Bank Math Master

Question: What will be the least number which when tripled will be exactly divisible by 8, 12, 15?

Solution: 
Prime factorization of, 
8 = 2 × 2 × 2
12 = 2 × 2 × 3
15 = 3 × 5

LCM = 2 × 2 × 2 × 3 × 5 = 120
So the smallest, 3n = 120
⇒ n = 120/3 
∴ n = 40

So the least number which when tripled will be exactly divisible by 8, 12, and 15 is 40.

Question: Two numbers are in the ratio 4 : 5 and their greatest common divisor (GCD) is 14. Find their least common multiple (LCM).

Solution:
Let the two numbers are,
First number = 4k
Second number = 5k
Since the numbers are in the ratio 4 : 5, and 4 and 5 are co-prime gcd(4,5) = 1,
the HCF of the two numbers is exactly k.

Given that, HCF = 14
Therefore, k = 14

So the actual numbers are:
First number = 4 × 14 = 56
Second number = 5 × 14 = 70

We know,
HCF × LCM = Product of the two numbers
⇒ 14 × LCM = 56 × 70
⇒ LCM = (56 × 70)/14
∴ LCM = 4 × 70 = 280

So the least common multiple (LCM) of the two numbers is 280.

Question: Which of the following is the largest?

Solution: 
ক) 9/12 = 0.75

খ) 10/15 = 0.66

গ) 5/8 = 0.62

ঘ) 7/10 = 0.70

So 9/12 = 0.75 is clearly the biggest.

Question: A cake is divided into 24 pieces. Rahim takes 1/4 of the cake. Karim takes 1/3 of the rest. How many pieces are left?

Solution:
Given that, 
A cake is divided into 24 pieces.

Now, 
Rahim takes 1/4 of the cake = 24 × (1/4) = 6 pieces
So Rahim takes 6 pieces.
∴ Remaining after Rahim = 24 - 6 = 18 pieces

And
Karim takes 1/3 of the rest = 18 × (1/3) = 6 pieces

∴ Pieces left after Karim = 18 - 6 = 12 pieces

So, 12 pieces are still left.

Question: (√6 + √6)² = ?

Solution: 
Given that, 
(√6 + √6)²
= (2√6)²
= 2² × (√6)²
= 4 × 6
= 24

Question: If p is odd and q is even, which expression is even?

Solution: 
We know, 
Odd + Even = Odd
And Odd × Even = Even

Now, let p = 3 and q = 4
ক) p + q = 3 + 4 = 7 ; Odd

খ) pq = 3 × 4 = 12 ; Even

গ) p + 2q + 1 = 3 + 2 × 4 + 1 = 4 + 8 = 12 ; Even

So, correct answer is Both b and c

Question: What is the minimum number of chocolates that must be added to an existing stock of 966 chocolates, so the total stock can be equally distributed among 6, 7, 8 or 9 person?

Solution: 
LCM of 6, 7, 8 or 9 is,
6 = 2 × 3
7 = 7
8 = 2 × 2 × 2
9 = 3 × 3
∴ LCM = 2³ × 3² × 7 = 8 × 9 × 7 = 504

Double of the chocolates = 2 × 504 = 1008
The chocolate to be added = 1008 - 966 = 42

So the minimum number of chocolates that must be added to the stock of 966 is 42.

Question: Which fraction is greater than 2/5 and less than 8/9?

Solution: 
2/5 = 0.40
8/9 = 0.88

Now,
4/9 = 0.44;  greater than 0.40, less than 0.88
3/4 = 0.75 → greater than 0.40, less than 0.88
5/12 = 0.41;  greater than 0.40, less than 0.88

All the given fractions lie between 2/5 and 8/9.

So correct answer: d) All of them 

Question: 


Solution: 

Question: The least number by which 180 must be multiplied to make it a perfect square is:

Solution: 
Prime factorization of 180 = 2 × 2 × 3 × 3 × 5
= 2² × 3² × 5¹
For a number to be a perfect square, every exponent in the prime factorization must be even.
2² ; even power 
3² ; even power 
5¹ ; odd power 
The only odd exponent is 5. To make it even, multiply by 5.
= 180 × 5
= 900
= 30²

So the least number by which 180 must be multiplied to make it a perfect square is 5.

Question: If the 5-digit number 750PQ is divisible by 3, 7 and 11, then what is the value of P + 2Q = ? 

Solution:
Given that,
Five-digit number 750PQ is divisible by 3, 7 and 11

Now, The LCM of 3, 7, and 11 is 231.

By taking the largest 5-digit number 75099 and dividing it by 231.
If we divide 75099 by 231 we get 325 as the quotient and 24 as the remainder.

Then, the five-digit number is 75099 - 24 = 75075.

The number = 75075 and P = 7, Q = 5

Now, P + 2Q = 7 + 2 × 5
= 7 + 10
= 17

∴ The value of P + 2Q is 17.

Question: Find the HCF of 4/9, 8/15, 12/25 is-

Solution:
We know,
HCF = (HCF of all numerators)/(LCM of all denominators)

Now, numerators: 4, 8, 12
4 = 2 × 2
8 = 2 × 2 × 2
12 = 2 × 2 × 3
∴ HCF = 2 × 2 = 4

And, denominators: 9, 15, 25
9 = 3 × 3
15 = 3 × 5
25 = 5 × 5
∴ LCM = 3 × 3 × 5 × 5 = 9 × 25 = 225

∴ HCF = (HCF of all numerators)/(LCM of all denominators) 
= 4/225

Question: {(2.39)² - (1.61)²}/(2.39 - 1.61) = ?

Solution: 
Let, 2.39 = a 
And 1.61 = b

Now, 
(a² - b²)/(a - b)
= (a + b)(a - b)/(a - b)
= a + b
= 2.39 + 1.61
= 4

Question: 

Solution: 

Question: The least perfect square number divisible by 3, 4, 5, 6 and 8 is:

Solution:
LCM of 3, 4, 5, 6, 8 is 120
Now, 120 = 2 × 2 × 2 × 3 × 5
To make it a perfect square, it must be multiplied by 2 × 3 × 5
So, required number
= 2² × 2² × 3² × 5²
= 4 × 4 × 9 × 25
= 3600

Question: If one-fifth of one-sixth of a number is 10, then what is 2/3 of that number?

Solution:
Let the number be x.
So one-sixth of the number = x/6

Now, one-fifth of one-sixth of the number = (1/5) × (x/6) = x/30

According to the question:
⇒ x/30 = 10
⇒ x = 10 × 30
∴ x = 300

∴ Now calculate 2/3 of the number = (2/3) × 300
= 2 × 100
= 200

Question: Four bells ring simultaneously at the start and then at intervals of 6 seconds, 12 seconds, 15 seconds, and 20 seconds respectively. How many times do they ring together in 2 hours?

Solution:
Given that,
Four bells ring simultaneously at starting and an interval of 6 sec, 12 sec, 15 sec and 20 sec respectively.

∴ LCM of (6, 12, 15, 20) = 60
All 4 bells ring together again after every 60 seconds

Now, In 2 Hours, they ring together = [(2 × 60 × 60)/60] times + 1 (at the starting)
= (120 + 1) times
= 121 times

∴ In 2 hours they ring together for 121 times

Question: What decimal of an hour is a second ?

Solution: 
There are 60 seconds in a minute and 60 minutes in an hour.
So, 1 hour = 60 × 60 = 3600 seconds
Therefore, 1 second = 1/3600 = 0.00027 of an hour.

Question: A man has Tk. 480 in the denominations of one-taka notes, five-taka notes and ten-taka notes. The number of notes of each denomination is equal. What is the total number of notes that he has ?

Solution: 
Let number of notes of each denomination be x.
Then x + 5x + 10x = 480
⇒ 16x = 480
∴ x = 30.

Hence, total number of notes = 3x = 90

Question: 5 - [4 - {3 - (3 - 3 - 6)}] is equal to:

Solution: 
Given that,
5 - [4 - {3 - (3 - 3 - 6)}]
= 5 - [4 - {3 - (-6)}]
= 5 - [4 - {3 +6}]
= 5 - [4 - {9}]
= 5 - [4 - 9]
= 5 - [-5]
= 5 + 5
= 10