Bank Math Master

Question: The value of (32)^0.08 × (32)^0.12 is -

solution:
Given that,
(32)^0.08 × (32)^0.12
= (32)^{0.08 + 0.12}
= (32)^0.2
= (2⁵)^2/10
= 2^10/10
= 2¹
= 2
∴ the value is 2

Question: Three numbers are in the ratio 2 : 5 : 8 and their average is 75. What is the difference between larger and smaller numbers?

Solution:
Let the three numbers are 2x, 5x, and 8x, where x is a constant multiplier.
The average of the three numbers is given as 75.
The average of the three numbers is,
⇒ (2x + 5x + 8x)/3 = 75
⇒ 15x/3 = 75
⇒ 5x = 75
⇒ x = 75/5
∴ x = 15
The first number is = 2x = 2 × 15 =30
The third number is = 8x = 8 × 15 = 120

∴ The difference between the larger and smaller numbers is, 120 − 30 = 90

Question: The ratio of the ages of a father and his son is 8 : 3 respectively. Six years ago, the ratio of their ages was 3 : 1 respectively. What was the father's present age?

Solution:
ধরি, পিতার বর্তমান বয়স = 8x বছর
এবং পুত্রের বর্তমান বয়স = 3x বছর

ছয় বছর আগে,
পিতার বয়স = 8x − 6 বছর
পুত্রের বয়স = 3x − 6 বছর

প্রদত্ত শর্ত,
(8x − 6)/(3x − 6) = 3/1
9x - 18 = 8x - 6
9x - 8x = 18 - 6
x = 12

∴ পিতার বর্তমান বয়স = 8x = 8 × 12 = 96 বছর

Question: log 2 + log 4 + log 8 + ............ Find the sum of first 15th term-

Solution:
given that,
log 2 + log 4 +log 8 + ............
= log 2 + log2² +log 2³ + ............
= log 2 + 2 log 2 + 3 log 2 + ............
= (1 + 2 + 3 + .........) log 2
The sum of the first 15 natural numbers is given by the formula:
Sum = n(n+1)/2
where n = 15
∴ Sum = 15(15 + 1)/2
= 15 × 8
= 120

So, the sum of the first 15 terms is: 120 log 2

Question: If the sum of (8, 12, 13, x) is 48, then the averahe of (8, 12, 13, x) is-

Solution:
Sum of the numbers are,
⇒ 8 + 12 + 13 + x = 48
⇒ x = 48 - 33
⇒ x = 15

The average of these four numbers is the sum of the numbers divided by 4,
Average = (8 + 12 + 13 + 15)/4
= 48/4
= 12
Thus, the average of the numbers 8, 12, 13, 15 is 12.

Question: Find the average of all the numbers between 7 and 46 which are divisible by 5 -

Solution:
First, let's list all the numbers between 7 and 46 that are divisible by 5.
So, the numbers divisible by 5 in this range are,
10, 15, 20, 25, 30, 35, 40, 45 

∴ The sum of these numbers is,
10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 = 220

∴ Average = 220/​8 = 27.5
Thus, the average of all numbers between 7 and 46 that are divisible by 5 is 27.5.

Question:2 log 4 - 3 log 2 + (1/3) log 1000 =?

Solution:
Given that,
2 log 4 - 3 log 2 + (1/3) log 1000 
= log 4² - log 2³ + log (10³)^1/3
= log 16 + log 10 - log 8
= log (16 × 10)/8
= log 20

Question: The difference of two numbers is 25% of the larger number. If the smaller number is 15, the larger one is-

Solution:
let,
the number = x
According to the question,
⇒ x - 15 = 25% of x
⇒ x - 15 = 25x/100
⇒ x - 15 = x/4
⇒ 4x - 60 = x
⇒ 4x - x = 60
⇒ 3x = 60
⇒ x = 60/3
∴ x = 20

So, the larger number is 20.

Question: A couple has a son and a daughter. The age of the father is five times that of the son and the age of the daughter is one-third of that of her mother. The wife is 8 years younger to her husband and the sister is 4 years older than her brother. The mother's age is?

Solution:
ধরি,
ছেলের বয়স = x বছর
পিতার বয়স  = 5x বছর

আবার, ধরি মায়ের বয়স = y বছর
মেয়ের বয়স = y/3 বছর

প্রশ্নমতে,
5x - y = 8 ........... (1)
এবং,
⇒ y/3 - x = 4
⇒ y/3 = x + 4
∴ y = 3x + 12 ...........(2)

(1) নং হতে পাই,
⇒ 5x - y = 8
∴ y = 5x - 8
এখন y এর মান (2) নং এ বসিয়ে পাই
⇒ 5x - 8 = 3x + 12
⇒ 5x - 3x = 12 + 8
⇒ 2x = 20
⇒ x = 20/2
∴ x = 10
∴ পিতার বয়স  = 5x = 5 × 10 = 50 বছর
∴ মায়ের বয়স = 50 - 8 = 42 বছর

Question: What is the value of [log₅(5log₅3125)]² ?

Solution:
Given that,
= [log₅(5log₅3125)]²
= [log₅(5log₅5⁵)]²
= [log₅(5 × 5log₅5)]²
= [log₅25]²  ; [ log₅5 = 1 ]
= [log₅5²]²
= [2log₅5]²
= 2²
= 4

Question: The difference between two integers is 6. If their product is 352, then the sum of two numbers is?

Solution:
Let the two integers be x and y
The difference between the two integers is 6,
∴ x - y = 6 and their product is, xy = 352

Now, We know
⇒ (x + y)² = (x - y)² + 4xy
⇒ (x + y)² = 6² + 4 × 352
⇒ (x + y)² = 36 + 1408
⇒ (x + y)² = 1444
⇒ x + y = √1444
∴ x + y = 38

∴ The sum of the two numbers is 38.

Question: The average of 4 positive integers 60. The highest integer is 83 and the lowest integer is 29. The difference between the remaining two integers is 28. Which of the following integers is higher of the remaining two integers?

Solution:
The average of 4 positive integers 60.
The sum of 4 positive integers is = 60 × 4 = 240
Given the two numbers are 83 and 29.
The difference between the remaining two integers is 28

Now,
Let higher integer number is x and other is (x - 28)

According to the question,
⇒ 83 + 29 + x + x - 28 = 240
⇒ 2x = 240 + 28 - 83 - 29
⇒ 2x = 156
⇒ x = 156/2
∴ x = 78
Thus, the higher of the remaining two integers is 78.

প্রশ্ন: x^2/3/45 = 4/x^1/3 , what is the value of x?

Solution:
Given that,
⇒ x^2/3/45 = 4/x^1/3
⇒ x^2/3 × x^1/3 = 4 × 45
⇒ x^{2/3 + 1/3} = 180
⇒ x^{(2 + 1)/3} = 180
⇒ x^3/3 = 180
∴ x = 180

Question: Three-fourth of a number is 70 more than its one-third. The number is -

Solution:
let,
the number = x
According to the question,
⇒ 3​x/4 = 1​x/3 + 70
⇒ 3​x/4 - 1​x/3 = 70
⇒ (9x - 4x)/12 = 70
⇒ 5x/12 = 70
⇒ 5x = 70 × 12
⇒ x = (70 × 12)/5
⇒ x = 14 × 12
∴ x = 168

Question: If log₈p = 25 and log₂q = 5 than-

Solution:
Given that,
⇒ log₈p = 25 and log₂q = 5
⇒ p = 8²⁵ and q = 2⁵
⇒ p = (2³)²⁵ and q = 2⁵ .........(1)
⇒ p = 2⁷⁵
⇒ p = (2⁵)¹⁵
∴ p = q¹⁵  ; [ From equation 1 ]

Question: log_3√3 27 = x², x =?

Solution:

∴ x = √2

Question: A, B, C and D are four consecutive even numbers respectively and their average is 75. What is the product of A and C?

Solution:
Let's define the four consecutive even numbers as A, B, C, and D. The average of these numbers is given as 75.
Than the sum is,
75 × 4 = 300
Now let the 1^st number is x. The next three will be x + 2, x + 4, x + 6

According to the question,
⇒ x + x + 2 + x + 4 + x + 6 = 300
⇒ 4x + 12 = 300
⇒ 4x = 300 - 12
⇒ 4x = 288
⇒ x = 288/4
∴ x = 72
So the 1^st number is A = 72
The next three numbers are B = 72 + 2 = 74, C = 72 + 4 = 76, D = 72 + 6 = 78

∴ Finally, the product of A and C is,
A × C = 72 × 76 = 5472
So, the product of A and C is 5472.

Question: (x/y)^{2p + 4} = (y/x)^{8 + p} , Find P =?

Solution:
Given that,
⇒ (x/y)^{2p + 4} = (y/x)^{8 + p}
⇒ (x/y)^{2p + 4} = {(x/y)^{- 1}}^{8 + p}
⇒ (x/y)^{2p + 4} = (x/y)^{- 8 - p}
⇒ 2p + 4 = - 8 - p
⇒ 2p + p = - 8 - 4
⇒ 3p = - 12
⇒ p = - 12/3
∴ p = - 4

Question: Jamil's average score in 4 tests was 85 out of a possible 100. If his scores in 2 of the tests were 70, and 80. What is the lowest that either of his other scores could have been?

Solution:
Average of 4 test was 85
Total marks = 85 × 4 = 340

Marks of other 2 subjects = (340 - 70 - 80)= 290
He can get highest 100 marks in one of the subjects.

∴ Lowest marks will be = (290 - 100) = 190