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('quantitative-techniques', 'Quantitative Techniques') 3% exam weight

Topic 3

Part of the GAT Pakistan study roadmap. ('quantitative-techniques', 'Quantitative Techniques') topic quanti-003 of ('quantitative-techniques', 'Quantitative Techniques').

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Average

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Average — Key Facts for GAT Pakistan

Basic Formula: $$\text{Average} = \frac{\text{Sum of all observations}}{\text{Number of observations}}$$

Shortcut for Equal Groups: $$\text{Average of Group} = \frac{\text{Sum of items in group}}{\text{Number of items}}$$

Simple Average Examples:

Example 1: Find average of 10, 20, 30, 40, 50
Sum = 10 + 20 + 30 + 40 + 50 = 150
Number = 5
Average = 150/5 = 30

Example 2: Average of first 10 natural numbers
Sum = n(n+1)/2 = 10 × 11/2 = 55
Average = 55/10 = 5.5

Example 3: Average of first n natural numbers = (n+1)/2
           (This is the middle number since they're equally spaced)

Common Average Values to Memorize:

NumbersAverage
1, 2, 3, …, n(n+1)/2
2, 4, 6, …, 2nn+1
1, 3, 5, …, (2n-1)n

⚡ GAT Exam Tip: For consecutive numbers, the average is always the middle number!

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Average — Detailed Study Guide

Weighted Average

Formula: $$\text{Weighted Average} = \frac{\sum (w_i \times x_i)}{\sum w_i}$$

Worked Examples:

Example 1: In a class, 20 students have average marks 70 and 30 students
have average marks 80. Find overall average.

Solution:
Sum of group 1 = 20 × 70 = 1400
Sum of group 2 = 30 × 80 = 2400
Total students = 50
Total sum = 1400 + 2400 = 3800
Overall average = 3800/50 = 76

Shortcut (using alligation concept):
Group 1: 20 students, avg 70
Group 2: 30 students, avg 80
Difference from overall: Group 1 gives 6, Group 2 gives 10
Ratio = 6:10 = 3:5
Overall average = (70 × 3 + 80 × 5)/8 = 410/8 = 51.25... 
Wait, that's not right either.

Let me recalculate properly:
Sum 1 = 20 × 70 = 1400
Sum 2 = 30 × 80 = 2400
Total = 1400 + 2400 = 3800
Average = 3800/50 = 76 ✓

⚡ Common Mistake: Don’t just average the averages! (70+80)/2 = 75 is WRONG because there are different numbers of students.

Average Speed

Average Speed Formula: $$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$$

For Two Equal Distances: $$v_{avg} = \frac{2v_1 v_2}{v_1 + v_2}$$

For Two Equal Times: $$v_{avg} = \frac{v_1 + v_2}{2}$$

Worked Examples:

Example 1: A car travels from A to B at 60 km/h and returns at 40 km/h.
Find average speed for the entire journey.

Solution (equal distances):
v₁ = 60, v₂ = 40
v_avg = (2 × 60 × 40)/(60 + 40) = 4800/100 = 48 km/h

Example 2: A person walks at 4 km/h for 2 hours and runs at 8 km/h for
3 hours. Find average speed.

Solution (equal times, but different):
Total distance = (4×2) + (8×3) = 8 + 24 = 32 km
Total time = 2 + 3 = 5 hours
Average speed = 32/5 = 6.4 km/h

Example 3: A man covers 1/3 distance at 30 km/h, 1/3 at 20 km/h,
and remaining 1/3 at 10 km/h. Find average speed.

Solution:
Let total distance = 3D
Time for first 1/3 = D/30
Time for second 1/3 = D/20
Time for third 1/3 = D/10
Total time = D/30 + D/20 + D/10 = (2D + 3D + 6D)/60 = 11D/60
Average speed = 3D / (11D/60) = 180/11 = 16.36 km/h

⚡ GAT PYQ: “A man covers half the distance at 12 km/h and the other half at 18 km/h. Find his average speed.” → Answer: 14.4 km/h

Missing Number Problems

Finding Missing Number:

Example: Average of 5 numbers is 20. If one number is removed and the
average becomes 18, what was the removed number?

Solution:
Sum of 5 numbers = 5 × 20 = 100
Sum of remaining 4 numbers = 4 × 18 = 72
Removed number = 100 - 72 = 28

Example: Average age of 8 people is 25 years. When a new person joins,
the average becomes 27 years. Find the age of the new person.

Solution:
Sum of 8 people = 8 × 25 = 200
Sum of 9 people = 9 × 27 = 243
New person age = 243 - 200 = 43 years

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Average — Complete Notes for GAT

Average with Replacement

Adding and Removing Items:

Example: Average weight of 15 students is 50 kg. When a new student joins,
the average becomes 51 kg. After 5 more students join (including the new one),
the average becomes 52 kg. Find weight of the new student.

Solution:
Total weight of 15 = 15 × 50 = 750 kg
After new student joins: 16 students, avg 51
Total = 16 × 51 = 816 kg
New student = 816 - 750 = 66 kg

After 5 more join (now 21 students), avg 52
Total = 21 × 52 = 1092 kg
Sum of additional 5 = 1092 - 816 = 276 kg
Average of 5 = 276/5 = 55.2 kg
So the remaining 4 students (not the new one) have average = (276 - 66)/4 = 52.5 kg

Temperature and Gradient Problems

Example: Average temperature on Monday, Tuesday, Wednesday is 35°C.
Average on Tuesday, Wednesday, Thursday is 37°C. Monday's temp is 32°C.
Find Thursday's temperature.

Solution:
M + T + W = 3 × 35 = 105
T + W + Th = 3 × 37 = 111
Subtracting: (T+W+Th) - (M+T+W) = 111 - 105
Th - M = 6
Th = 6 + M = 6 + 32 = 38°C

GAT-Style Practice Questions

1. The average of 20 numbers is 15. If each number is multiplied by 3,
   what is the new average?
   (a) 15 (b) 30 (c) 45 (d) 18

   Answer: (c) 45
   Solution: New average = Old average × Multiplication factor
             = 15 × 3 = 45

2. Average of first 20 even numbers is:
   (a) 20 (b) 21 (c) 40 (d) 42

   Answer: (b) 21
   Solution: First 20 even numbers: 2, 4, 6, ..., 40
             Sum = n(n+1) = 20 × 21 = 420
             Average = 420/20 = 21

3. In a class, average height of 30 boys is 160 cm and of 20 girls is
   150 cm. Find average height of all 50 students.
   (a) 155 cm (b) 156 cm (c) 154 cm (d) 157 cm

   Answer: (b) 156 cm
   Solution: Boys total = 30 × 160 = 4800
             Girls total = 20 × 150 = 3000
             Combined total = 7800
             Average = 7800/50 = 156 cm

4. The average of 10 numbers is 24. If one number 'a' is removed,
   average becomes 25. Find 'a'.
   (a) 10 (b) 15 (c) 20 (d) 25

   Answer: (b) 15
   Solution: Sum of 10 numbers = 240
             Sum of 9 numbers = 225
             Removed number a = 240 - 225 = 15

5. A man covers 200 km at 50 km/h and 200 km at 100 km/h.
   Find average speed.
   (a) 75 km/h (b) 66.67 km/h (c) 70 km/h (d) 80 km/h

   Answer: (b) 66.67 km/h
   Solution: Equal distances, so use formula:
             v_avg = 2v₁v₂/(v₁+v₂) = 2×50×100/(150) = 66.67 km/h

⚡ GAT Strategy: For average speed with equal distances, use 2v₁v₂/(v₁+v₂). For equal times, use (v₁+v₂)/2.

Mean Deviation and Range

Mean Deviation: $$\text{Mean Deviation from Mean} = \frac{\sum |x_i - \bar{x}|}{n}$$

Example: Find mean deviation of 3, 6, 9, 12, 15

Mean = (3+6+9+12+15)/5 = 45/5 = 9

|x_i - mean|: |3-9|=6, |6-9|=3, |9-9|=0, |12-9|=3, |15-9|=6
Sum of deviations = 18
Mean deviation = 18/5 = 3.6

Range: $$\text{Range} = \text{Maximum} - \text{Minimum}$$

Standard Deviation (brief): For grouped data or small samples: $$\sigma = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n}}$$

For population data: $$s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}$$

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