ব্যাংক নিয়োগ প্রস্তুতি ⎯ লং কোর্স

প্রশ্ন: The average of runs of a cricket player of 10 innings was 35. How many runs must he make in his next innings so as to increase his average of runs by 5?

Solution: 
Total runs =35 × 10 = 350
Now increase in average is 5 runs
so, 
New average =35 + 5 = 40 runs
Total runs = 40 × 11= 440
Runs made in the 11th inning =440 - 950 = 90

Question: A father said to his son, 'I was as old as you are at the present at the time of your birth'. If the father's age is 48 years now, the son's age five years back was-

Solution:
let, at present son is x years old
so, at time of his birth the father was x years old

∴ at present the age of father is = x + x year
= 2x year

2x = 48
⇒ x = 48/2
= 24 year

∴ the son's age five years back was = 24 -5 
= 19 year

প্রশ্ন: The average of 15 numbers is zero. Of them, at the most, how many may be greater than zero?

সমাধান: ধরি, সংখ্যাগুলি হল a₁​, a₂​, ..., a₁₅
দেয়া আছে,  সংখ্যাগুলোর গড় শূন্য

তাহলে, (a₁​ + a₂​ +...+ a₁₅)/15 = 0
⇒ a₁​ + a₂​ +...+ a₁₅​ = 0
⇒ a₁​ + a₂​ +...+ a₁₄ ​= −a₁₅
∴ সর্বোচ্চ ১৪ টি সংখ্যার মান শুন্য থেকে বড় হতে পারে।

Question: A is two years older than B who is twice as old as C. If the total of the ages of A, B and C be 42, the how old is B?

Solution: 
Let,
C is = x years old
B is 2x years old
A is (2x + 2) years old

Now,
2x + 2x + 2 + x = 42
⇒ 5x + 2 = 42
⇒ 5x = 40
∴ x = 8
B is (2 × 8) = 16 years old

প্রশ্ন: The average of 4 consecutive numbers is 5.5 . The largest of these numbers is:

সমাধান:
৪ টি সংখ্যার গড় ৫.৫
৪ টি সংখ্যার সমষ্টি = ৫.৫ × ৪ = ২২

ধরি, সংখ্যাগুলি হল a, a + ১, a + ২, a + ৩
প্রশ্নমতে,
a + a + ১ + a + ২ + a + ৩ = ২২
⇒ ৪a + ৬ = ২২
⇒ ৪a = ১৬
⇒ a = ৪

∴ বড় সংখ্যাটি হল a + ৩ 
= ৪ + ৩
= ৭

Question: The ages of Akash and Belal are in the ratio 6 : 5 and the sum of their ages is 44 years. What will be the ratio of their ages after 8 years ?

Solution:
The ages of Akash and Belal are in the ratio 6 : 5 
let, akash is 6x years old and belal is 5x years old
5x + 6x = 44
⇒ 11x = 44
∴ x = 4 years old

So, Akash is 24 years old and belal is 20 years old

After 8 years, Akash is (24 + 8) = 32 years old and belal is (20 + 8) years old
∴ their ratio = 32 : 28
= 8 : 7

প্রশ্ন: A pupil's marks were wrongly entered as 63 instead of 43. Due to that the average marks for the class got increased by half. The number of pupils in the class is

সমাধান:
ধরি, মোট ছাত্র সংখ্যা x
একজন বাদে বাকি ছাত্রদের নম্বরের সমষ্টি y

(y + 63)/ x = a + 0.5 ....(Ⅰ)
(y + 43)/ x = a.........( Ⅱ)

(Ⅰ) - ( Ⅱ) থেকে পাই,
{(y + 63)/ x} - {(y + 43)/ x } = a + 0.5 - a
⇒ (y + 63 - y - 43)/x = 0.5
⇒ 20/x = 0.5
⇒ x = 40

∴ ক্লাসে ছাত্র সংখ্যা ৪০ জন।

Question: The age of a mother today is thrice that of her daughter. After 10 years , the age of the mother will be twice that of her daughter. The present age of the daughter is =?

Solution:
let, age of daughter today is x and age of mother is 3x
After 10 years, age of daughter x + 10 and age of mother is 3x + 10

So,
3x + 10 = 2 (x + 10)
⇒ 3x + 10 = 2x + 20
⇒ 3x - 2x = 10
⇒ x = 10
The present age of the daughter is 10 years

Question: Rajan got married 10 years ago. His present age is (6/5) times of his age at the time of his marriage.  Rajan's sister was 10 years younger to him at the time of his marrige. The present age of Rajan's sister is = ? 

Solution:
Present age of Rajan is = x year

x = (6/5) (x - 10)
⇒ 5x = 6x - 60
⇒ 6x - 5x = 60
⇒ x = 60 years

at time of marriage, rajan's age is 60 - 10 = 50 years
at time of marriage, rajan's sister age is = 50 - 10 years = 40 years

∴ The present age of rajan's sister age is = 40 + 10 = 50 year

Question: Of the four numbers, whose average is 80, the first is one-fourth of the sum of the last three. What is the value of the first number?

Solution:
Average of four numbers is 80
The sum of four numbers is = (4 × 80) = 320

let, the first number is x
then sum of last three is 4x

x + 4x = 320
⇒ 5x = 320
∴ x = 64

Question: The average salary of all the workers in a workshop is Tk. 6000. The average salary of 7 technicians is Tk. 12000 and the average salary of the rest is Tk. 5000. How many workers are there?

Solution:
let, there are x number of workers
 The average salary of all the workers in a workshop is Rs. 6000
∴ total salary = 6000 × x = 6000x

The average salary of 7 technicians is Tk. 12000
∴ salary of 7 technicians is = (12000 × 7) = 84000 tk

the average salary of the rest is Tk.  5000
∴ salary of the rest = 5000 × (x - 7) tk

 5000 × (x - 7) +  84000 = 6000x
⇒ 5000x - 35000 + 84000 = 6000x
⇒ 1000x = 49000
∴ x = 49

so, there are 49 workers

Question: The ages of Saha and Akash are in the ratio 9 : 8 respectively. After 5 years, the ratio of their ages will be 10 : 9. What is the difference in their ages?

Solution:
The ages of Saha and Akash are 9x and 8x
After 5 years, the ratio of their ages will be (9x + 5) and (8x + 5)

(9x + 5) /(8x + 5) = 10/9
⇒ 81x + 45 = 80x + 50
⇒ x = 5

∴  the difference in their ages is = 9x - 8x
= x
= 5 years

Question: The average age of a husband and his wife was 25 years at the time of their marriage. After five years they have a three-year old child. What is the average age of the family now? 

solution:
The average age of a husband and his wife was 25 years at the time of their marriage
Sum of age = 25 × 2 = 50

After five years, sum of their age = 50 + 5 + 5 = 60
Sum of age of the family = 60 + 3 = 63 years

∴the average age of the family is = 63/3 = 21 years

Question: The average of five numbers is 27. If one number is excluded, the average becomes 20. The excluded number is :

Solution:
The average of five numbers is 27
The sum of five numbers is (27 × 5)
= 135

let, the excluded number is x

so,
(135 - x)/4 = 20
⇒ 135 - x = 80
⇒ x = 135 - 80
∴ x = 55

Question: Three years ago, the average of A, B, and C was 27 years and that of A and C, 5 years ago was 20 years. B’s present age is-

Solution:
Three years ago, the average of A, B, and C was 27 years old
total age 3 years ago = (27 × 3) = 81 years
So, at present their total age is = 81 + 3 + 3 + 3 = 90 years

 average of A and C, 5 years ago was 20 years
total age 5 years ago = (20 × 2) = 40 years
At present, total age of A and C is = 40 + 5 + 5 = 50 years

∴ B's present age is = 90 - 50 years = 40 years

Question: If the mean of a, b, c is 4 and ab + bc + ca = 0, then the mean of a², b² and c² is-

Solution:
Given that 
(a + b + c)/3 = 4
a + b + c = 12

We know,
(a + b + c)² = a² + b² + c² + 2 (ab + ba + ca)
⇒ (a + b + c)² = a² + b² + c² + 2 × 0
⇒ (a + b + c)² = a² + b² + c²
⇒ a² + b² + c² = 12²
⇒ a² + b² + c² =144

∴the mean of a², b² and c² is = ( a² + b² + c²)/3
= 144/3
= 48

Question: The average of ten number is 7. If each number is multiplied by 10, then the average of the new set of number is-

Solution:
let, 7 numbers are a₁, a₂, a₃,.......,a₇
so, (a₁ + a₂ + a₃+.......+a₇)/10 = 7
a₁ + a₂ + a₃+.......+a₇ = 70 

If each number is multiplied by 10, Then sum = (a₁ × 10) + (a₂ × 10) + (a₃ × 10) + .... + (a₇ × 10)
= 10 (a₁ + a₂ + a₃+.......+a₇)
= 10 × 70
= 700

then average will be = 700/10
= 70

Question: The average price of three items of furniture is Tk. 30000. If their prices are in the ratio 3 : 5 : 7, what is the price of the cheapest item?

Solution:
The average price of three items of furniture is Rs. 30000.
total price = (30000 × 3) = 90000
their prices are in the ratio 3 : 5 : 7

∴ the price of the cheapest item is = 90000 × 3/15
= 18000 tk

Question:  The average of 7 consecutive numbers is a. If the next two numbers are included, the average will be-

Answer:
let first number is n
so, the numbers are n, n+1, n+2,......, n+6
(n + n + 1 + .... + n + 6)/7 = a
⇒ 7n + (1 + 2 + 3 +....+ 6) = 7a
⇒ 7n + (6 × 7)/2 = 7a
⇒ 7n + 21 = 7a
⇒ 7n = 7a - 21
⇒ n = (7a - 21)/7
∴ n = 7(a - 3)/7 = a - 3

If the next two numbers are included, the average will be
= (n + n + 1 + .... + n + 8)/9
= {9n +(1 + 2 + 3 +.....+8)}/9
= (9n + 36)/9
= n + 4
= a - 3 + 4
= a + 1

So, the average will increase by 1

Question: A student was asked to find the arithmetic mean of the numbers 3, 12, 7, 9, 15, 13, 8, 18, 17, 21, 14 and x. He found the mean to be 12. What should be the number in place of x?

Solution:
the arithmetic mean of the numbers 3, 12, 7, 9, 15, 13, 8, 18, 17, 21, 14 and x 
= (3 + 12 + 7 + 9 + 15 + 13 + 8 + 18 + 17 + 21 + 14 + x )/12
= (137 + x)/12

∴  (137 + x)/12 = 12
⇒ 137 + x = 144
⇒ x = 7

Question: The average of first five multiples of 5 is-

Solution:
first five multiples of 5 are
= (5 × 1)
= (5 × 2)
= (5 × 3)
= (5 × 4)
= (5 × 5)

∴ The average of first five multiples of 5 is = {(5 × 1) + (5 × 2) + (5 × 3) +  (5 × 4) + (5 × 5)}/5
= 5 (1 + 2 + 3 + 4 + 5)/5
= 1 + 2 + 3 + 4 + 5
= 15

Question: The average of 4 terms is 20 and the last term is 1/3 of the remaining terms. What will be the last number?

Solution:
The average of 4 terms is 20 
total sum = (4 × 20) = 80

let, last term is x 
remaining terms is 3x

3x + x = 80
⇒ 4x = 80
∴ x = 20
The last number is 20

Question: The average monthly income of P and Q is Tk. 6,000; that of Q and R is 5,250; and, that P and R are Tk. 5,500. What is P’s monthly income?

Solution:  
Average monthly income of P and Q = Tk. 6000
Average monthly income of Q and R = Tk. 5250
Average monthly income of P and R = Tk. 5500

Total income of P + Q = Tk. 2 × 6000 = Tk. 12000 .........(i)
Total income of Q + R = Tk. 2 × 5250 = Tk. 10500 .........(ii)
Total income of P + R = Tk. 2 × 5500 = Tk. 11000 ...........(iii)

On adding equation (i) + (ii) + (iii), we get
2 (P +Q + R) = 12000 + 10500 + 11000
P + Q + R = 33500/2
P + Q + R = Tk 16750 ........(iv)
by equation (iv) - (ii)
P's income = (16750 - 10500) = Tk. 6250.

Question: The average age of 8 members of a group is 36 years. The age of the eldest member is 43 years. If we exclude that eldest member from the group, then what will be the average age (in years) of the remaining members?

Solution:
Total age = (36 × 8)
= 288
After excluding the eldest member, total age = 288 - 43
= 245 year

∴Now the average age of remaining members = 245/7
= 35 years

Question: The average ages of the players on team A and team B are 20 and 30 years, respectively. The average age of the players on the teams together is 26. If the total number of players on the two teams is 100, then how many players does team A have?

Solution:
Let, team A has x players and team B has y players.
total age of team A = (20 × x) = 20x year
total age of team B = (30 × y) = 30y year

he average age of the players on the teams together is 26
total age of the team = 26(x + y) 

So,
20x + 30y = 26(x + y) 
⇒ 20x + 30y = 26x + 26y
⇒ 6x = 4y
∴ y = 3/2 x

Again, x + y = 100
⇒ x + (3/2)x = 100
⇒ (2x + 3x) = 200
⇒  5x = 200
⇒ x = 40