Here’s the whole story visually:

The rest is details.Most everyone in math knows these tricks — here’s a summary of techniques used to make quick * mental* calculations of averages.

** ^{†}** In the problems below, consider the residual,

R is defined as the remainder above or below the guessed average, aka, the residual.

RS is defined as the residual sum.

- In school, I have 8 exam scores; what’s my average?

90

92

88

95

97

81

85

100

~~~

**STEP by STEP (with explanation)**

, then mentally calculate the running residual**Guess maybe 90**sum using^{†}:**two or three numbers at a time**

say(for 90: 90 is neither above or below the guessed average ∴R=0; ∴RS = 0),**zero**

say(for 92 and 88: the average of 92 and 88 is neither above or below the guessed average ∴R=0; ∴RS = 0),**zero**

say(for 95, 97, and 81: ∴R = 5 + 7 – 9; ∴RS = 3),**3**

say(for 85 and 100: ∴R= -5 + 10 = 5; ∴RS = 8).**8**

Answer: The average is 90 + 8/8 = 91.

- Same problem — different guess. In school, I have 8 exam scores; what’s my average?

90

92

88

95

97

81

85

100

~~~

**STEP by STEP (with explanation)**

, then mentally calculate the running residual**Guess maybe 85**sum using^{†}:**two or three numbers at a time**

say(for 90, 92, 88: ∴R = 5 + 7 + 3 = 15; ∴RS = 15),**15**

say(for 95 and 97: ∴R = 10 + 12 = 22; ∴RS = 37),**37**

say(for 81, 85, and 100: ∴R = -4 + 0 + 15 = 11; ∴RS = 48),**48**

Answer: The average is 85 + 48/8 = 91.

- Find average of 5 ‘hard numbers’ (with a lot of 9s, 8s, and 7s):

987

965

882

949

767

~~~

**STEP by STEP (with explanation)**

, mentally calculate the running residual**Assume a basis of 1000**sum:^{†}

say(for 987: ∴R = -13; ∴RS = 13 down),**13**

say(for 965: ∴R = -35; ∴RS = 48 down),**48**

say(for 949: ∴R = -51; ∴RS = 99 down),**99**

say(for 882: ∴R = -118; ∴RS = 217 down),**217**

say(for 767: ∴R = -233; ∴RS = 450 down),**450**

Answer: The average is 1000 – 450/5 = 910.

- Incrementing an average, for example, I have 8 exam scores:

90

92

88

95

97

81

85

100

with an average of 91 (see Problem 1). Now I have a score of 78; what is my new average?**without re-summing**

Answer: It is simply 91 + (78 – 91)/9 = 91 – 13/9 = 89 5/9 = 89.55

Formula: **[Powerful Formula]**Incrementing an average, for example, I have 100 exam scores with an average of 87. Now I have a score of 100; what is my new average?**without re-summing**

Answer: It is simply 87 + (100 – 87)/101 = 87.1287

- Incrementing an average, for example, I have 10,000 exam scores with an average of 87. Now I have a score of 100; what is my new average
?**without re-summing**

Answer: It is simply 87 + (100 – 87)/10001 = 87.0013 **[Very Powerful Formula]**Incrementing an average, for example, I have 100 exam scores with an average of 87. Now I have 3 scores of {85, 80, 75}; what is my new average?**without re-summing**

87 – 85 = 2

87 – 80 = 7

87 – 75 = 12

Answer: It is simply 87 – (2 + 7 + 12)/103 = 86.7961

- Determine the lowest allowable score to maintain a certain average. Say I have 8 scores with an average of 94. What is the least score on my 9th exam to maintain at least an average of 90?

Answer: It is 94 + (90 – 94) · 9 = 58.

Formula: **[Very Very Powerful Formula]**Determine the lowest allowablescore to maintain a certain average. Say I have 8 scores with an average of 94. What is the least score on my final exam to maintain at least a course grade of 90? Assume that the final exam is 40% of my course grade.**final exam**

Answer: It is [90 – (94 x 0.6)]/0.4 = 84.

Formula:**[Wicked Powerful Formula]**Alternative and easier formula:- So, how much does a 100 raise your average?

Formula:

Enjoy!