Average

"Required Run rate = ( 98 - ( 1.9×30 ) )/ (20) = ( 41) /(20) = 2.05 "

No. of persons = 9
Let the average expenditure of 9 persons be x
Now according to question,
12 × 8 + (x+8) / 9 = x
96 = x + 8 = 9x
8x = 104
x = 13
Total money spent by 9 persons = 9x
9×13 =₹117

By alligation method,
Ratio of A : B : C = 3 : 4 : 5
Average of all 3 groups = [(93×3) + (86×4) + (95×5)]/(3 + 4 + 5)
= (279 + 344 + 475)/12
= 1098/12
= 91.5
∴ average of all three groups = 91.5:

Let the present ages of P, Q, R and S be P, Q, R, and S respectively.
(P + R) / 2 = 33
⇒ P + R = 66 ⇒ eqn (i)
Q = R + 18 ⇒ eqn (ii)
Q = S - 6 ⇒ eqn (iii)
P = S - 33(1/3)% of S
⇒ P = S - (100/3)% of S
⇒ P = S - [(100/3)/100] × S
⇒ P = S - (1/3) × S
⇒ P = (3S - S) / 3
⇒ P = 2S/3 ⇒ eqn (iv)
Put the value of eqn (iv) in eqn (i), we get
(2S/3) + R = 66 ⇒ eqn (v)
Put the value of Q from eqn (ii) in eqn (iii), we get
R + 18 = S - 6
⇒ S - R = 18 + 6
⇒ S - R = 24 ⇒ eqn (vi)
On Adding eqn (v) and eqn (vi)
(2S/3) + S = 66 + 24
⇒ (2S + 3S)/3 = 90
⇒ 5S = 270
⇒ S = 54 which is the present age of S.
Now, from eqn (iii), the present age of Q = S - 6 = 48
Now, from eqn (ii), the present age of R = Q - 18 = 30
Thus, Required average = (48 + 30 + 54)/3
= 13$$\frac{2}{3}$$
= 44 years

Let daughter's present age be Z years.
Then, Shalini's present age = 5Z years
⇒ Ratio of Shalini daughter's present age to Shalini's present age = Z : 5Z = 1 : 5
Ratio of Shalini's present age to her father's present age = 2 : 5
Then, Ratio of present age of Daughter: Shalini : Father = 2 : 10 : 25
⇒ age of Daughter 6 years hence = 2x + 6
age of Shalini 6 years hence = 10x + 6
age of Father 6 years hence = 25x + 6
average age of all the three 6 years hence = 43 years
⇒ (2x + 6 + 10x + 6 + 25x + 6) / 3 = 43
⇒ (37x + 18) = 43 × 3
⇒ 37x = 129 - 18
⇒ 37x = 111
⇒ x = 111/37
⇒ x = 3
Then, present age of Daughter = 2x = 2 × 3 = 6 years
present age of Shalini = 10x = 10 × 3 = 30 years
present age of Father = 25x = 25 × 3 = 75 years
Required Ratio = 6 : (75 - 30)
= 6 : 45
= 2 : 15

Let the age of Sachin = S;
Naveen = G;
Kaif = K;
Dhoni = D;
Irfan = I
Average age of Sachin and Naveen = 35 years
⇒ (S + G)/ 2 = 35
Now, Total age of Sachin and Naveen = 35 × 2
⇒ S + G = 70 ⇒ eqn (1)
If Kaif replaces Sachin, the avcrage age = 32 years
⇒ (K + G)/ 2 = 32
Now, Total age of Kaif and Naveen = 32 × 2
⇒ K + G = 64 ⇒ eqn (2)
If Kaif replaces Naveen, then the average age = 38 years
⇒ (S + K)/ 2 = 38
Now, Total age of Sachin and Kaif = 38 × 2
⇒ S + K = 76 ⇒ eqn (3)
Adding eqn (1), (2) and (3), we get
2S + 2G + 2K = 70 + 64 + 76
-⇒ 2 (S + G + K) = 210
-⇒ (S + G + K) = 210/2
-⇒ (S + G + K) = 105
average age of Dhoni and Irfan = ($$\frac{1}{2}$$) × average age of Sachin, Naveen and Kaif
⇒ (D + I)/2 = ($$\frac{1}{2}$$) × (S + G + K)/3
⇒ (D + I)/2 = ($$\frac{1}{2}$$) × (105/3)
⇒ (D + I)/2 = ($$\frac{1}{2}$$) × 35
⇒ (D + I) = ($$\frac{1}{2}$$) × 35 × 2
⇒ (D + I) = 35
Now, Average age of all the five people = (S + G + K + D + I)/5
= (105 + 35)/5
= 140/5
= 28 years

Let age of Shilpa = x, So age of Mother = 2x+5, David = x+4, Father = (2x+5)+2 = 2x+7
(2x+5 + 2x+7)/2 = 46
(4x+12) = 92
4x = 80
So, x = 20
So, (x+4)/x = 24/20 = 6/5

Given, average age of 25 students = 15 yrs
Average age = Total age/Number of students
Total age of 25 students = 15×25 = 375 yrs
Average age of first 12 students = 14 yrs
So, the total age of first 12 students = 14×12 = 168 yrs
Average age of last 12 students = 16 yrs
So, total age of last 12 students = 12×16 = 192 yearss
∴ age of 13th student = Total age of 25 students - (Total age of first 12 students + Total age of last 12 students)
Age of 13th student = 375 - (168 + 192) = 15 yrs
Age of Amrutha = 5 + age of 13th student = 5 + 15
Amrutha's age = 20 yrs.

Runs in 26th inning = Runs total after 26 innings - Runs total after 25 innings
= [ 26×( 62+1) ] - [ 25×62 ]
= (26×63) - (25×62)
= 1638 - 1550 = 88
Hence, option (b) is the correct answer.

Average ages of couple = 25 years
So, the sum of ages of couple = 25×2 = 50 years
When 1st child born the average age of family = 18 years
∴ the total age of family(3 members) = 18 × 3 = 54 years
(At that time child age would be zero, so the age of father and mother would have increased by same.)
it's increased by 2 years for each = 50 + 2 + 2 = 54 years
When second child born the average age of family = 15 years
∴ the total age of family(4 members) = 15 × 4 = 60 years
(So, at that time second child age would be zero, the age of father, mother and first child would have increased by same.)
So it's increased by 2 years for each = 54 + 2 + 2 + 2 = 60 years.
Therefore after 4 years, the total age of couple (from age of 50) = 50 + 4 + 4 = 58 years
So, the average age of couple = 58/2 = 29 years.