The average weight of 21 boys was recorded as $64$ kg. If the weight of the teacher was added, the average increased by $1$ kg. What was the teacher's weight?

### Concept & Strategy This is an **Average Inclusion** problem. When a new entity joins a group and raises the overall average, they are not only bringing the original average value but also enough extra value to distribute among all members (including themselves) to raise the average. $$ \text{New Value} = \text{Old Average} + [ \text{Total New Members} \times \text{Increase in Average} ] $$ ### Step-by-Step Solution * **Given:** * Initial number of boys = $21$ * Initial average weight = $64$ kg * New total members (boys + teacher) = $21 + 1 = 22$ * Increase in average = $1$ kg * New average weight = $64 + 1 = 65$ kg * **Calculation / Deduction:** Let's use the standard summation method first to see the underlying math. 1. Find the total weight of the $21$ boys: $\text{Total weight} = 21 \times 64 = 1344$ kg 2. Find the new total weight of all $22$ people: $\text{New total weight} = 22 \times 65 = 1430$ kg 3. Calculate the teacher's weight by finding the difference: $\text{Teacher's weight} = 1430 - 1344 = 86$ kg ### Exam Strategy & Shortcut Never multiply $21 \times 64$ in a competitive exam! Use the **Transfer of Value** logic. For the average to become $65$ kg, the teacher must bring at least $64$ kg (the old average) to avoid pulling it down. Then, the teacher must also provide an extra $1$ kg for every single person in the new group ($22$ people) to raise the average. Teacher's weight $= \text{Old Average} + (\text{New Group Size} \times \text{Increase})$ Teacher's weight $= 64 + (22 \times 1) = 64 + 22 = 86$ kg. You can do this in your head in $3$ seconds. ### Common Pitfall The most common trap is multiplying the increase ($1$ kg) by the *old* group size ($21$) instead of the *new* group size ($22$). Remember, the teacher's own weight in the new average is also $65$ kg, so they must supply that extra $1$ kg for themselves too! ### Final Answer Therefore, the correct answer is **86 kg**.