Average

 
Hello friends...

Today we are not going much details in the topic of average.

As we all know standard definition of average, & which is simple to know what is average but I’m not using that definition as this leads to very confusion when there is matter of generation of first expression.

We all know, for solving numerical ability examples in banking exams, we have to construct first expression like if twice of no.is 44 then what is that no ?

So, let no.be x, then first expression will be 2x = 44, after solving this first expression we get x = 22, but as I have already discussed this thing in previous topics that constructing this first expression is very time consuming & if you make any fault in this expression then obviously your answer will be wrong. 

Most of the people do not take care of this important factor & that’s why they says “I could not complete in time”.

I always deliberately pay much attention on such topics & that’s why I changed definition of average little bit. Don’t worry you will easily grasp this definition as you don’t have to remember this at all. This definition is just so obvious that you would forget standard definition of average. I will tell you it with following examples. Just empty your mind, & concentrate on concept. As there are no tricks in maths, If there any, then that must be another application of math itself.

Suppose, you, he & I have 1, 2, 3 bats.

Now we all have 6 bats but I can’t carry 3 bats to playground far away from our home, since my bag small & I’m too small. Then what we will do, we will give my bat to one who has less bat than me to carry.

Everybody must be clear up to this…as this is general practice, if I can’t afford so much weight( of 3 bats) then I have to reduce my bats. But is mandatory to take 6 bats to ground & this one bat I will have to give another buddy. But whom?

Now this will start Application of Average. Average will tell us to whom I should hand over my one bat.

Let’s construct our questions again

1.    I have some extra bats that I can’t carry so I will have to avoid those extra bats & 

2.    To whom I should hand it over, either you (carrying 1 bat) or him( carrying 2 bats)

So we will start average here,

According to average 1+2+3/3 is average that is 2. This will tell me that each one has 2 bats. & problem end…

How problem ends?

1.    you have 1 bat, so take 1 bat more from anybody & start walking

2.    he has 2 bats already so you & he will walk immediately

3.    & I have 2 bats now, you know who took his bat, so I walk to play ground

“One last thing, from here after forget who had how many bats, from now everybody has 2 bats.”…..this quoted this is the conclusion so you have to remember that

“Average is what everybody have (for banking examples only)”

Suppose, A,B,C has 12, 30, 15 pens then everybody has 12/3+30/3+15/3 = 4 + 10 + 5 = 19 pens. (I deliberately didn’t calculate sum)

Important Formulas - Average
1. Average = $\frac{\text{Sum of observations}}{\text{/Number of observations}}$

2. Average Speed :
If a car covers a certain distance at X kmph and an equal distance at Y kmph. Then, the average speed of the whole journey =2/{ 1/X + 1 / Y} = 2XY$\frac{}{\mathrm{/\; X+Y}}$ kmph

Now go to examples…

1.   There were 4 ppl in room having average age 44. Then one left & new average age is again 44 then age of man left is ?

==> Simple, All 4 men before have age = 44 ( see explanation above)

then one left & average remains same, means all remaining 3 has age = 44

As

People	Before	After
1	44	44
2
3
4		You can tell easily how much it should be

Obviously if we add 44 to 44( of after’s column) then we will get 44 as new average.

So ans is 44.

Note : We are not sure about values of remaining 3 ppls age but we are here sure about the age of man left.

2.   The average of 25 result is 18. The average of 1st 12 of them is 14 & that of last 12 is 17. Find the 13th result.

==> Clearly 13th result=(sum of 25 results)-(sum of 24 results)

=(18*25)-(14*12)+(17*12)

=450-(168+204)

=450-372

=78. ( Sorry I forgot source, google it & mention it in comment)

Wow, such a simple example, but If you didn’t get it!, forget it. This approach fortunately not using our technique.

So Out technique is

Numbers	statement 1st	statement 2nd	statement 3rd
First 12	All are 18 (I don’t know anything)	14	14(we are sure, from stmt 2)
13th		18	??? We were assumed it	18 + 12 * 5
Last 12		Really don’t know yet(K)	17

For above (k)

where I have written really don’t know (k), I was little wrong since all have average 18 then last 12 must be such that all will become 18

That no.is 22, So I know average of last 12 is 22, but till the 2^nd statement in example only. In 3^rd statement they has said “last 12 average is 17”, ohhh 5 less than our answer, so these last no.s are 17 not 22( See table above, column 3)

Now there is only place to make correction is 13^th no. So we make changes in that to adjust our average. Ohh this previous statement was damn important. Since average is adjustments, later examples will teach you that, So let’s adjust 18

As our last 12s average changes from 22 to 17 we need to add those numbers to 18 so that adjustment will success.

Note:

1. If 18 would have average, we would have simply added 5(= 22-17) to it to 18 + 5 = 22,

2. But 18 is number, that is concrete (statistical) reading so we will have to add 5 X 12, that means, we took 5 from all 12 numbers of last 12s

So ans is 18 + 12 * 5 = 78

Hope if I can calculate it in my mind , you can too…

One more think, we have digged too much the concept of average so that our study won’t find fake to anybody & one last thing this is not trick

3.      A batsman makes a score of 87 runs in the 17th inning and thus increases his average by 3 Find his average after 17th inning.

Ans.==>   Till 16^th inning batsman has some score & some average. Forget score just consider average. That average was increased by 3 in 17^th inning. Meaning

Innings Span	Average	Score
till 16th (not only 16th )	x	some score ?
17th	x+ 3	87

Obviously   87 - 16*3 is the score till 16^th innings & it is 87-48 = 39

Adjustment : as we have increased our average by 3 we subtract 16*3 from new reading which is 87, its average was considered by us was x (but we didn’t use it) 

Q. 4  : There were 35 students in a hostel. Due to the admission of 7 new students the expenses for the mess were increased by 42/- per day while the average expenditure per head diminished by Re 1 .what was the original expenditure of the mess.

Ans : Let the initial average expenditure per head be x,
Then the initial total expenditure= 35x

After the admission of 7 new students, number of students will become 42 

average expenditure per head = (x-1)
New total expenditure = 42(x-1)

Given that Expenses for the mess were increased by 42
42(x-1) - 35x = 42
7x = 84
x = 12

Original expenditure = 35x = Rs.420

Q. 5 : The average of 20 numbers is zero. Of them, How many of them may be greater than zero, at the most?
A. 1	B. 20
C. 0	D. 19

Explanation:
Average of 20 numbers = 0

=> $\frac{\text{Sum of 20 numbers}}{}$
$\frac{\mathrm{-------------------------\; =\; 0}}{}$
$\frac{20}{}$

=> Sum of 20 numbers = 0

Hence at the most, there can be 19 positive numbers.
(Such that if the sum of these 19 positive numbers is x, 20th number will be -x)

Other Solved Examples : 
1. The mean weight of a group of seven boys is 56 kg. The individual weights (in kg) of six of them are 52, 57, 55, 60, 59 and 55. Find the weight of the seventh boy.
Solution:
Mean weight of 7 boys = 56 kg.
Total weight of 7 boys = (56 × 7) kg = 392 kg.
Total weight of 6 boys = (52 + 57 + 55 + 60 + 59 + 55) kg
= 338 kg.
Weight of the 7th boy = (total weight of 7 boys) - (total weight of 6 boys)
= (392 - 338) kg
= 54 kg.
Hence, the weight of the seventh boy is 54 kg. 




2. A cricketer has a mean score of 58 runs in nine innings. Find out how many runs are to be scored by him in the tenth innings to raise the mean score to 61.

Solution:
Mean score of 9 innings = 58 runs.
Total score of 9 innings = (58 x 9) runs = 522 runs.
Required mean score of 10 innings = 61 runs.
Required total score of 10 innings = (61 x 10) runs = 610 runs.
Number of runs to be scored in the 10th innings
= (total score of 10 innings) - (total score of 9 innings)
= (610 -522) = 88.
Hence, the number of runs to be scored in the 10th innings = 88.




3. The mean of five numbers is 28. If one of the numbers is excluded, the mean gets reduced by 2. Find the excluded number.
Solution:
Mean of 5 numbers = 28.
Sum of these 5 numbers = (28 x 5) = 140.
Mean of the remaining 4 numbers = (28 - 2) =26.
Sum of these remaining 4 numbers = (26 × 4) = 104.
Excluded number
= (sum of the given 5 numbers) - (sum of the remaining 4 numbers)
= (140 - 104)
= 36.
Hence, the excluded number is 36.





4. The mean weight of a class of 35 students is 45 kg. If the weight of the teacher be included, the mean weight increases by 500 g. Find the weight of the teacher.
Solution:
Mean weight of 35 students = 45 kg.
Total weight of 35 students = (45 × 35) kg = 1575 kg.
Mean weight of 35 students and the teacher (45 + 0.5) kg = 45.5 kg.
Total weight of 35 students and the teacher = (45.5 × 36) kg = 1638 kg.
Weight of the teacher = (1638 - 1575) kg = 63 kg.
Hence, the weight of the teacher is 63 kg.



5. The average height of 30 boys was calculated to be 150 cm. It was detected later that one value of 165 cm was wrongly copied as 135 cm for the computation of the mean. Find the correct mean.
Solution:
Calculated average height of 30 boys = 150 cm.
Incorrect sum of the heights of 30 boys
= (150 × 30)cm
= 4500 cm.
Correct sum of the heights of 30 boys
= (incorrect sum) - (wrongly copied item) + (actual item)
= (4500 - 135 + 165) cm
= 4530 cm.
Correct mean = correct sum/number of boys
= (4530/30) cm
= 151 cm.
Hence, the correct mean height is 151 cm.




6. The mean of 16 items was found to be 30. On rechecking, it was found that two items were wrongly taken as 22 and 18 instead of 32 and 28 respectively. Find the correct mean.
Solution:
Calculated mean of 16 items = 30.
Incorrect sum of these 16 items = (30 × 16) = 480.
Correct sum of these 16 items
= (incorrect sum) - (sum of incorrect items) + (sum of actual items)
= [480 - (22 + 18) + (32 + 28)]
= 500.
Therefore, correct mean = 500/16 = 31.25.
Hence, the correct mean is 31.25.




7. The mean of 25 observations is 36. If the mean of the first observations is 32 and that of the last 13 observations is 39, find the 13th observation.
Solution:
Mean of the first 13 observations = 32.
Sum of the first 13 observations = (32 × 13) = 416.
Mean of the last 13 observations = 39.
Sum of the last 13 observations = (39 × 13) = 507.
Mean of 25 observations = 36.
Sum of all the 25 observations = (36 × 25) = 900.
Therefore, the 13th observation = (416 + 507 - 900) = 23.
Hence, the 13th observation is 23.




8. The aggregate monthly expenditure of a family was $ 6240 during the first 3 months, $ 6780 during the next 4 months and $ 7236 during the last 5 months of a year. If the total saving during the year is $ 7080, find the average monthly income of the family.
Solution:
Total expenditure during the year
= $[6240 × 3 + 6780 × 4 + 7236 × 5]
= $ [18720 + 27120 + 36180]
= $ 82020.
Total income during the year = $ (82020 + 7080) = $ 89100.
Average monthly income = (89100/12) = $7425.
Hence, the average monthly income of the family is $ 7425.