Bank Math Master

Question: What will be the least number which when doubled will be exactly divisible 14, 18, 21, 28?

Solution:
Given the numbers 14, 18, 21, 28
First, factorize the numbers,
14 = 2 × 7
18 = 2 × 3 × 3
21 = 3 × 7
28 = 2 × 2 × 7

∴ LCM is = 2 × 2 × 3 × 3 × 7 = 252
So the number will be = 252/2 = 126
The least number is 126

Question: Which of the following is the largest?

Solution:
14/18 = 7/9 = 0.78
15/25 = 3/5 = 0.6
3/4 = 0.75
12/14 = 6/7 = 0.86

Question: The ratio of two numbers is 5 : 8 and their H.C.F is 4. Their L.C.M is-

Solution:
Let, the numbers are 5x and 8x 
And given H.C.F is x = 4

We know,
⇒ L.C.M × H.C.F = Product of numbers
⇒ L.C.M × 4 = 5x × 8x 
⇒ L.C.M × 4 = 5 × 4 × 8 × 4
⇒ L.C.M = 5 × 4 × 8 = 160

∴ The L.C.M of the two numbers is 160.

Question: A cake is divided into 24 pieces. If Kamal takes 1/4 of the cake and Jamal takes 1/3 of the rest that are left, how many pieces are still left?

Solution:
The cake is divided into 24 pieces, and Kamal takes 1/4 of the cake is = (1/4) × 24 = 6 pieces
After Kamal takes 6 pieces, there are,
24 − 6 = 18 pieces.

Jamal takes 1/3 of the remaining pieces,
(1/3) × 18 = 6 pieces.
So Jamal takes 6 pieces.

So the cake are left = 24 - 6 - 6 = 12 pieces.

∴ After Kamal and Jamal take their portions, 12 pieces of the cake are still left.

Question: 
= ?

Solution:

Question: The average of the first six multiples of 3 is-

Solution:
The first six multiples of 3 are,
3, 6, 9, 12, 15, 18

The sum of these multiples is,
3 + 6 + 9 + 12 + 15 + 18 = 63

∴ Average = 63/6​ = 10.5
The average of the first six multiples of 3 is 10.5

Question: Which of the following is divisible by 3?

Solution:
We have to simply find the sum of the digits for each option and check if the sum is divisible by 3....

34616 = 3 + 4 + 6 + 1 + 6 = 20
46904 = 4 + 6 + 9 + 0 + 3 = 23
10984 = 1 + 0 + 9 + 8 + 4 = 22
14589 = 1 + 4 + 5 + 8 + 9 = 27

Hence we can see that only 27 is divisible by 3. Thus, 14589 is divisible by 3.

Question: If a/b = 5/4, then (4a + 3b)/(4a - 3b) =?

Solution:

Question: x^1/2/27 = 12/x^3/2, What is the value of x is-

Solution:
Given that,
⇒ x^1/2/27 = 12/x^3/2
⇒ x^1/2 × x^3/2 = 27 × 12
⇒ x^{{(1/2) + (3/2)}} = 324
⇒ x^4/2 = 324
⇒ x² = 324
⇒ x = √324
∴ x = 18

Question: (5√5)³ =?

Solution: 
Given that,
= (5√5)³
= 5√5 × 5√5 × 5√5
= (5 × 5 × 5)(√5 × √5 × √5)
= 125 × 5√5
= 625√5

Question: The difference between the present ages of Arun and Ashok is 12 years. Seven years ago, the ratio of their ages was 6 : 8 respectively. What is the Ashok's present age?

Solution:
Let, 7 years ago
Arun's age was 6x and Ashok's age was 8x

At present,
Arun's age is (6x + 7) years and Ashok's age is (8x + 7) years

According to the question,
⇒ (8x + 7) - (6x + 7) = 12
⇒ 8x + 7 - 6x - 7 = 12
⇒ 2x = 12
⇒ x = 12/2
⇒ x = 6

∴ Ashok's present age = 8x + 7 = 8 × 6 + 7 = 55 years

Question: The value of log₂{log₄(log₄256⁴)} =?

Solution:
= log₂{log₄(log₄256⁴)}
= log₂{log₄(log₄(4⁴)⁴}
= log₂{log₄(16log₄4)}
= log₂{log₄4²}
= log₂2(log₄4)
= log₂2
= 1
∴ The value of the expression is 1.

Question: The least number, Which when divided by 15, 20, 30, 40 leaves in each case a remainder of 6 is-

Solution:
We first find the LCM of 15, 20, 30, and 40 by factoring them,
15 = 3 × 5
20 = 2 × 2 × 5
30 = 2 × 3 × 5
40 = 2 × 2 × 2 × 5

So, the LCM is = 2 × 2 × 2 × 3 × 5 = 120

∴ The least number is = 120 + 6 = 126

Question: The least number by which 320 must be multiplied to make it perfect square, is-

Solution: 
Here,  
320 = 2 × 2 × 2 × 2 × 2 × 2 × 5 = 2⁶ × 5
To make it perfect squre, it must be multiplied by 5 

Therefore, the least number by which 320 must be multiplied to make it a perfect square is 5.

Question: The average of fourteen numbers is 20 and the average of the first eight is 16. What is the average for the rest?

Solution:
The average of the fourteen numbers is = 20
Sum of 14 numbers = 14 × 20 = 280

The average of the first 8 numbers is = 16
Sum of first 8 numbers = 8 × 16 = 128

Total of remaining six numbers = 280 - 128 = 152
Average of the rest = 152/6 = 25.33

Question: The ages of Moni and Roni are in the ratio 6 : 5 respectively. After 10 years, the ratio of their ages will be 8 : 7. What is the difference in their ages now?

Solution:
Let the present ages of Moni and Roni be 6x and 5x respectively, where x is a common factor.

After 10 years, the ratio of their ages will be 8 : 7. then,
⇒ (6x + 10) : (5x + 10) = 8 : 7
⇒ 6x + 10/5x + 10 = 8/7
⇒ 7(6x + 10) = 8(5x + 10)
⇒ 42x + 70 = 40x + 80
⇒ 42x - 40x = 80 - 70
⇒ 2x = 10
⇒ x = 10/2
∴ x = 5

Now,
Moni's age = 6x = 6 × 5 = 30 years
Roni's age = 5x= 5 × 5 = 25 years

∴ The difference in their ages now = 30 − 25 = 5 years

Question: If 3√3 × 3³ ÷ 3^1/3 = 3^a/6 then the value of a is-

Solution:
Given that,
⇒ 3√3 × 3³ ÷ 3^1/3 = 3^a/6
⇒ 3¹ × 3^1/2 × 3³  ÷ 3^1/3 = 3^a/6
⇒ 3^{{1 + (1/2) + 3 - (1/3)}} = 3^a/6
⇒ {1+ (1/2) + 3 - (1/3)} = a/6
⇒ (6 + 3 +  18 - 2 )/6 = a/6
⇒ 25/6 = a/6
∴ a = 25

Question: If a and b are even numbers, Which number is odd?

Solution:
We know,
even × even = even
even + even + 2 = even
even + even = even
even × even - 1 = odd

Question: The difference of 13/12 and its reciprocal is equal to-

Solution:
The reciprocal of 13/12​ is 12/13
Now, calculate the difference between 13/12​ and 12/13
= (13/12)​ - (12/13)
= (169 - 144)/156
= 25/156

∴ The difference between 13/12​ and its reciprocal is 25/156.

Question: log₁₀(26/51) - log₁₀(13/32) + log₁₀(119/91) - log₁₀(64/39) is equal to-

Solution:
= log₁₀(26/51) - log₁₀(13/32) + log₁₀(119/91) - log₁₀(64/39)
= log₁₀(26/51)+ log₁₀(119/91) - log₁₀(13/32)  - log₁₀(64/39)
= log₁₀{(26/51) × (119/91)} - { log₁₀(13/32)  + log₁₀(64/39)}
= log₁₀2/3 - log₁₀{(13/32) × (64/39)}
= log₁₀(2/3) - log₁₀(2/3)
= 0