Bank Math Master

Question: If 283M456 is divisible by 3, what is the value of M? 

Solution:
 একটি সংখ্যা ৩ দ্বারা বিভাজ্য হবে যদি সংখ্যাটির অঙ্কগুলোর সমষ্টি ৩ দ্বারা বিভাজ্য হয়। 

২ + ৮ + ৩ + ৪ + ৫ + ৬ = ২৮; এর সাথে ২ যোগ করলে ৩০ হয়, যা ৩ দ্বারা বিভাজ্য।
∴ M = ২

Question: Three numbers are in the ratio of 3 : 4 : 5 and their H.C.F. is 20. Their L.C.M. is -

Solution: 
সংখ্যা তিনটির গ. সা. গু ২০

সংখ্যা তিনটি হচ্ছে (৩ × ২০) বা ৬০, (৪ × ২০) বা ৮০, (৫ × ২০) বা ১০০ 

৬০ = ৩ × ২ × ২ × ৫ 
৮০ = ২ × ২ × ২ × ২ × ৫
১০০ = ২ × ২ × ৫ ×৫

ল. সা. গু = ২ × ২ × ২ × ২ × ৩ ×  ৫ × ৫ 
= ১২০০ 

প্রশ্ন:  If 8 × 2^x = (1/16), what is the value of x? 

সমাধান: 
8 × 2^x = (1/16)
⇒ 2^x = (1/16 × 8)
⇒ 2^x = 1/(2⁴ × 2³)
⇒ 2^x = 1/2⁷
⇒ 2^x = 2 ^{- 7}
∴ X = -7

Question: The ages of Anish and Belal are in the ratio 6 : 5 and the sum of their ages is 44 years. What will be the ratio of their ages after 10 years ?

Solution:
The ages of Anish and Belal are in the ratio 6 : 5 
let, Anish is 6x years old and belal is 5x years old
5x + 6x = 44
⇒ 11x = 44
∴ x = 4 years old

So, Anish is 24 years old and belal is 20 years old

After 10 years, Anish is (24 + 10) = 34 years old and belal is (20 + 10) years or 30 years old
∴ their ratio = 34 : 30
= 17 : 15

Question: The average of the first five multiples of 13 is-

Solution: 
The  first five multiples of 13: (13 × 1), (13 × 2), (13× 3), (13 × 4), (13 × 5)
their sum = (13 × 1) + (13 × 2) + (13 × 3) + (13 × 4) + (13 × 5)
= 13 (1 + 2 + 3 + 4 + 5)
= 13 × 15

∴ average = (13 × 15)/5
= 39

প্রশ্ন:

সমাধান: 

প্রশ্ন: If 5^a = 125, then 5^{a - 3} =?

সমাধান: 
5^a = 125
⇒ 5^a 5 ^-3 = 5³.5^-3
⇒  5 ^{a - 3} = 1

Question: Three times the first of three consecutive odd integers is 3 more than twice the third. The first integer is:

Solution: 
ধরি, প্রথম বিজোড় সংখ্যা x 
দ্বিতীয় বিজোড় সংখ্যা x + 2
তৃতীয় বিজোড় সংখ্যা x + 4

প্রশ্নমতে, 
3x = 2(x + 4) + 3
⇒ 3x = 2x + 8 + 3
⇒ x = 11

প্রথম বিজোড় সংখ্যা ১১।

Question: What will be the least number which when halved will be exactly divisible by 12, 18, 21 and 30?

Solution:
12 = 2 × 2 × 3
18 = 2 × 3 × 3
21 = 3 × 7
30 = 2 × 3 × 5

L.C.M. of 12, 18, 21 30 = 2 × 3 × 2 × 3 × 7 × 5 = 1260
Required number = 1260 × 2 = 2520 

Question: If 32 × 4 ^{a + 2} = 2 ^b, then b =?

Solution: 
32 × 4 ^{a + 2} = 2 ^b
⇒ 2⁵ × 2^{2(a + 2)} = 2 ^b
⇒  2⁵ × 2^{2a + 4} = 2b
⇒ 2 ^{5 + 2a + 4} = 2^b
⇒ 2 ^{2a + 9} = 2^b
⇒ b = 2a + 9

Question: The average age of three boys is 20 years. If their ages are in ratio 3 : 5 : 7, the age of the eldest boy is-

Solution: 
The average age of three boys is 20 years.
sum of three boys = (20 × 3)
= 60 years

their ages are in ratio 3 : 5 : 7
so, there ages are 3x, 5x, 7x

3x + 5x + 7x = 60
⇒ 15x = 60
∴ x = 4

age of the eldest boy is (7 × 4) years 
= 28 years

Question: If log_x(xy) = m, the value of log_y(xy) =?

solution: 
log_x(xy) = m
⇒ x^m = xy
⇒ x^m/x = xy/x
⇒  x ^{m - 1} = y 
∴ x = y ^{1/m -1} 

log_y(xy)
= log_yx + log_yy
= log_y y^{1/(m -1)} + 1
= 1/(m - 1) log_yy + 1
= {1/(m - 1)} + 1
= (1 + m - 1)/(m - 1)
= m/(m - 1)

প্রশ্ন:

সমাধান: 

Question: The difference between a number and its three-fifths is 240, What is the number?

Solutionh: 
let the number x

x - 3x/5 = 240 
⇒ (5x - 3x)/5 = 240 
⇒ 2x = 1200
∴ x = 600

Question: If x² = 2.89, what is the value of x? 

Solution: 
x² = 2.89
⇒ x = √2.89 
= √(289/100)
= 17/10
= 1.7

Question: A is two years older than B who is twice as old as C. If the average of the ages of A, B and C be 9, the how old is B?

Solution:
তিনজনের বয়সের গড় ৯ বছর
বয়সের সমষ্টি = (৯ × ৩) বছর
= ২৭ বছর 

ধরি, C এর বয়স x বছর 
B এর বয়স 2x বছর 
A এর বয়স 2x + 2

x + 2x + 2x + 2 = 27
⇒ 5x + 2 = 27 
⇒ 5x = 25
∴ x = 5

B এর বয়স = 2x
= (2 × 5) বছর 
= 10 বছর 

Question: The average of 4 terms is 30 and the last term is 1/3 of the remaining terms. What will be the last number?

Solution:
The average of 4 terms is 30 
total sum = (4 × 30) = 120

let, last term is x 
remaining terms is 3x

3x + x = 120
⇒ 4x = 120
∴ x = 30
The last number is 30

Question: One-fourth of the sum of prime numbers, greater than 4 but less than 16, is =?

Solution: 
prime numbers, greater than 4 but less than 16 is = 5,  7, 11, 13

One-fourth of the sum = (5 + 7 + 11 + 13)/4
= 36/4
= 9

প্রশ্ন: What is the cube root of 0.008?

সমাধান:

Question:

Solution: