       # Shortcut of Average-2 Shortcut 20:

Weighted average $A=\frac{A_1N_1+A_2N_2+..+A_nN_n}{N_1+N_2+...+N_n}$

Question:
Average marks scored by students in three classes are 40, 50, and 60. If there are 10, 20 and 30 students in the classes respectively, find the overall average of the three classes.
A1 = 40; A2 = 50; A3 = 60
N1 = 10; N2 = 20; N3 = 30
Substitute in the above formula. We get,
=[(40×10) + (50×20) + (60×30)]/[10+20+30]
=[400 + 1000 + 1800]/60
= 53.33

Shortcut 21:

Finding the middle subject mark, if the middle subject overlaps. $M=A_1N_1+A_2N_2+..+A_0N_0$

Question:
Average marks scored by Sai in 11 subjects is 60. Average marks scored by her in first 6 subjects is 50 and average marks scored by her in last 6 subjects is 62. Find the mark scored by Sai in 6th subject.
Ao = 60; No = 11
A1 = 50; N1 = 6
A2 = 62; N2 = 6
Substituting the values in the formula,, we get
M = 50×6 + 62×6 – 60×11
= 300 + 372 – 660
= 12
Marks scored by Sai in 6th subject = 12

Shortcut 22:

Finding the middle subject mark if the middle subject is left out $M=A_0N_0-A_1N_1-A_2N_2$

Question:
The average marks scored by Shradha in 9 subjects is 75. The average marks in first 4 subjects is 69 and average marks in last 4 subjects is 78. Find the marks scored by her in 5th subject.
Ao = 75; No = 9
A1 = 69; N1 = 4
A2 = 78; N1 = 4
Substituting the above values in the formula, we get
M = 75×9 – 69×4 – 78×4
= 675 – 276 – 312
= 87
Marks scored by Sradha in 5th subject = 87

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