Averages & Mixtures • Topic 4 of 4

Change-in-Average Problems

These questions ask how the mean shifts when you add a value, remove one, or swap one out — and the fast way is to track the total, not to recompute the whole average. Adding a new value that is above the current average pulls the mean up; below it, down; equal to it, no change. The size of the shift is governed by how far the new value sits from the old mean, spread across the new count. A cleaner tool for "one value replaced by another" is to work with the change in sum: if a value goes up by 12 across a set of 6, the average rises by 12/6 = 2. The GRE’s classic version is the cricket-or-batting-average style: "a player averaging 40 over 10 innings scores 84 in the 11th — new average?" — total goes from 400 to 484 over 11 innings, giving 44. Always convert the average condition into a statement about the total first; the rest is one subtraction and one division.

✅ Solved examples

1. A student averages 72 over 4 tests. What must she score on a 5th test to raise the average to 75?
Needed total = 75 × 5 = 375. Current total = 72 × 4 = 288. Fifth score = 375 − 288 = 87.
2. The average of 10 numbers is 30. If one number, 18, is removed, what is the new average?
Total = 300; remove 18 → 282 over 9 numbers. New average = 282/9 = 31.33… ≈ 31.3.
3. A player averages 40 over 10 innings, then scores 84 in the 11th. New average?
Total = 400 + 84 = 484 over 11 innings ⇒ 484/11 = 44.
4. In a set of 6 numbers averaging 25, one value is increased by 30. By how much does the average rise?
The sum rises 30, spread over 6 numbers: 30/6 = 5, so the average rises by 5 (to 30).

✏️ Practice — try these, take hints as needed

1. A batsman averages 36 over 5 innings. He scores 60 in the 6th. New average?
Old total = 36 × 5 = 180.
New total = 180 + 60 = 240.
240/6.
40
2. The average of 8 numbers is 21. Removing one number raises the average of the rest to 22. What was the removed number?
Old total = 168.
New total = 22 × 7 = 154.
Removed = 168 − 154.
14
3. Six people average 28 kg. A 7th person joins and the average becomes 30 kg. Find the new person’s weight.
Old total = 168.
New total = 30 × 7 = 210.
210 − 168.
42 kg
4. A student needs a 90 average over 5 quizzes. After four she has 88, 92, 85 and 91. What must the fifth be?
Target total = 90 × 5 = 450.
Four sum to 356.
450 − 356.
94
5. In a class the average of 20 scores is 68. A score of 40 was wrongly entered as 80. Corrected average?
The sum is 40 too high.
Correct sum = 68·20 − 40.
(1360 − 40)/20.
66

📝 Topic test — 8 questions

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