ব্যাংক নিয়োগ প্রস্তুতি ⎯ লং কোর্স

Question: If the average of 5, 11, 18, and 'x' is 14, what is the value of 'x'?

Solution:
According to the question,
(5 + 11 + 18 + x) / 4 = 14
⇒ 34 + x  = 56
⇒ x  = 22

Question: Find the average of all prime numbers between 20 and 50?

Solution:
Prime number between 20 and 50 =  23, 29, 31, 37, 41, 43, 47

∴ average = (23 + 29 + 31 + 37 + 41 + 43 + 47) / 7 = 35.86

Question: Out of three numbers, the first is twice the second and is half of the third. If the average of the three numbers is 56, then the average of the first and the third number is -

Solution:
Let, the second number be = x
So, the first number is = 2x
and the third number is = 2x × 2 = 4x

According to the question,
2x + x + 4x = 56 × 3
⇒ 7x = 168
⇒ x = 24

So, the first number is = 2 × 24 = 48
and the third number is = 4 × 24 = 96

Then, the average of the first and the third number is = (48 + 96)/2 = 72

Question: Find out the geometric mean of '3, 25, and 45'.

Solution:
The geometric mean of '3, 25, and 45'

= (3 × 25 × 45)^1/3
= (3375)^1/3
= (15³)^1/3
= 15

Question: The average of 20 numbers is 40. If the numbers 30 and 50 are discarded, then the average of the remaining numbers is - 

Solution:
Sum of 20 numbers = 20 × 40 = 800
Sum of remaining 18 numbers = 800 - (30 + 50) = 720
Average of the remaining numbers = 720/18 = 40

Question: A is 60 years old and B is 40 years old. How many years ago was the ratio of their ages 5 : 3?

Solution: 
Let, 'x' years ago the ratio of their ages was 5 : 3

According to the question,
(60 - x) : (40 - x) = 5 : 3
⇒ (60 - x) / (40 - x) = 5 / 3
⇒ 180 - 3x = 200 - 5x
⇒ 2x = 20
⇒ x = 10

Question: The sum of the ages of a man and his son is 45 years. Five years ago, the product of their age was 34. What is the age of the son?  

Solution: 
Let, the son's age be = x years
So, the man's age is = (45 - x) years

ATQ,
(x - 5) (45 - x - 5) = 34
⇒ (x - 5) (40 - x) = 34
⇒ 40x - x² - 200 + 5x - 34 = 0
⇒ 45x - x² - 234 = 0
⇒ x² - 45x + 234 = 0
⇒ x² - 39x - 6x + 234 = 0
⇒ (x -39) (x - 6) = 0

So, x = 6, or 39
Son's age must be less than father's age.

∴ x = 6

Question: The ratio of Rifat and Rafat's ages is 4 : 3. Rifat will be 30 years old after 6 years. How old is Rafat now?

Solution:
Let, Rifat's age is = 4x years
and Rafat's age is = 3x years

ATQ,
4x + 6 = 30
⇒ 4x = 24
⇒ x = 6

So, Rafat's age is = (3 ×6) = 18 years

Question: The ratio of the ages of father and son now is 6 : 1. After 5 years the ratio will become 7 : 2. What is the sum of their ages now?

Solution:
Let, the ages of the father = 6x
and the ages of the son = x

ATQ,
(6x + 5) / (x +5) = 7/2
⇒ 12x + 10 = 7x + 35
⇒ 5x = 25
⇒ x = 5

So, the ages of the son = 5 years
and the ages of the father = 6 × 5 = 30 years

∴ the sum of their ages now = 30 + 5 = 35 years

Question: The average of 7 consecutive odd numbers is 33. What is the average of the first and last numbers?

Solution:

The middle number is = 33
So, the other numbers are = 27, 29, 31, 33, 35, 37, 39

The average of the first and last numbers is = (27 + 39)/2 = 66/2 = 33

Question: The average of ten numbers is 8. If each number is multiplied by 10, what is the new average? 

Solution:
The sum of ten numbers = 10 × 8 = 80

ATQ,
The average of ten numbers is = The sum of ten numbers / 10

Now, 
If each number is multiplied by 10, the new average is
= (The sum of ten numbers × 10) / 10
= (80 × 10) / 10
= 80

Question: The Average age of 20 students and a teacher is 15 years. When the teacher's age is excluded, the average decreases by 1. What is the age of the teacher?

Solution:
20 boys + 1 teacher's total age is = 21 × 15 = 315 years
Without the teacher, 20 boys total age = 20 × (15 - 1) = 280 years

So, the teacher's age is = (315 - 280) years = 35 years

Question: What is the average of the sum of the first 20 natural numbers?

Solution:
The average is = {20(20 + 1)} / (2 × 20)
= 420/40
= 10.5

Question: The average of 80 numbers is 42. When 5 more numbers are included, the average of 85 numbers becomes 45. Find the average of 5 numbers.

Solution:
Sum of 80 numbers= 80 × 42 = 3360
Now, Sum of 85 numbers = 85 × 45 = 3825
Hence, the sum of 5 numbers = 3825 - 3360 = 465
Average of five numbers = 465/5 = 93

Question: A is two years older than B who is twice as old as C. If the total ages of A, B, and C is 27, how old is C?

Solution:
Let, C's age be x years.
Then, B's age = 2x years.
And A's age = (2x + 2) years. 

ATQ,
2x + 2 + 2x + x = 27
⇒ 5x = 25
⇒ x = 5

Question: What is the average of 1², 2², 3², 4², 5², 6^2, 7²?

Solution:
Average = (1² + 2² + 3² + 4² + 5² + 6²+ 7²)/7
= (1 + 4 + 9 +16 + 25 + 36 + 49)/7
= 140/7
= 20

Alternative Solution: 
Sum = {n(n + 1)(2n + 1)}/6
= {7(7 + 1)(14 + 1)}/6
= (7 × 8 × 15)/6
= 140

Average = 140/7 = 20

Question: The average of two numbers is Q. If one number is P, then the other is -

Solution:
Sum of two numbers is = 2Q
∴ the other is = 2Q - P

Question: In seven given numbers, the average of the first four numbers is 4 and the last four numbers is also 4. If the average of these given seven numbers is 3, the fourth number is - 


Solution:
Sum of first four numbers = 4 × 4 = 16
Sum of last four numbers = 4 × 4 = 16
Sum of first four numbers and last four numbers = 16 + 16 = 32

Again, Sum of seven numbers = 7 × 3 = 21

So, fourth number is = 32 - 21 = 11

Question: The average age of three boys is 20 years and their age ratio is 4 : 5 : 6. What is the age of the youngest boy?

Solution:
Sum of the three boy's age = 20 × 3 = 60 years
So, age of the youngest boy is = 60 × (4/15) = 16 years

Question: A passenger travels from Dhaka to Cumilla at a speed of 30 kmph and returns with a speed of 60 kmph. What is the average speed?

Solution:
Average Speed
= 2xy/(x + y)
= (2 × 60 × 30)/(60+30) kmph
= 3600/90 kmph
= 40 kmph

Question: The average of first five multiples of 3 is -

Solution: 
First five multiples of 3 are = 3, 6, 9, 12, 15.

∴ Average = (3 + 6 + 9 + 12 + 15)/5
= 45/5
= 9

Question: A batsman makes a score of 82 runs in the 15th innings and thus increases his average by 2. Find his average after the 15th innings.

Solution:
Let, the average after the 15th innings be = x
Then, average after 14th innings = (x - 2)

ATQ,
14 × (x - 2) + 82 = 15x
⇒ 14x - 28 + 82 = 15x
⇒ x = 54

Question: The present age of son is half of the present age of his mother. Ten years ago, his mother's age was thrice the age of her son. What is the present age of the mother?

Solution:
Let, the mother's age be = 2x years
Then, the son's age = x years

∴(2x − 10) = 3(x − 10)
⇒2x − 10 = 3x −30
⇒x = 20

Son's age = 20 years
Mother's age = (2 × 20) = 40 years

Question: The ratio of a man's age and his son's age is 7 : 3 and the product of their ages is 756. The ratio of their age after 6 years will be - 

Solution: 
Let, man's age be = 7x years 
Then, son's age = 3x years

ATQ,
7x × 3x  = 756
⇒ 21x² = 756
⇒ x² = 36
⇒x = 6

The ratio of their ages after 6 years = (7 × 6 + 6) : (3 × 6 + 6)
= 48 : 24
= 2 : 1

Question: Ten years ago A was half of B in age. If the ratio of their present ages is 3 : 4, what is the age present of A?

Solution:
Let, B's age 10 years ago = 2x years
and A's age 10 years ago = x years

Now,
A's age at present = (x + 10) years
B's age at present = (2x + 10) years

ATQ,
(x + 10) / (2x + 10) = 3/4
⇒ 4X + 40 = 6X + 30
⇒ 2x = 10
⇒ x = 5

∴ A's age at present = (5 + 10) = 15 years

Question: The average age of parents is 38 years. The average age of parents and a child is 30 years. What is the age is the child?

Solution:
Total age of parents = 38 × 2 = 76 years
Total age of parents and the child = 30 × 3 = 90 years

So, Child's age = 90 - 76 = 14 years

Question: The daughter's age is mean proportional to age of the father and son. The age of the father and son is 50 years and 8 years respectively. What is the age of the daughter?

Solution: 
Let,
1st proportional = Father's age
and 3rd proportional = Son's age

We know,
Mean Proportional = √(1st proportional × 3rd proportional)

So, daughter's age = √(50 × 8) years
= √400 years
= 20 years

Question: The sum of Moni and Mimi's ages is 50 years and the difference between their ages is 6 years. What is the product of their ages?

Solution: 
Let, Moni's age be = x years
and Mimi's age be = y years

ATQ,
x + y = 50.........(i)
x - y = 6.............(ii)

Now,
(i) + (ii),
x + y + x - y = 50 + 6
⇒ 2x = 56
⇒ x = 28

So, y = 50 - 28 = 22

∴ product of their ages = 28 × 22 = 616 years

Question: Rahim's present age is 9 times as Karim's age. After 9 years, Rahim's age would be 3 times as Karim's age.  What is the present age of Karim?

Solution:
Let, Karim's present age be x years
and Rahim's present age is = 9x years

Now,
Karim's age after 9 years = (x + 9) years
Rahim's age after 9 years = (9x + 9) years

ATQ,
9x + 9 = 3(x + 9)
⇒ 9x + 9 = 3x + 27
⇒ 6x = 18
⇒ x = 3