**Examples**

**1.** A batsman has a certain average runs for 16 innings. In the 17th inning he made a score of 85 runs thereby his average is increased by 3. What is his average after 17th inning?Examples

**Sol:** The average for 17th inning has been increased by 3.

The total increase in the runs for 17th inning = 17 × 3

= 51

But the batsman scores 85.

Average runs in his 16th innings = 85 – 51

= 34.

Hence the average of runs after 17th innings = 34 + 3

= 37

**2.** A man has 7 children. When their average age was 12 years, the child who was 6 years of age, died. What was the average age of surviving children 5 years after the death of the above child?**Sol:** Average age of 7 children = 12 years

Total age of 6 children = 12 × 7 = 84 years

Total age of 6 children after the death of a child aged 6 years = 84 – 6 = 78

Hence the average age of the surviving children = _{}

After 5 yrs. = 13 + 5 = 18 yrs.

**3. ** If the weights of 5 students of a class are 49.6 kg, 39.8 kg, 45.2 kg and 24.6 kg respectively then what is their average weight?

Sol: Total weight of 5 students = 49.6 + 39.8 + 40.8 + 45.2 + 24.6

= 200 kg.

∴ Their average weight = _{}

= 40 kg.

**4. ** The average temperature for Monday, Tuesday and Wednesday was 36^{o}C. The average temperature for Tuesday, Wednesday and Thursday was. If the temperature for Thursday was 37^{o}C, what was the temperature on Monday?**Sol: ** Average temperature for Monday, Tuesday and Wednesday = 36^{o}C

Total temperature for Monday, Tuesday and Wednesday = 36 × 3

= 108^{o}C

∴ Average temperature for Tuesday, Wednesday and Thursday = 38^{o}C

∴ Total temperature for Tuesday, Wednesday and Thursday = 38 × 3

= 114^{o}C

∴ Total temperature for Tuesday and Wednesday only = 114 – 37

= 77^{o}C

∴ Temperature for Monday only = 108 – 77

= 31^{o}C

**5.** A train covers the first 16 km at a speed of 20 km per hour another 20 km at 40 km per hour and the last 10 km at 15 km per hour. Find the average speed for the entire journey.

**6. ** A vehicle travels from A to B at the speed of 40 km/hr, but from B to A at the speed of 60km/hr. what is its average speed during the whole journey?**Sol:** Let the distance from A to B be x km

**7. ** The average age of a class of 40 boys is 16.95 years. A new boy joins the class and the average age now is 17 years. What is the age of the new boy?**Sol: **The average age of 40 boys = 16.95 years

Total are of 40 boys = 16.95 × 40

= 678 years

The average age of 41 boys = 17 years

Total age of 41 boys = 17 × 41

= 697 years

Age of the new boy = 697 – 678

= 19 years

The average of the given quantities is calculated by dividing the sum of the quantities by their number.

If the average of ‘x’ quantities is ‘a’ and average of ‘y’ quantities is ‘b’, then average of x and y is

Weighted Average

Let x_{1}, x_{2}, x_{3}, …..x_{n} be the quantities and w_{1}, w_{2}, w_{3}, ….wn be the weights attached to them then Weighted Average

** SOLVED EXAMPLES**

**Example 1. **A cricketer has completed 18 innings and his average is 26.5 runs. How many runs must he make in his next innings so as to raise his average to 27?

(1) 36 (2)46 (3)38 (4) 40 (5) None of these**Explanation:**

(1) Formula = n(y – x) + y

n = 18

x = 265

y = 27

= 18(27 – 26.5) + 27

= 9 + 7

= 36

**Example 2. **The average of 13 numbers is 30. The average of 1st 7 of these numbers is 32 and the last 7 of these numbers is 22. Find the 7th number.

(1) 15 (2) 16 (3) 12 (4) 20 (5) None of these**Explanation:**

(3) Total number = 13 × 30 = 390

1st 7 numbers = 7 × 32 = 224

Last 7 number = 7 × 22 = 154

7th number = 390 – (224 + 154)

= 12

**Example 3. **The average of marks obtained by 65 candidates in a certain examination is 25. If the average marks of passed candidates is 27 and that of the failed candidates is 14, what is the no. of candidates who passed the examination?

(1) 53 (2) 64 (3) 55 (4) 70 (5) None of these**Explanation:**

(3) Passed candidates = x

Failed candidates = 65 – x

Total marks ∴ 25 × 65 = 27x + (65 – x)14

⇒ 1625 = 27x + 910 – 14x

⇒ 13x = 715

⇒ x = 55

**Example 4. **In a class there are 30 students whose average is decreased by6 months, when 4 students aged 16, 17, 18 and 19 years respectively are replaced by the some no. of students. Find the average of the new students.

(1) 16.25 (2) 15.25 (3) 16 (4) 15.75 (5) None of these

**Example 5. **The average age of 11 persons in a committee is increased by 2 years when three men aged 32 years, 33 years and 34 years are substituted by three women. Find the average age of these three women.

(1) 40 1/3 (2) 40 (3) 41 1/3 (4) 41 years (5) None of these

**Example 6. **The average of 40 numbers is 405. If each of the numbers is divided by 15. Find the average of new set of numbers.

(1) 27 (2) 28 (3) 21 (4) 26 (5) None of these

**Explanation:**

405/15 = 27

**Example 7. **The average age of 80 boys in a class is 15. The average age of a group of 15 boys in the class is 16 and the average age of another 25 boys in the class is 14. What is the average age of remaining boys in the class. (approx.)

(1) 20 (2) 15 (3) 18 (4) 21 (5) None of these**Explanation:**

(2) Remaining boys average age = x

⇒ 15 × 16 + 25 × 14 + 40x = 80 × 15

⇒ 240 + 350 + 40x = 1200

⇒ 40x = 1200 – 590

x = 610/40

x = 15. 25

**Example 8. **A batsman in his 12th innings makes a score of 63 runs and there by increases his average score by ‘2’. The average of his score after 12th innings is.

(1) 39 (2) 41 (3) 37 (4) 45 (5) None of these**Explanation:**

(2) Average score = x till 11 innings

Total runs = 11x

Total runs after 12th inning = 11x + 63

Average after 12th inning = ( x + 2)

Total run = 12 ( x + 2)

12 ( x + 2) = 11x + 63 ⇒ x = 63 – 12 × 2 = 39

Average of his score = 39 + 2 = 41

**Example 9. **A certain amount was to be distributed among the A, B and C in the ratio 2 : 3 : 4 but was erroneously distributed in the ratio 7 : 2 : 5 as a result of this ‘B’ received Rs. 40 less. What is the actual?

(1) 210 (2) 270 (3) 230 (4) 280 (5) None of these

**Example 10. **Two number are respectively 20% and 50% more than a third number. The ratio of the two numbers is.

(1) 2 : 5 (2) 4 : 5 (3) 3 : 5 (4) 6 : 7 (5) None of these

**Exercise-2**

**1.** Deepak obtained 70, 63, 72, 81 and 74 marks (out of 100) in Mathematics, Physics, Chemistry, English and Hindi. What is his average marks?

a) 70 b) 72 c) 73 d) 71**Ans-**b;

Average = (70 + 63 + 72 + 81 + 74) / 5

= 360/5 = 72 marks

**2.** Find the average of all prime numbers between 20 and 40?

a) 30 b) 32 c) 31 d) 28**Ans-**a;

There are four prime numbers between 20 and 40.

Required average = (23 + 29 + 31 + 37) / 4

120/4 = 30

**3.** The second of the three numbers is twice the first and third is 4 times the first. If the average of the three numbers is 7, then find the smallest of the three numbers?

a) 1 b) 3 c) 7 d) 6**Ans-**b;

Let x be the first number, then 2x and 4x will be second and third respectively.

Average = (x + 2x + 4x) / 3 = 7

⇒ 7x = 21 ⇒ x = 3

Smallest number is 3.

**4.** The average of 15, x, 17 and 19 is 18. What is the value of x?

a) 22 b) 20 c) 18 d) 21**Ans-**d;

Average = (15 + x + 17 + 19) / 4 = 18

⇒ (51 + x) / 4 = 18 ⇒ x = 72 − 51 ⇒ x = 21

**5.** Ramesh receives the following scores in his maths test: 78, 92, 83 and 99. What score does he need in the next test in order to have an average of 90?

a) 92 b) 94 c) 97 d) 98**Ans-**d;

Let x be the next test score. Then,

The average = (78 + 92 + 83 + 99 + x) / 5 = 90

⇒ (352 + x) / 5 = 90 ⇒ x = 98

**6.** The average weight of A, B and C is 40 kg. If the average weight of A and B is 35 kg and that of B and C is 36 kg. Then find the weight of B?

a) 22 kg b) 23 kg c) 28 kg d) 30 kg**Ans-**a;

A + B + C = 40 × 3 = 120 kg

Weight of A and B = 35 × 2 = 70 kg

Weight of B and C = 36 × 2 = 72 kg

∴ Weight of B = (A + B) + (B + C) − (A + B + C)

= 70 + 72 − 120 = 22 kg

**7.** Mean of 20 observations was 63 but later it was found that the observation 69 was misread as 96. Find the correct mean?

a) 62 b) 61.65 c) 62.65 d) 64.35**Ans-**b;

As 69 is misread as 96, we have taken extra 27 value in account which is supposed to be taken away from the total of 20. So, the new average will be

63 − 27/20 = 63 − 1.35 = 61.65**8.** The average age of 25 students in a class is 15 years. The average age of 16 of those students is 13 years. What is the average age (in years) of the remaining 9 students?

a) 16 b) 19 c) 18.5 d) 17.5**Ans-**c;

Sum of ages of 9 students = 25 × 15 − 16 × 13 = 167 years

Required average = 167/9 = 18.5 years.**9.** The mean of 50 numbers is 30. Later it was discovered that two entries were wrongly entered as 82 and 13 instead of 28 and 31. Find the correct mean?

a) 36.12 b) 30.66 c) 29.28 d) 38.21**Ans-**c;

Mean of 50 numbers = 30

Sum of 50 numbers = 30 × 50 = 1500

Wrongly entered numbers are 82 and 13

∴ 1500 − (82 + 13) = 1500 − 95 = 1405

Correctly entered numbers = 1405 + (28 + 31)

= 1405 + 59 = 1464

Required mean = 1464/50 = 29.28

**10.** Out of 4 numbers, whose average is 60, the first one is one-fourth of the sum of the last three. The first number is?

a) 15 b) 45 c) 48 d) 60**Ans-**c;

Let the four numbers be a, b, c and d.

∴ (a + b + c + d) / 4 = 60

⇒ a + b + c + d = 240 and a = (a + b + c + d) / 4

⇒ a = (240 − a) / 4 ⇒ 5a / 4 = 60 ⇒ a = 60 × 4/5 = 48

**11. **The average of 25 observations is 13. It was later found that an observation 73 was wrongly entered as 48. The new average is?

a) 12.6 b) 14 c) 15 d) 13.8**Ans-**b;

Total = 13 × 25 = 325

According to the question,

Total = 325 − 48 + 73 = 350

Average of 25 numbers = 350/25 = 14

**12. **The average weight of a group of 20 boys was calculated to be 89.4 kg and it was later discovered that one weight was misread as 78 kg instead of 87 kg. The correct average weight is?

a) 88.95 kg b) 89.25 kg

c) 89.55 kg d) 89.85 kg**Ans-**d;

Total weight of 20 boys = 89.4 × 20 = 1788 kg

Now, 1788 − 78 + 87 = 1797 kg

The correct average weight = 1797/20= 89.85 kg

**13.** The average of three consecutive odd numbers is 12 more than one third of the first of these numbers. What is the largest of the three numbers?

a) 15 b) 17 c) 19 d) Data inadequate**Ans-**c;

Let consecutives odd number be x, x + 2, x + 4.

(x + x + 2 + x + 4) / 3 = 12 + x/3

⇒ (3x + 6) / 3 = (36 + x) / 3

⇒ 3x + 6 = 36 + x ⇒ 2x = 30 ⇒ x = 15

∴ Largest number = x + 4 = 15 + 4 = 19

**14. **The average of the first 100 positive integers is....

a) 100 b) 151 c) 50.5 d) 49.5**Ans-**c;

1 + 2 + 3 + ... + n = [n(n + 1)] / 2

∴ Average of these numbers = (n + 1) / 2

∴ Required average = (100 + 1) / 2 = 50.5

**15.** The average of 30 numbers is 12. The average of the first 20 of them is 11 and that of the next 9 is 10. The last number is....

a) 60 b) 45 c) 40 d) 50**Ans-**d;

Sum of 30 numbers = 30 × 12 = 360

Sum of first 20 numbers = 20 × 11 = 220

Sum of next 9 number = 9 × 10 = 90

∴ Last number = 360 − (220 + 90) = 50

**16. **The average of 5 numbers is 46 and that of the first four of them is 45. The fifth number is....

a) 9 b) 45 c) 46 d) 50**Ans-**d; Fifth number = 5 × 46 − 4 × 45 ⇒ 230 − 180 = 50

**17.** In a class there are 12 boys and 24 girls. Average weight of boys and girls is 50 kg and 40 kg respectively. Then what is the average weight of the class?

a) 42.33 kg b) 43.33 kg

c) 44.33 kg d) 45.33 kg**Ans-**b; The average weight of the class is

(12 × 50 + 24 × 40) / 36 = (50 + 2 × 40) / 3

= 43.33 kg.

**18. **Average rainfall from Monday to Saturday is 5 inches and the average for the whole week is 8 inches. What is the rainfall on Sunday?

a) 23 inches b) 25 inches

c) 27 inches d) 26 inches**Ans-**d; The rain fall from Monday to Saturday

= 5 × 6 = 30 inches.

The rain fall for the whole week = 7 × 8

= 56 inches

∴ The rain fall on Sunday = 56 − 30

= 26 inches

**19.** The average of ‘n’ numbers is x. When 36 is subtracted from two of the numbers, the new average becomes (x – 8). The value of ‘n’ is?

a) 6 b) 8 c) 9 d) 72**Ans-**c; Average of ‘n’ numbers = x

∴ Sum of ‘n’ numbers = n × x = nx

∴ (nx − 36 − 36) / n = x - 8

⇒ x − 72/n = x − 8 ⇒ 72/n = 8

⇒ n = 72/8 = 9

**20.** Four years ago average age of A, B and C was 25 years. Five years ago the average age of B and C was 20 years. A’s present age is....

a) 60 years b) 37 years

c) 62 years d) 15 years**Ans-**b; Let present ages of A, B and C be x years, y years and z years respectively.

∴ (x − 4) + (y − 4) + (z − 4) = 3 × 25 = 75

⇒ x + y + z = 75 + 12 = 87 ...(i)

Again, (y − 5) + (z − 5) = 2 × 20 = 40

⇒ y + z = 40 + 10 = 50 ...(ii)

∴ Present age of A = (x + y + z) − (y + z)

= 87 − 50 = 37 years

**21.** Among three numbers, the first is twice the second and thrice the third. If the average of the three numbers is 49.5, then the difference between the first and the third number is....

a) 54 b) 28 c) 39.5 d) 41.5**Ans-**a; Let the first number be x. Then, the second and third number will be x/2 and x/3 respectively.

∴ (x + x/2 + x/3) / 3 = 49.5

⇒ (6x + 3x + 2x) = 49.5 × 3

⇒ 11/6 x = 49.5 × 3

⇒ x = (49.5 × 3 × 6) / 11 = 81

Therefore, first number is 81 and third number is 27

Difference between first and third number = 81 − 27 = 54

**22. **The average age of 3 girls is 20 years and their ages are in the ratio 3 : 5 : 7. The age of the youngest girl is?

a) 4 years b) 8 years

c) 12 years d) 5 years**Ans-**c; Let the ages of 3 girls be 7x, 5x and 3x years respectively.

∴ 7x + 5x + 3x = 3 × 20 = 60

⇒ x = 60/15 = 4

∴ Age of the youngest girl = 3x

= 3 × 4 = 12 years.

**23. **A batsman scored 98 runs in his 26th innings thus increased his average by 2 runs. What was his average before that match?

a) 46 b) 36 c) 28 d) 72**Ans-**a; Let the earlier average be x runs per match.

Then, 25x + 98 = 26(x + 2)

⇒ x = 46

** PRACTICE EXERCISE**

**1.** The average of marks obtained by 120 candidates in a certain examination is 35. If the average marks of passed candidates is 39 and that of the failed candidates is 15. What is the number of candidates who passed the examination?

(1) 90 (2) 85 (3) 100 (4) 120 (5) None of these**Ans:** 100

**2.** The average salary of all workers in workshop is 8000. The average salary of 7-technicians is Rs. 12000 and the average salary of the rest is Rs. 6000. The total number of workers in the work shop is.

(1) 20 (2) 21 (3) 23 (4) 22 (5) None of these**Ans:** 21

**3.** In a competitive examination, the average marks obtained was 45. It was later discovered that there was some error in computerization and the marks of 90 candidates had to be changed from 80 to 50, and the average come down to 40 marks. The total number of candidates appeared in the examination.

(1) 500 (2) 600 (3) 540 (4) 580 (5) None of these**Ans:** 540

**4.** The average weight of A, B, C is 84 kg. If D joins the group the average weight of the group becomes 80 kg. If another man ‘E’ who weights is 3kg more than D replace A, then average of B, C, D and E becomes ‘79’ kg. What is the weight of ‘A’

(1) 64 (2) 72 (3) 75 (4) 100 (5) None of these**Ans:** 75

**5.** 1/3rd of certain journey is covered at the rate of 25kmph. 1/4th at the rate of 30 kmph and the respect 50 kmph. Find the average speed for the hole journey .

(1) 33.33 kmph (2) 36 kmph (3) 42 kmph (4) 27 kmph (5) None of these**Ans:** 33.33 kmph

**6. **Deepak’s opinion, his weight is greater than 65 kg but less than 72 kg. His brother does not agree with Deepak and he thinks that Deepak’s weight is greater than 60 kg but less than 70 kg. His mother’s view is that his weight cannot be greater than 68 kg. If all are them are correct in their estimation, what is the average of different probable weights of Deepak.

(1) 67 kg (2) 68kg (3) 69 kg (4) 70 kg (5) None of these**Ans:** 67 kg

**7.** The average of marks obtained by 120 candidates in a certain examination is 35. If the average marks of passed candidates is 39 and that of the failed candidates is 15. What is the number of candidates who passed the examination?

(1) 90 (2) 85 (3) 100 (4) 120 (5) None of these**Ans:** 100

**8.** The average age of Raju & Aditya is 35 years. If Raghu replaces Raju. The average age becomes 32 years and if Raghu replaces Adithya then the average becomes 32 years. If the average age of Naveen and Ramu behalf of the average age of Raju, Aditya and Raghu. Then the average age of all the people-?

(1) 28 (2) 32 (3) 35 (4) 30 (5) None of these**Ans:** 28

**9.** The average salary is being paid to all it’s employees by a company is 15,500. The average salary of the senior employees is 18000 per month and the average salary of the junior employees is 12000 per month. If there are only two levels of employees viz junior and senior levels, then what fraction of the total employees is the too junior level employees are

(1) 7/5 (2) 7/12 (3) 5/12 (4) 5/7 (5) None of these

Ans: 7/12

**10.** The average age of ‘8’ persons in a committee is increased by 2 years when two men aged 35 years and 45 years are substituted by two women. Find the average age of these two women.

(1) 48 (2) 36 (3) 42 (4) 29 (5) None of these**Ans:** 48

**The average of the given quantities is calculated by dividing the sum of the quantities by their number.**

Average = Sum of the quantities **/** Number of quantities

**Weighted average:** Let x_{1}, x_{2}, x_{3}, ...... x_{n} be the

quantities and w_{1}, w_{2}, w_{3}, ...... wn be the weights attached to them

then Weighted Average = w_{1}x_{1} + w_{2}x_{2} + w_{3}x_{3} + ........ + w_{n}x_{n /} w_{1} + w_{2} + w_{3} + ........ + w_{n}

**MODEL QUESTIONS****1. **The average of seven numbers is 18. The average of first three numbers is 14 and the average of last three numbers is 19. What is the middle number?

A) 9 B) 27 C) 32 D) 35

**2. **The average of 17 results is 60. If the average of first 9 results is 57 and that of the last 9 results is 65. What is the value of 9th result? (SSC-CGL – 2017)

A) 39 B) 78 C) 117 D) 156

**3. **In a class, the average age of 23 boys is 12 years and the average age of 17 girls is 11 years. What is the average age of the whole class?

A) 11.5 years B) 11.245 years C) 11.575 years D) 11.525 years

**4. **In a class, the average score of thirty students on a test is 69. Later it was found that the score of one of the students was wrongly read as 88 instead of 58. The actual average score is (SSC-CGL – 2019)

A) 71 B) 78 C) 65 D) 68

**5. **The average of 23 numbers is 28. Later, it was pointed out that value of one the numbers was wrongly taken as 78 instead of 87. What is the correct average?

A) 31.25 B) 27.15 C) 28.39 D) 32.65

**6. **Out of 6 numbers the sum of the first 5 numbers is 7 times the 6th number. If their average is 136, then the 6th number is.... (SSC-CGL – 2019)

A) 102 B) 96 C) 84 D) 116

**7. **A cricketer’s average for 11 matches is 25 runs. In the next four matches he scored 72, 34, 51 and 48 runs. Find his new average for the 15 matches?

A) 39 B) 32 C) 36 D) 28

**8. **The average age of a class of 24 students is 11 years. If the teacher’s age is also included, the average increases by one year. Find the age of the teacher?

A) 35 B) 39 C) 34 D) 36

**9. **The average age of six members of a family is 20 years. If the age of the servant is included, then the average age increase by 25%. What is the age of the servant (in years)? (SSC-CGL – 2017)

A) 30 B) 55 C) 50 D) 35

**10.** The average weight of 8 persons increases by 1.5 kg when a person weighting 65 kg is replaced by a new person. What could be the weight of the new person?

A) 55 kg B) 57.5 kg C) 75 kg D) 77 kg

**11. **The average of twelve numbers is 45.5. The average of the first four numbers is 41.5 and that of the next five numbers is 48. The 10th number is 4 more than the 11th number and 9 more than the 12th number. What is the average of the 10th and 12th number? (SSC-CGL – 2019)

A) 46.5 B) 46 C) 47.8 D) 47

**12. **5 kg sugar is purchased at Rs.10 per kg, 6kg at Rs.11 and 9kg at Rs.12 per kg. What is the weighted average price of sugar per kg?

A) 8.8 B) 10.8 C) 11.2 D) 11.8

**13.** The average of 12 numbers is 9. If each number is multiplied by 2 and added to 3, the average of the new set of numbers is (SSC-CGL – 2016)

A) 9 B) 18 C) 21 D) 27

**14. **In a class, average height of all students is 'a' cms. Among them, average height of 10 students is 'b' cms and the average height of the remaining students is 'c' cms. Find the number of students in the class. (Here a > c and b > c) (SSC-CGL – 2016) **15.** The average weight of some students in a class is 68.5 kg. If four new students of 72.2 kg, 70.8 kg, 70.3kg and 66.7 kg are enrolled in the class, the average weight of students increases to 300 g. Initially how many students were there in the class? (SSC-CGL – 2018)

A) 11 B) 26 C) 21 D) 16

**ANSWERS: 1-B; 2-B; 3-C; 4-D; 5-C; 6-A; 7-B; 8-D; 9-B; 10-D; 11-A; 12-C; 13-C; 14-D; 15-D.**

**Explanations
1. **The total of seven numbers = 7 × 18 = 126

The total of first 3 and last 3 numbers is = 3 × 14 + 3 × 19 = 99

∴ Middle number is (126 − 99) = 27

**Shortcut: **3 × (−4) + 3 × (+1) = −12 + 3 = −9

∴ Middle number = 18 + 9 = 27

**2. **The total of 17 numbers = 17 × 60 = 1020

The total of first 9 and last 9 numbers is = 9 × 57 + 9 × 65 = 1098

∴ Middle number is (1098 − 1020) = 78

**Shortcut:** 9 × (−3) + 9 × (+5) = −27 + 45 = +18

∴ Middle number = 60 + 18 = 78

**3. **Total age of 40 students = (23 × 12 + 17 × 11) = 463

Average age = 463/40 = 11.575 years

**Shortcut:** Let all the students average be 11 years

Extra age by all boys = 23 × 1 yr. = 23 years

Extra average for all 40 students = 23/40 = 0.575

∴ Actual Average = 11 + 0.575 = 11.575

**4. **The Correct Average

**Shortcut: **

Additional score taken = 88 − 58 = 30

Additional average = 30/30 = 1

∴ New average is 69 − 1 = 68

**5. Correct Average **

**Shortcut: **Value less taken = 87 − 78 = 9

Average less taken = 9/23 = 0.39

∴ New average is 28 + 0.39 = 28.39

**6. **Let the 6th number be ‘X’ ∴ 7x = (136 × 6) − x

⇒ x = 102

**7. **Total runs in 11 matches = 11 × 25 = 275 runs

Total runs in the next 4 matches

= 72 + 34 + 51 + 48 = 205 runs

= 32 runs

**Shortcut:** Let the average of all 15 matches be 25 Extra runs in 4 matches

= (72 − 25) + (34 − 25) + (51 − 25) + (48 − 25) = 105

Extra Average = 105 /15 = 7

∴ New Average = 25 + 7 = 32

**8.** Total of 24 students is 24 × 11 = 264 years

Total age with teacher is 25 × 12 = 300 years

∴ Teacher’s age is 300 − 264 = 36 years

**Shortcut:** Teacher’s age = 11 + (25 × 1) = 36 years

**9. **Total age of the family is 6 × 20 = 120 years

Average age after servant’s inclusion

= 125% of 20 = 25 years

Total age with servant = 7 × 25 = 175 years

∴ Servant’s age is 175 − 120 = 55 years

**Shortcut:** Servant’s age = 20 + (7 × 5) = 55 years

**10. **Let the average weight of 8 persons be ‘x’ years and the age of the new person be ‘y’ years

**Shortcut:** 65 + 8 × 1.5 = 77 kg

**11. **The total of 12 numbers = 12 × 45.5 = 546

The total of first 4 and next 5 numbers is

= 4 × 41.5 + 5 × 48 = 406

Total of 9th, 10th & 11th numbers = 546 − 406 = 140

Let the 10th number be x

∴ x + x − 4 + x − 9 = 140

⇒ x = 51

**Shortcut: **4 × (− 4) + 5 × (+ 2.5) = − 16 + 12.5 = − 3.5

∴ Total of 9th, 10th & 11th numbers

= (45.5 × 3) + 3.5 = 140

Let the 10th number be x

∴ x + x − 4 + x − 9 = 140

⇒ x = 51

= 46.5

**12. **Weighted average

**Shortcut:** Let the assumed average be Rs. 10

Additional average

**13. **New average **Shortcut: **New average = (9 × 2) + 3 = 21

**14.** Let the number of students be ‘x’

**15. **Let the number of students be ‘x’

⇒ 68.5x + 280 = 68.8(x + 4)

⇒ 68.5x + 280 = 68.8x + 275.2

⇒ 0.3x = 4.8