ব্যাংক ডেইলি কুইজ [লং কোর্সের অংশ]

Question: The average (arithmetic mean) of 10, 30, and 50 is 5 more than the average of 20, 40, and ?

Solution:
The average of 10, 30, and 50 is = (10 + 30 + 50)/3 = 30
The average of 20, 40 and x must be 30 - 5 = 25

ATQ,
(20 + 40 + x)/3 = 25
⇒ 60 + x = 75
∴ x = 15

Question: The ratio of the current ages of Momen and his brother is 15:14. The ratio of the ages of Momen and his sister in 6 years will be 18:19. The gap between Momens age and his sister is the same as the gap between Momens age and his brother. What is Momens current age?

Solution:
Let's say Momen's current age is x
From the first ratio (Momen : Brother = 15 : 14) If Momen's age is x, then his brother's age is (14x/15)
Let's say Momen's sister's age is y

Given that the gap between Momen and sister is same as gap between Momen and brother
|x - y| = |x - (14x/15)|
|x - y| = |x - 14x/15|
|x - y| = |15x/15 - 14x/15|
|x - y| = x/15  ......  (1)

After 6 years, the ratio of Momen:Sister will be 18:19
(x + 6)/(y + 6) = 18/19
19(x + 6) = 18(y + 6)
19x + 114 = 18y + 108
19x = 18y - 6
y = (19x + 6)/18 .......... (2)


From equations 1 and 2 ⇒ 
x - y = x/15 (since y is less than x)
x - (19x + 6)/18 = x/15
18x - 19x - 6 = 18x/15
-x - 6 = 18x/15
-15x - 90 = 18x
-33x = 90
x = 30

Therefore, Momen's current age is 30 years.

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. Determine how old Q is right now.

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,

we can write,
(3x - 5) : (4x - 5) = 5 : 7
⇒ (3x - 5)/(4x - 5) = 5/7
⇒ 21x - 35 = 20x - 25
⇒ x = 10

∴ their present age of Q = 4X = 40

Question: Five years ago, Rahul was half as old as Maya. If Rahul is 15 years old now, how old is Maya now?

Solution:
Let,
Age of Maya now = x years
ATQ,
(15 - 5)/2 = (x - 5)
⇒ 5 = (x - 5)
⇒ x = 10

∴ The age of Maya now is 25 years

Question: Find the average of the first 24 consecutive natural numbers.

Solution:
The average of the first n consecutive natural numbers is
=(n + 1)/2

So, average = (24 + 1)/2 [Here, n=24]
= 25/2
= 12.5

Question: A number p equals 3/2 the average of 10, 12, and q. What is q in terms of p?

Solution:
Average = (10 + 12 + q)/3
= (22 + q)/3

Now, p is 3/2 the average
So, p = (3/2) × {(22 + q)/3}
⇒ p = (22 + q)/2
⇒ 22 + q = 2p
∴ q = 2p - 22

Question: The ratio between the present ages of A and B is 3 : 5. If the ratio of their ages five years hence becomes 13 : 20, then the present age of B is:

Solution:
Let's say A's present age is 3x and B's present age is 5x
After 5 years,
A's age will be = 3x + 5
B's age will be = 5x + 5

ATQ,
(3x + 5)/(5x + 5) = 13/20
⇒ 20(3x + 5) = 13(5x + 5)
⇒ 60x + 100 = 65x + 65
⇒ 60x - 65x = 65 - 100
⇒ - 5x = - 35
∴ x = 7

Since B's present age = 5x = 5 × 7 = 35 years

Question: The captain of a cricket team of 11 members is 26 years old, and the wicket-keeper is three years older than the captain. If the ages of captain and wicketkeeper are excluded, the average age of the remaining players of the team is one year less than the average age of the whole team. What is the average age of the team?

Solution:
Let the average age of the whole team by x years.
Total age of the whole team 11x years
Age of the captain 26 years
Age of the wiket-keeper 26 + 3 = 29 years

The average age of the remaining players after excluding the ages of captain and wicketkeeper = x - 1
∴ Total age of the players without the ages of captain and wicketkeeper = 9(x - 1)

We can say,
11x - (26 + 29) = 9(x - 1)
⇒ 11x - 55 = 9x - 9
⇒ 11x - 9x = - 9 + 55
⇒ 2x = 46
∴ x = 23 Years.

Question: The mean of five numbers is 28. If one of the numbers is excluded, the mean gets reduced by 2. Find the excluded number.

Solution:
Mean of 5 numbers = 28.
Sum of these 5 numbers = (28 × 5) = 140.

Mean of the remaining 4 numbers = (28 - 2) =26.
Sum of these remaining 4 numbers = (26 × 4) = 104.

Excluded number
= (sum of the given 5 numbers) - (sum of the remaining 4 numbers)
= (140 - 104)
= 36. 

Hence, the excluded number is 36.

Question: The average salary of 30 officers in a firm is Tk 120 and the average salary of laborers is Tk 40. Find the total number of laborers if the average salary of the firm is Tk 50.

Solutions:
The sum of the salary of officers will be = 30 × 120 = 3600
Let the number of laborers = x
ATQ,
3600 + 40x = 50(30 + x)
⇒ 3600 + 40x = 1500 + 50x
⇒ 2100 = 10X
⇒ x = 210

Question: The average of Amit's five quiz scores is 88. What score does Amit need to get on a sixth quiz to raise his average for all six quizzes to 90?

Solution:
Sum of 5 scores = 88 × 5 = 440
Sum of 6 scores = 90 × 6 = 540
Sixth quiz score = 540 - 440 = 100

Question: Father is aged three times more than his son Ronit. After 8 years, he would be two and a half times of Ronit's age. After further 8 years, how many times would he be of Ronit's age?

Solution:
Let,
Ronit's present age be x years.
Then, father's present age =(x + 3x) years
= 4x years.

ATQ,
(4x + 8) = {5(x + 8)}/2
⇒ 8x + 16 = 5x + 40
⇒ 3x = 24
∴ x = 8

Now, (4x + 16)/(x + 16) = 48/24 = 2
∴ Father would be 2 times of Ronit's age.

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

Solution:
Average of 4 terms = 20
Hence, the total sum of 4 terms = 80
Let terms be A, B, C & D

So,
The sum will be A + B + C + D =80
Given, 3A = B + C + D
So, 4A = 80,
A = 20

Question: The age of a man is 4 times of his son. Five years ago, the man was nine times old as his son was at that time. The present age of man is?

Solution:
Let the son's age be x years and the father's age be 4x years

ATQ,
4x - 5 = 9(x - 5)
⇒ 4x - 5 = 9x - 45
⇒ 5x = 40
∴ x = 8

∴ Present age of the father = 4 × 8 = 32 years

Question: If the average of three consecutive even numbers is 34, find the smallest of these numbers.

Solution:
Let the first number is x, then the next two even numbers would be x + 2, x + 4

As per question;
(x + x + 2 + x + 4)/3 = 34
⇒ (3x + 6)/3 = 34
⇒ 3x + 6 = 102
⇒ 3x = 96
∴ x = 32

Smallest number would be = 32