Average is one of the important topics of arithmetic and the questions from this area are asked in CAT almost every year. In this article we will discuss about averages in details.
Average means balancing of the terms. It means that when we have certain numbers with us and we want to make them all equal without changing their sum, then that equal number will be the average of all the numbers.
Mathematically, Average = Sum of terms/ Number of terms
Or, we can have Sum of terms = Average × Number of terms.
The above result is applicable in all the questions of average.
Ex. Find the average of 12, 15, 21, 28, 34.
Sol: Here, the average of the numbers = Sum of terms/ Number of terms = (12 + 15 + 21 + 28 + 34)/5 = 110/5 = 22
Ex. The average weight of 15 men in a group is increased by 1 kg when one of the men who weighs 60 kg is replaced by a new man. Find the weight of new man.
Sol: We know that, Sum of terms = Average × Number of terms. Now, this formula can be used to get the answer.
Let the average weight of the original 15 men is x, so the average weight of the final 15 men will be x + 1. Let the weight of the new man is y kg
Using the above result, we have, 15x – 60 + y = 15 (x + 1)
⇒ 15x – 60 + y = 15x + 15 ⇒ y = 75 kg.
Note: Following points should be kept in mind while solving the questions of average.
I. If the difference in the terms is same and the total number of terms are odd (say ‘n’), then the average will be the middle term, i.e., (n +1)/2th term.
E.g., If we have the numbers 4, 7, 10, 13, 16, 19, 22 then we can see that the difference in these numbers is same. In this case we can directly say that the average will be the middle term. Here, number of terms are 7. So, n = 7. Hence, the middle term is (7 + 1)/2 = 4th term which is 13. So, the average of the given numbers is 13.
II. If the difference in the terms is same and the total number of terms are even (say ‘n’), then there will be two middle terms n/2th term and (n/2 + 1)th term and the average of all the numbers will be equal to the average of these two middle terms.
E.g., If we have the numbers 4, 7, 10, 13, 16, 19, 22, 25, then again, the difference in the terms is same and there are two middle terms here, i.e., 13 and 16. So, the average of all the terms is equal to the average of 13 and 16 which is 14.5.
Note: In the above 2 pointers, you can see that the terms are in AP and hence the average can be found as (First Term + Last term)/2.
Ex. Consider a sequence of seven consecutive integers. The average of the first five integers is n. Find the average of all the seven integers.
1. n 2. n + 1 3. n + (2/7) 4. (n + 2)/7
Sol: Here, we have consecutive integers, which means the average will be the middle term. Now the middle term of the first five integers will be the 3rd term and it is the average which is given to be equal to ‘n’. So, the 4th term will be n + 1. Now the middle term of all the 7 integers will be the 4th term and it will be the required average. Hence, the average of all the seven integers is n + 1.
III. If same number is added to each of the given numbers, then the average also increases by the same amount.
E.g., Average of 10, 20, 30, 40 and 50 is 30. Now If we add same number say 5 to every number, then the numbers will become 15, 25, 35, 45 and 55 and their average will be 35 which is 5 more than the original average.
IV. If the same number is subtracted from the given numbers or same number is multiplied to each of the given numbers, then the average also decreases/multiplies by the same amount. The same happens with the division also.
Let us now consider what happens to the average of the given numbers if a number is increased or decreased by some amount.
Let us take an example where there are five children A, B, C, D and E having 10, 20, 30, 40 and 50 marbles respectively. The average number of marbles will be 30 with them. Now, what does it mean? As discussed at the start of this article that average simply means balancing of the terms which means that if we want to equate the number of marbles with these children then everyone will get 30 marbles. Now how will that happen? Here, C has his quota of 30 marbles. B has 10 marbles less than the average and D has 10 marbles more than the average. So, D will give his additional 10 marbles to B and both will have 30 marbles each. Similarly, E will give his additional 20 marbles to A and both will have 30 marbles each. Now, consider that say A got 20 more marbles from his parents. So, what will he do with these additional 20 marbles. Obviously, as they have earlier distributed the marbles equally, so he will divide these additional 20 marbles with the other children. So, everyone will get 20/5 = 4 additional marbles. Therefore, the number of marbles with each of them will be 34 and so as the average. Now, what do we learnt from here?
“If a number is increased by some amount, then that increase will be distributed on all the given numbers equally and whatever is received by each number, the average also increases by the same amount.”
You can see that the same happened in the above example also, where the additional 20 marbles obtained by A got distributed equally among all the five children and the average also increased by 4.
Now, what will happen If say E lost his 20 marbles? In this case the loss of E will be shared by all the five children. So, 20/5 = 4 is the number of marbles that will be given to E by A, B, C and D so that everyone has 26 marbles now and that is also the average numbers of marbles with them.
E.g., Let us consider the numbers 13, 16, 20, 21, 35. The average of these numbers is 21. Now if we replace the number 13 with the numbers say 18, then what will be the average?
Well, the answer is very easy. Replacing the number 13 by 18 means that 13 is increased by 5. Now, this increase will be equally divided among all the 5 numbers. So, every number will get 5/5 = 1, so the average will also increase by 1. Therefore, the new average will be 21 + 1 = 22.
Hence, we can say that “If a number is decreased by some amount, then that decrease will be distributed on all the given numbers equally and whatever is reduced from each number, the average also decreases by the same amount.”
The above discussed two points are useful when we do the questions without making the equations and that really saves the time in the competitive exams like CAT, XAT etc.
In the next part, we will discuss about the change in average if a number is deleted from the group of numbers.
Well, let us again take the same example as above where the children A, B, C, D and E have marbles with them and the average marbles with them is 30. Now, what will happen if say A leaves the group. Will the average increase or decrease?
Remember that as per the concept of average, each one of them has 30 marbles. Now when A will leave the group, then he will return the 20 marbles back to E. So, B, C, D and E will have 20 marbles as surplus with them and they will divide them equally. So, each will get additional 20/4 = 5 marbles and total marbles with them will be 35 each and average will also become 35. So, in this case, the average increased.
Let us now consider another case where E leaves the group. In this case, A will lose the 20 marbles he got from E. Now this loss of A will be shared by A, B, C and D equally. So, everyone will share 20/4 = 5 marbles and they will have 25 marbles each. In this case the average will decrease to 25.
It means that if a number smaller than the average is deleted, then the average will increase and if a number greater than average is deleted then the average will decrease. This increase or decrease in the average is equal to the difference of the deleted number from the average divided by the remaining numbers.
E.g., Let us consider the numbers 13, 16, 20, 21, 35. The average of these numbers is 21. Now if we delete the number 13, then what will be the new average?
Well here the difference of 13 from the average is 8. So, average will increase by 8/4 = 2. The new average will be 21 + 2 = 23.
Weighted Average: The weighted average is the average where the numbers in the data have different degree of importance. While calculating the weighted average, each number is multiplied by the weight assigned to it and then all the values are added.
Let there are n values x1, x2, x3, …….., xn having the weights w1, w2, w3,………, wn respectively.
The weighted average = (w1x1 + w2x2 + w3x3 + ………… + wnxn)/(w1 + w2 + w3 + ….wn)
Ex. A test was given to three sections A, B and C of class 9. The average scores of the sections A, B and C are 30, 35 and 40 respectively. The number of students in sections A, B and C are 20, 25 and 30 respectively. Find the average score of the class 9.
Sol: Here the number of students in each section is different, so the formula of weighted average will be used to find the average of the class.
Weighted average = (30 × 20 + 35 × 25 + 40 × 30)/(20 + 25 + 30) = 2675/75 = 35.67. Therefore, the average of the class is 35.67.
Let us now discuss few examples:
1. The average of Rahul after 50 one day matches is 70. If Rahul’s target is to increase his average to 72 after the next one-day match how much should Rahul score in the next match.
1. 165 2. 172 3. 160 4. 176
Sol: Total score of Rahul after 50 matches = 50 × 70 = 3500
If his average after 51 matches is 72, then the total will be = 51 × 72 = 3672
Hence, Rahul should score 3672 – 3500 = 172 runs in the next match.
2. From a group of 16 students with average weight 42.5 kg, two students leave the group and because of that the average of the group decreases to 41.25 kg. Find the average weight of the two students who left the group.
1. 51.25 2. 51.75 3. 50.5 4. 52.25
Sol: Total weight of 16 students = 16 × 42.5 = 680 kg
Total weight of the remaining 14 students = 14 × 41.25 = 577.5
Therefore, the total weight of the two students who left the group = 680 – 577.5 = 102.5 kg
Hence, the average weight of the two students = 102.5/2 = 51.25.
3. If a student weighing 70 kg joins a group of n students, the average weight of the group increases by 1 kg. If the new student weighed 55 kg, the average weight of the group would have declined by 2 kg. Find n.
1. 3 2. 4 3. 5 4. 6
Sol: Let the average weight of the group is x kg
Now, in first case nx + 70 = (x + 1) × (n + 1)
⇒ nx + 70 = nx + n + x + 1
⇒ n + x = 69 ……..(i)
In the second case, nx + 55 = (x – 2) (n + 1)
⇒ nx + 55 = nx + x – 2n – 2
⇒ x – 2n = 57 …..(ii)
Solving (i) and (ii), we get 3n = 12 or n = 4.
4. The average of 13 consecutive numbers is 324. If there is another series of consecutive numbers which starts with the same number as the previous series, then what is the average of first 20 numbers of this series.
1. 327 2. 326.5 3. 327.5 4. 328.5
Sol: Since the numbers are consecutive the average would be the same as the middle term. The middle term of 13 numbers is the 7th number which is 324. Since the next series start with the same number, the 10th number is 327 And 11th number is 328. So, the average of 20 consecutive numbers is the average of 10th and 11th number i.e., average of 20 numbers is equal to (327 + 328)/2 = 327.5.
5. The average age of 20 women is 49 years. The age of the youngest woman is 32 years and that of the oldest woman is 56 years. If two women with ages 45 and 50 years leave the group and three women join the group, the average remains unchanged. What is the average age (in years) of the three women who join the group later?
1. 48 2. 32 3. 49 4. 45
Sol: Total age of the group of 20 women = 49 × 20 = 980 years.
Total age of the women who left the group = 45 + 50 = 95 years.
Let, the average age of 3 women who joined later be x years.
Hence, the sum of their ages = 3x.
Since the average age after the leaving and joining of women is still 49 years, therefore, 980 + 3x – 95 = 49 × 21
⇒ 3x + 885 = 1029 ⇒ 3x = 144 ⇒ x = 48 years.
6. A class consists of 20 boys and 30 girls. In the mid-semester examination, the average score of the girls was 5 higher than that of the boys. In the final exam, however, the average score of the girls dropped by 3 while the average score of the entire class increased by 2. The increase in the average score of the boys is:
1. 9.5 2. 10 3. 4.5 4. 6
Sol: The average score of 30 girls decreased by 3, so, the total score of the girls decreased by 30 × 3 = 90. The average of the class still increased by 2, i.e., the total score of the class increased by 50 × 2 = 100 and this increase happened in spite of the decrease in the marks of the girls. It means that all the increase happened because of the marks of the boys and they have compensated for the decrease in marks of the girls also. Hence, the marks of boys must have been increased by 90 + 100 = 190 and their average increased by 190/20 = 9.5.
7. In an apartment complex, the number of people aged 51 years and above is 30 and there are at most 39 people whose ages are below 51 years. The average age of all the people in the apartment complex is 38 years. What is the largest possible average age, in years, of the people whose ages are below 51 years?
1. 26 2. 27 3. 28 4. 25
Sol: Let there are ‘n’ people below the age of 51 years.
The total age of all the people = (30 + n) × 38 years.
Now, the average age of the people below the age of 51 years will be maximum if the average age of the people above 51 years of age is minimum i.e., their average age is 51 years.
Hence, the total of the age of people below 51 years = (30 + n) × 38 – 51 × 30 = 38n – 390
Average age of the people below 51 years of age = (38n – 390)/n = 38 – 390/n
This average will be maximum if 390/n is minimum i.e., n is maximum and maximum value of n = 39
Hence, the maximum average age = 38 – 390/39 = 28 years.