Bank Math Master

Question: Present ages of Rakib and Tanvir are in the ratio of 7 : 5 respectively. Two years hence, the ratio of their ages will become 15 : 11 respectively. What is Rakib's present age in years?

Solution:
Let the present ages of Rakib and Tanvir be 7x and 5x respectively.
ATQ,
(7x + 2)/(5x + 2) = 15/11
⇒ 77x + 22 = 75x + 30
⇒ 2x = 8
⇒ x = 4

So, Rakib's present age = 7 × 4 = 28 years

Question: If log_n2 = p and log_n5 = q, then log_n250 = ?

Solution:
Given, 
log_n2 = p and log_n5 = q

∴ log_n250 = log_n(2 ×125)
= log_n(2 × 5³)
= log_n2 + log_n5³
= log_n2 + 3 log_n5
= p + 3q

Question: If doubling a number and adding 16 to the result gives the same answer as multiplying the number by 8 and taking away 8 from the product, the number is:

Solution:
Let the number be a
ATQ,
2a + 16 = 8a - 8
⇒ 16 + 8 = 8a - 2a
⇒ 6a = 24
∴ a = 4

Question: If log_a324 = 4, what is the value of the base?

Solution:
log_a324 = 4
⇒ a⁴ = 324
⇒ a⁴ = 4 × 81
⇒ a⁴ =2² × 3⁴
⇒ a⁴ = ((√2)²)² × 3⁴
⇒ a⁴ = (√2)⁴ × 3⁴
⇒ a⁴ = (3√2)⁴
∴ a = 3√2

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 is 32, how old is A?

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

∴ (2a + 2) + 2a + a = 32
⇒ 5a = 30
⇒ a = 6
Hence, A's age = (2 × 6) + 2 = 14 years.

Question: If (5^{3x - 5} b^{2x - 6})/5^{x + 1} = a^{2x - 6} where a>0, b>0 and 5b ≠ a then what is the value of x?

Solution:
(5^{3x - 5} b^{2x - 6})/5^{x + 1} = a^{2x - 6}
⇒ 5^{3x - 5}/5^{x + 1} = a^{2x - 6}/b^{2x - 6}
⇒ 5^{3x - 5 - x - 1} = (a/b)^{2x - 6}
⇒ 5^{2x - 6} = (a/b)^{2x - 6}
⇒ 5^{2x - 6}/(a/b)^{2x - 6} = 1
⇒ {5/(a/b)}^{2x - 6} = {5/(a/b)}⁰  [∵ {5/(a/b)}⁰ = 1]
⇒ 2x - 6 = 0
⇒ 2x = 6
∴ x = 3
 

Question: The average age of a group of 15 people is 30 years. If the age of each person in the group is decreased by 5 years, what will the new average age be?

Solution:
Here,
Sum of the age/Total number of people = 30
⇒ Sum of the age/ 15 = 30
∴ Sum of the age = 450

Decrease each age by 5 years, the total decrease = 5 × 15 = 75
New sum = 450 - 75 = 375
Therefore, the new average age = 375/15 = 25 years

Question: The value of log₅ (125 × 625)/25 is equal to-

Solution:
log₅ {(125 ⋅ 625)/25}
= log₅ {(5³ ⋅ 5⁴)/5²}
= log₅ 5^{3 + 4 - 2}
= log₅ 5⁵
= 5 log₅ 5
= 5 ⋅ 1
= 5

Question: Topu's age is (1/6)th of her father's age. Topu's father's age will be twice of Arif's age after 10 years. If Arif's 8th birthday was celebrated 2 years ago,then what is Topu's present age?

Solution:
Arif's present age
= (8 + 2) years
= 10 years
After 10 year's Topu's father's age = 2(10 + 10) = 40 years
Topu's father's current age = 40 - 10 = 30 years
Topu's age = (1/6) × 30 = 5 years

Question: (0.04)^{- 1.5} = ?

Solution:
(0.04)^{- 1.5} 
= (4/100)^{- 1.5}
= (1/25)^{- 1.5}
= (1/25)^{- (3/2)}
= (25)^3/2
= (5²)^3/2
= 5^{2 × (3/2)}
= 5³
= 125

Question: The sum of five consecutive numbers is 85. What is the difference between thrice the smallest number and twice the largest number?

Solution:
Let the smallest number be x
Then the five consecutive numbers are: x, x + 1, x + 2, x + 3, x + 4
ATQ,
x + (x + 1) + (x + 2) + (x + 3) + (x + 4) = 85
⇒ 5x + 10 = 85
⇒ 5x = 75
⇒ x = 15

∴ Difference between thrice the smallest number and twice the largest number
= (x × 3) - {(x + 4) × 2}
= (15 × 3) - {(15 + 4) × 2}
= 7

Question: What is the value of log_√3 81?

Solution:
log_√3 81 
= log_√3 3⁴
= 4 log_√3 3
= 4 log_√3 (√3)²
= 2 ⋅ 4 log_√3 √3
= 2 ⋅ 4 ⋅ 1
= 8

Question: The average temperature for Wednesday, Thursday and Friday was 40°C. The average for Thursday, Friday and Saturday was 41°C. If temperature on Saturday was 42°C, what was the temperature on Wednesday?

Solution:
Here,
Total temperature for Wednesday, Thursday and Friday = 3 × 40° = 120°C
Total temperature for Thursday, Friday and Saturday = 3 × 41° = 123°C

Now,
(Thursday + Friday + Saturday) - (Wednesday + Thursday + Friday) = 123° - 120°
⇒ Saturday - Wednesday = 3°
∴ Wednesday = 42° - 3° = 39°C

Question: If √2 = 1.414, the value of 2√32 - 3√128 + 4√50 = ?

Solution:
2√32 - 3√128 + 4√50
= 2 ⋅ 4√2 - 3 ⋅ 8√2 + 4 ⋅ 5√2
= 8√2 - 24√2 + 20√2
= 4√2
= 4 × 1.414
= 5.656

Question: If log_a (1/16) = - 2, then what is the value of a?

Solution:
log_a (1/16) = - 2
⇒ a^-2 = 1/16
⇒ a^-2 = 1/4²
⇒ a^-2 = 4^-2
∴ a = 4

Question: The difference between two numbers is 16. If one-third of the smaller number is greater than one-seventh of the larger number by 4, then the two numbers are:

Solution:
Let the numbers be p and (p + 16)
ATQ,
(p/3) - {(p + 16)/7} = 4
⇒ (7p - 3p - 48)/21 = 4
⇒ 4p = 81 + 48
⇒ p = 132/4
∴ p = 33

So, the numbers are 33 and (33 + 16) = 49

Question: A man was asked to state his age in years. His reply was, "Take my age 3 years hence, multiply it by 3 and then subtract 3 times my age 3 years ago, and you will know how old I am". What is the age of the man?

Solution:
Let the age of the man be = p years
Then,
⇒ 3(p + 3) - 3(p - 3) = p
⇒ 3p + 9 - 3p + 9 = p
⇒ p = 18
∴ The age of the man is 18 years.

Question: log2 + log4 + log8 + ................. Find the sum of first 7 term.

Solution:
log2 + log4 + log8 + .................+ first 7 term
= log2 + log 2² + log 2³ + ................. + first 7 term
= log2 + 2log2 + 3log2 + ................. + first 7 term
= (1 + 2 + 3 + ...... + 7)log2
=   [{7(7 + 1)}/2] × log2
=  28log2 

Question: a^{(x - y)(x + y)} × a^{(z - x)(z + x)} × a^{(y - z)(y + z)} = ?

Solution: