c) is the formula for even numbers (\( n + 1 \)).

Q2. A batter has a certain average after 15 innings. She scores 64 in the 16th innings, which raises her average to 34. By how many runs did the average increase?

(a) 1 (b) 2 (c) 3 (d) 4

Answer: (b) 2

New total \( = 34 \times 16 = 544 \). Old total \( = 544 - 64 = 480 \), old average \( = 480 \div 15 = 32 \). Increase \( = 34 - 32 = 2 \).

Shortcut: the 16th score exceeds the new average by \( 64 - 34 = 30 \); that excess is spread over the 15 earlier innings: \( 30 \div 15 = 2 \).

Q3. In a school, section A has 30 students with an average score of 60, and section B has 20 students with an average score of 70. The average score of all 50 students is:

(a) 62 (b) 64 (c) 65 (d) 66

Answer: (b) 64

Weighted average \( = \frac{30\times 60 + 20\times 70}{50} = \frac{3200}{50} = 64 \). Option (c), 65, is the unweighted mean of 60 and 70 — wrong because the sections are of unequal size.

Q4. A bus travels from Raipur to Bilaspur at 40 km/h and returns by the same route at 60 km/h. Consider the following statements:

1. The average speed for the round trip is 50 km/h.
2. The average speed for the round trip is 48 km/h.
3. The average speed for the round trip does not depend on the distance between the two cities.

Which of the statements given above is/are correct?

(a) 1 only (b) 2 only (c) 2 and 3 only (d) 1 and 3 only

Answer: (c) 2 and 3 only

For equal distances, average speed \( = \frac{2xy}{x+y} = \frac{2\times 40\times 60}{100} = 48 \) km/h, so statement 2 is correct and statement 1 is wrong. Since the formula contains no distance term, statement 3 is correct: for any one-way distance \( D \), total time \( = \frac{D}{40} + \frac{D}{60} = \frac{D}{24} \) and \( \frac{2D}{D/24} = 48 \).

Q5. The average of 50 observations was found to be 30. Later it was discovered that one observation, 82, had been wrongly recorded as 28. The correct average is:

(a) 28.92 (b) 30.54 (c) 31.08 (d) 31.80

Answer: (c) 31.08

The sum was short by \( 82 - 28 = 54 \). Correct average \( = 30 + \frac{54}{50} = 30 + 1.08 = 31.08 \). Option (a) applies the correction in the wrong direction (\( 30 - 1.08 \)); option (b) divides the error by 100 instead of by the 50 observations.

Level 3: Speed Drill — Solve in 60 Seconds

Prelims Paper II is a speed test; the reasoning-and-mathematics part (Part 2) of Mains Paper IV (GS-II) expects the same arithmetic with clear steps. Solve each in under 60 seconds with the shortcuts above.

1. The average weight of 8 people rises by 2.5 kg when a person weighing 65 kg is replaced. Weight of the newcomer?
2. The average of five consecutive even numbers is 36. The largest number?
3. The average age of 30 students is 14 years; including the teacher, it becomes 15. The teacher’s age?
4. The average of 11 results is 50; the first six average 49 and the last six average 52. The sixth result?
5. The average temperature of Monday–Wednesday is 37 °C and of Tuesday–Thursday is 34 °C. Thursday’s temperature is 31 °C. Monday’s temperature? (practice figures)

Answers with the shortcut used

1. 85 kg — total rises by \( 8 \times 2.5 = 20 \); \( 65 + 20 \).
2. 40 — the middle number is 36; go up two steps of 2.
3. 45 years — \( 15 \times 31 - 14 \times 30 = 465 - 420 \).
4. 56 — \( 294 + 312 - 550 \); the sixth result is counted twice.
5. 40 °C — Mon − Thu \( = 3 \times 37 - 3 \times 34 = 9 \); Mon \( = 31 + 9 \).

Quick Revision Table

Idea	Formula / Fact
Average	Sum ÷ Number; Sum = Average × Number
AP / consecutive numbers	\( \frac{\text{first} + \text{last}}{2} \)
First \( n \) natural / even / odd	\( \frac{n+1}{2} \) / \( n+1 \) / \( n \)
Weighted average	\( \frac{n_1 a_1 + n_2 a_2}{n_1 + n_2} \)
Replacement, average up by \( d \)	New \( = \) old \( + nd \)
Wrong entry	Old average \( + \frac{\text{correct} - \text{wrong}}{n} \)
Average speed, equal distances	\( \frac{2xy}{x+y} \)
Average speed, equal times	\( \frac{x+y}{2} \)

Key Takeaways

What’s Next

• Basic Numeracy — Quick Revision — percentages, ratios and averages on one page.
• Data Interpretation — averages across years and categories are a staple of DI tables.
• General Mental Ability — series and puzzles that lean on the same number sense.
• Logical Reasoning — the other scoring block of CSAT.
• Ratio and Proportion — revisit if weighted averages felt slow.