2.4 Weighted Average

Figure: A weighted average is a see-saw — it always leans toward the group with more members.

When groups of different sizes are combined, or values carry different “weights” (marks with credit hours, prices with quantities, ages with number of people), the simple average is invalid. Use:

Weighted Average = (w_1x_1 + w_2x_2 + … + w_nx_n)/(w_1 + w_2 + … + w_n)

where x_i are the values and w_i are their respective weights (group sizes, quantities, etc.)

Solved Example 10: In a class, 40 students scored an average of 60 marks and 60 students scored an average of 70 marks. Find the average marks of the whole class (100 students).

Weighted average = (40 × 60 + 60 × 70) / (40 + 60) = (2400 + 4200) / 100 = 6600/100 = 66.

Solved Example 11: A shopkeeper mixes 20 kg of rice at ₹40/kg with 30 kg of rice at ₹60/kg. Find the average price per kg of the mixture.

Weighted average = (20 × 40 + 30 × 60)/(20+30) = (800 + 1800)/50 = 2600/50 = ₹52/kg.

Solved Example 12: The average age of a group of 12 men is 30 years. A group of 8 women has an average age of 25 years. Find the average age of all 20 people combined.

Weighted average = (12×30 + 8×25)/20 = (360+200)/20 = 560/20 = 28 years.

Solved Example 12(a): In an exam, a student scored 60 marks in a subject worth 4 credit hours and 80 marks in a subject worth 6 credit hours. Find his weighted average score across the two subjects.

Weighted average = (4×60 + 6×80)/(4+6) = (240+480)/10 = 720/10 = 72.

Notice that this is structurally identical to Example 10 and Example 11 — whether the “weight” is number of students, quantity in kg, or credit hours, the formula does not change. Recognising this pattern across differently-worded questions is what separates fast solvers from slow ones in this topic.

Solved Example 12(b): Three sections of a college have 10, 15, and 25 students with average marks of 50, 60, and 70 respectively. Find the average marks of all 50 students, using the weighted average formula directly (this is the three-group extension of Example 10).

Weighted average = (10×50 + 15×60 + 25×70)/(10+15+25) = (500+900+1750)/50 = 3150/50 = 63.

Solved Example 12(c): The average score of a class of 40 students is 66. Some of these students, having an average score of 72, form one subgroup, while the remaining students have an average score of 60. If the overall class average is 66, find how many students are in the 72-average subgroup.

Let the number of students with average 72 be x; the rest, (40-x), have average 60. 72x + 60(40−x) = 66×40 72x + 2400 − 60x = 2640 12x = 240 x = 20 students. (This is the reverse of the usual weighted-average question — here the unknown is a weight (a headcount), not a value, which is exactly the kind of twist RRB NTPC likes to slip in.)