Nearly every average question becomes easy the moment you convert it to a total. "The average of 10 numbers is 35" is hard to manipulate; "the ten numbers add to 350" is not.
The second idea worth having is the deviation method: instead of recomputing an average, track how much the change moves the total, then divide by the count once at the end.
Try each one before opening the answer. A solution you have read is not a solution you can produce under time pressure, which is the only kind that counts.
Q1 The average age of 30 students is 14 years. When the teacher’s age is included the average rises to 15. How old is the teacher?
45.
Students total 30 × 14 = 420. With the teacher, 31 people average 15, so they total 465. The teacher is 465 − 420 = 45. The shortcut: the teacher must be the new average plus the number of students times the rise, 15 + 30 × 1 = 45.
Q2 A batsman has an average of 36 after 16 innings. How many must he score in the 17th to raise his average to 38?
70.
Sixteen innings total 576. To average 38 over 17 innings he needs 646. The difference is 70. By deviation: he needs the new average plus the rise applied to every earlier innings, 38 + 16 × 2 = 70.
Q3 A class of 20 boys averages 60 marks and 30 girls average 70. What is the class average?
66.
Totals, not averages: 20 × 60 = 1200 and 30 × 70 = 2100, so 3300 over 50 students, which is 66. Averaging 60 and 70 to get 65 is the error this question exists to catch - the groups are different sizes, so the answer leans towards the larger one.