ব্যাংক নিয়োগ প্রস্তুতি ⎯ লং কোর্স

Question: The average of runs of a cricket player of 10 innings was 32. How many runs must he make in his next innings so as to increase his average of runs by 4?

Solution:
Average = 32
So, total = 32 × 10 = 320.

Now increase in average = 4
So, new average = 32 + 4 = 36

Total runs after next innings =  36 × 11 = 396
∴ Runs made in the 11th inning = 396 - 320 = 76

Question: The average of 7 consecutive numbers is 21. The largest of these numbers is-

Solution:
Let the numbers be x, x + 1, x + 2, x + 3, x + 4, x + 5 and x + 6,
Then,
(x + x + 1 + x + 2 + x + 3 + x + 4 + x + 5 + x + 6)/7 = 21
⇒ 7x + 21 = 147
⇒ 7x = 126 
∴ x = 18

∴ Largest number = x + 6 = 18 + 6 = 24

Question: The ratio of the present ages of P and Q is 3 : 4. Five years ago, the ratio of their ages was 5 : 7. Find their present ages.

Solution:
As the ratio of their present ages is 3 : 4 ,
let their present ages be 3x and 4x.
So, 5 years ago, as the ratio of their ages was 5 : 7,

According to the question,
(3x - 5) : (4x - 5) = 5 : 7
⇒ (3x - 5)/(4x - 5) = 5/7
⇒ 21x - 35 = 20x - 25
⇒ x = 10

Hence, their present ages are 3x = 30 and 4x = 40

Question: In a class average age of 15 boys is 11. If 5 boys each of age 9 years are added, what would be the new average?

Solution:
Sum of ages of 15 boys = 15 × 11= 165
Sum of ages of 5 boys = 5 × 9 = 45
Total age of 20 boys = 165 + 45 = 210
∴ Average of ages of 20 boys = 210/20 = 10.5 years

Question: Product of present age of Rahim and Latif is 2223 years and their present age ratio is 19 : 13 find the difference age of Rahim and Latif.

Solution:
Let, age of Rahim be 19x and age of Latif be 13x
Then, product of their ages = 19x × 13x = 247x²

Now,
247x² = 2223
⇒ x² = 9
∴ x = 3

Hence, required difference = 19x - 13x = 6x = 6 × 3 = 18 years

Question: The average of six consecutive numbers A, B, C, D, E and F is 62. What is the sum of B and F?

Solution:
Let the Numbers A, B, C, D, E, F be x, x + 1, x + 2, x + 3, x + 4, x + 5.
According to question, 
x + x + 1 + x + 2 + x + 3 + x + 4 + x + 5 = 62 × 6
⇒ 6x + 15 = 372
⇒ 6x = 372 - 15 = 357
⇒ 6x = 357
⇒ x = 59.5

∴ B = x + 1 = 60.5 
∴ F = x + 5 = 64.5

∴ B + F = 60.5 + 64.5 = 125

Question: The average of nine numbers is 60, that of the first five numbers is 55 and the next three is 65. The ninth number is 10 less than the tenth number. Then, tenth number is-

Solution:
Average of nine numbers = 60
Average of first five numbers = 55 and
average of next three numbers = 65
Tenth number = Ninth number + 10 

The sum of nine numbers = 60 × 9 = 540
The sum of the first five numbers = 55 × 5 = 275
The sum of the next three numbers = 65 × 3 = 195
Ninth number = (540 - 275 - 195) = (540 - 470) = 70

∴ Tenth number = 70 + 10 = 80

Question: If 5 years ago, the ratio of age of Hasan and Rina was 1 : 2 and after 15 years from present their ratio would be 5 : 6. Find the age of Rina after 20 years.

Solution:
Let, present age of Hasan be x
and present age of Rina be y.

Then, according to question
(x - 5)/(y - 5) = 1/2
⇒ 2x - 10 = y - 5
⇒ x = (y + 5)/2 .............(1)

Also,
(x + 15)/(y + 15) = 5/6
⇒ 6x + 90 = 5y + 75
⇒ 6x + 15 = 5y

Putting value of x from equation 1, we get
3y + 15 + 15 = 5y
⇒ 2y = 30
⇒ y = 15

∴ Age of Rina after 20 years = 15 + 20 = 35 years.

Question: The average of x and y is 45, and the average of y and z is 50. If y = 42, then find the value of (x + z).

Solution:
Given y = 42

Average of x and y = 45
∴ x + y = 45 × 2
⇒ x + y = 90
⇒ x + 42 = 90
⇒ x = 90 - 42
⇒ x = 48

Average of y and z = 50
∴ y + z = 50 × 2
⇒ y + z = 100
⇒ 42 + z = 100
⇒ z = 100 - 42
⇒ z = 58

Now, x + z = 48 + 58 = 106

Question: The present age of a son is one-fourth of the present age of his father. Eight years ago,  the father's age was seven times the age of his son. What is the present age of the son?

Solution: Let the father's age be 4x years
Then, Son's age = x years

ATQ,
4x - 8 = 7(x - 8)
or, 4x - 8 = 7x - 56
or, -8 + 56 = 7x - 4x
or, 48 = 3x
or, x = 16

∴ The present age of the son = 16 years.

Question: The average of six numbers is A and the average of three of these is B. If the average of the remaining three is C, then which one is correct?

Solution:
Total sum of six numbers = 6A
Total sum of three numbers = 3B
Total sum of the other numbers = 3C

Now,
6A = 3B + 3C
or, A = 3(B + C)/6
or. A = (B + C)/2
∴ 2A = B + C

Question: The average age of the children in a tour group is 12 years and that of the adults is 32 years. If the average age of the entire tour group is 20 years, find the ratio of children to adults in the group.

Solution: Average age of children = 12 years
Average age of adults = 32 years
Average age of the entire group = 20 years

Let, the number of adults = A
and, the number of children = C

Then, the total number of people in the group is (C + A)

ATQ,
12C + 32A = 20(C + A)
Or, 12C + 32A = 20C + 20A
Or, 32A - 20A = 20C - 12C
Or, 12A = 8C
Or, C : A = 12 : 8
Or, C : A = 3 : 2

∴ The ratio of children to adults in the group is 3 : 2.

Question: The present ages of three cousins are in the ratio of 4 : 5 : 6. Five years ago, their total age was 60 years. In three years, what will be the age of the youngest cousin?

Solution: Present age ratio of three cousins is 4 : 5 : 6
Let, their age is 4X, 5X and 6X respectively.

ATQ,
4X - 5 + 5X - 5 + 6X - 5 = 60
Or, 15X - 15 = 60
Or, 15X = 75
Or, X = 5

∴ The present age of the youngest cousin is = 4X = 4 × 5 = 20 years.
In three years, his age will be = (20 + 3) = 23 years.

Question: The average age of a group of 15 employees is 28 years. When 5 more employees join the group, the average age increases by 2 years. The average age of the new employees is?

Solution: Average age of 15 employees = 28 years
Total age of 15 employees = 15 × 28 = 420 years

After 5 more employees join:
Total number of employees = 15 + 5 = 20 employees
New average age = 28 + 2 = 30 years
Total age of 20 employees = 20 × 30 = 600 years

Total age of 5 new employees = 600 - 420 = 180 years

Average age of 5 new employees = 180 ÷ 5 = 36 years

Question: The average age of Arif, Tania, Joya, and Rafi is 22 years. The average age of Arif, Tania, and Joya is 20 years and the average age of Tania, Joya, and Rafi is 24 years. Find the average age of Tania and Joya.

Solution:
Given,
Arif + Tania + Joya + Rafi = 22 × 4 = 88 years ……(1)
Arif + Tania + Joya = 20 × 3 = 60 years ……(2)
Tania + Joya + Rafi = 24 × 3 = 72 years ……(3)

From (1) - (2)⇒
Rafi’s age = 88 − 60 = 28 years

From (3)⇒
Tania + Joya = 72 - 28 = 44 years

∴ Average age of Tania and Joya = 44/2 = 22 years

Question: The average of 3, 8, 7, and x is 6 and the average of 19, 2, 7, x and y is 9. What is the value of y?

Solution: 
Given that,
The average of 3, 8, 7, x is 6

Therefore,
6 = (3 + 8 + 7 + x​)/4
⇒ 24 = 18 + x
⇒ x = 24 - 18
∴ x = 6

Therefore,
9 = (19 + 2 + 7 + x + y​)/5
⇒ 45 = 28 + 6 + y
⇒ y = 45 - 34 
∴ y = 11

Question: A is 50 years old and B is 35 years old. How many years ago was the ratio of their ages 3 : 2?

Solution: 
Let,
'x' years ago, the ratio of their ages was 3 : 2

ATQ,
(50 - x) : (35 - x) = 3 : 2
⇒ (50 - x)/(35 - x) = 3/2
⇒ 105 - 3x = 100 - 2x
⇒ 3x - 2x = 105 - 100
∴ x = 5

Question: Find the average of all the numbers between 10 and 50 which are divisible by 4.

Solution:
Numbers between 10 and 50 divisible by 4 are = 12, 16, 20, 24, 28, 32, 36, 40, 44, 48.

Required average = (12 + 16 + 20 + 24 + 28 + 32 + 36 + 40 + 44 + 48​)/10
= 300/10
= 30

Question: The sum of the present ages of a son and his father is 70 years. After 5 years, the age of the father will be three times that of the son. At present, their ages are?

Solution:
Let,
The son's age is x years.
Then, father's age = (70 - x) years.

ATQ,
(70 - x) + 5 = 3(x + 5)
⇒ 75 - x = 3x + 15
⇒ 4x = 60
∴ x = 15

∴ Son's age = 15 years
And the father's age = (70 - 15) = 55 years.

Question: If the average of 'p' numbers is 3q² and the average of 'q' numbers is 3p², what is the average of the combined (p + q) numbers?

Solution:
দেওয়া আছে, 'p' সংখ্যার গড় = 3q²
∴ p সংখ্যার সমষ্টি = p × 3q²

'q' সংখ্যার গড় = 3p²
∴ 'q' সংখ্যার সমষ্টি = q × 3p²

∴ মোট সমষ্টি = (p × 3q²) + (q × 3p²)
= 3pq² + 3p²q
= 3pq(q + p)

∴ তাদের গড় = মোট সমষ্টি / (p + q)
= 3pq(p + q) / (p + q)
= 3pq

Question: The average temperature on Monday, Tuesday, and Wednesday was 26°C. The average temperature on Tuesday, Wednesday, and Thursday was 25°C. If the temperature on Monday was 28°C, what was the temperature on Thursday?

Solution:
The total temperature on Monday, Tuesday, and Wednesday = 26° × 3 = 78°
The total temperature on Tuesday, Wednesday, and Thursday = 25° × 3 = 75°
 
ATQ,
(Mon + Tue + Wed) - (Tue + Wed + Thu) = 78° - 75°
⇒ Mon - Thu = 3°
⇒ Thu = Mon - 3°
⇒ Thu = 28° - 3°
∴ Thu = 25°

∴ The temperature on Thursday = 25°

Question: The average weight of 25 students in a class was found to be 48 kg. Later it was discovered that one student's weight was recorded as 35 kg instead of 60 kg. What is the correct average weight of the 25 students?

Solution:
Given,
The average weight of 25 students in a class was found to be 48 kg
Incorrect sum of the weight of 25 students = (48 × 25) kg
= 1200 kg

Correct sum of the weight of 25 students = (incorrect sum) - (wrongly copied item) + (actual item)
= (1200 - 35 + 60) kg
= 1225 kg

Correct mean = correct sum/number of students
= (1225/25) kg
= 49 kg

∴ Hence, the correct mean weight is 49 kg

Question: A couple has a son and a daughter. The age of the father is three times the daughter's age and the age of the mother is twice the son's age. The father is 9 years older than the mother and the daughter is 6 years younger than the son. What is the age of the mother?

Solution:
Let the ages of father, mother, son, and daughter be F, M, S, and D respectively.

According to the question (ATQ):
F = 3D .............(i)
M = 2S .............(ii)
F = M + 9 ...........(iii)
S = D + 6 ...........(iv)

From (ii) we get,
M = 2S
Or, M = 2(D + 6) [from (iv)]
Or, M = 2{(F/3) + 6} [from (i)]
Or, M = (2/3) × (F + 18)
Or, M = (2/3) × (M + 9 + 18) [from (iii)]
Or, M = (2/3) × (M + 27)
Or, 3M = 2M + 54
Or, M = 54

∴ The age of the mother is 54 years.

Question: The average of 10 numbers is 23. If each number is increased by 4, what will the new average be?

Solution:
Given,
Average of 10 numbers = 23
⇒ Sum/Total numbers = 23
⇒ Sum/10 = 23
∴ Sum of the 10 numbers = 230

If each number is increased by 4, the total increase = 4 × 10 = 40
New sum = 230 + 40 = 270

Therefore, the new average = 270/10 = 27

Question: Of three numbers, the average of the first and second numbers is 12 more than the average of the second and third numbers. What is the difference between the first and third numbers?

Solution: 
Let, these numbers are x, y and z respectively.

ATQ,
{(x + y)/2} - {(y + z)/2} = 12
⇒ {(x + y) - (y + z)}/2 = 12
⇒ (x + y - y - z)/2 = 12
∴ x - z = 24

∴ The difference between the first and the third number is = 24.