Bank Math Master

Question: The average of five consecutive numbers is x. If the next two numbers are also included, how shall the average vary?

Solution:
Let the five consecutive numbers be 5, 6, 7, 8, and 9 respectively.

So, the average is = (5 + 6 + 7 + 8 + 9)/5 = 35/5 = 7
Suppose, the average is, x = 7

If the next two numbers are added, then average is = (5 + 6 + 7 + 8 + 9 + 10 + 11)/7 = 56/7 = 8
So, the new average is, x = 7 + 1

So, the average increased by 1

Question: A two-digit number becomes five-sixth of itself when its reversed. Two digits differ by one. The number is -

Solution:
(যেহেতু অঙ্ক উল্টানোর পর সংখ্যাটি ছোট হয়, তাই মূল সংখ্যার একক স্থানীয় অঙ্কটি ছোট এবং দশক স্থানীয় অঙ্কটি বড়।

ধরি,
একক স্থানীয় অঙ্ক = x
প্রশ্নমতে, দশক অঙ্ক = x + 1

সংখ্যাটি = 10(x + 1) + x = 11x  + 10
সংখ্যাদ্বয় স্থান বিনিময় করলে, নতুন সংখ্যা = 10x + (x + 1) = 11x  +1

প্রশ্নমতে,
(5/6)(11x + 10) =11x + 1
⇒ 55x + 50 = 66x + 6
⇒ 11x = 44
⇒ x = 4

সংখ্যাটি  = 11 × 4 + 10 = 54

Question: The average of a class of 39 students is 15 years. If the age of the teacher is included, then the average increase's by 3 months. Find the age of the teacher.

Solution:
Total age of 39 persons  = 39 × 15 = 585 years
The average age of 40 persons = 15 years + 3 months = 15 + (3/12) = 61/4 years

Total age of 40 person = (61/4) × 40 = 610 years

So, the age of the teacher = 610 - 585 = 25 years

Question: If the average of 6, 15, 22, and 'x' is 15, what is the value of 'x'?

Solution:
According to the question,
(6 + 15 + 22 + x) / 4 = 15
⇒ 43 + x  = 60
⇒ x  = 17

Question: Rahim's average score in 4 tests was 80 out of a possible 100. If his scores in two of the tests were 60 and 70, what is the lowest that either of his other scores could have been?

Solution: 
Total marks of 4 tests = 4 x 80 = 320

Marks of the other two subjects = (320 - 60 - 70) = 190 marks
He can get the highest  100 marks in one of the subjects.

∴ The lowest mank = 190 - 100 = 90 marks

Question: A father is three times as old as his son. After fifteen years father will be twice as old as his son age that time. What is the present age of the son?

Solution:
Let, the son's age be x years
and the father's age is = 3x years

ATQ,
3x + 15 = 2(x + 15)
⇒ 3x + 15 = 2x + 30
⇒ x = 15

Question: If a number is multiplied by two-thirds of itself the value obtained is 384. What is the number?

Solution:
Let, the number be x

ATQ,
x × (2x/3) = 384
⇒ 2x² = 1152
⇒ x² = 576
⇒ x = 24

Question: A man's present age is two-fifth of the age of his mother. After 8 years, he will be one-half of the age of his mother. What is the average of their present ages?

Solution:
Let, the mother's age be = x years
man's age = (2x/5) years

According to the question,
(2x/5) + 8 = (x + 8)/2
⇒ (2x + 40)/5 = (x + 8)/2
⇒ 5x + 40 = 4x + 80
⇒ x = 40

mother's age = 40 years
man's age = (2 x 40)/5 = 16 years

the average of their present ages = (40 + 16)/2 = 28 years

Question: log4 + log16 + log64 +………. What is the sum of the first 20 terms?

Solution:
log4 + log16 + log64 + ……….
= log4 + log4² + log4³+ ……...
= log4 + 2log4 + 3log4 + .........
= log4 (1 + 2 + 3 +…….)
= {20(20 + 1) / 2} log4
= 210 log4
= 210 log2² 
= (210 × 2) log2
= 420 log2

Question: If log_2√264 = x² then, what is the value of x?

Solution:
  log_2√264 = x²
⇒ log_2√22√2⁴ = x²
⇒ 4log_2√22√2 = x²
⇒ x² = 4
⇒ x² = 2²
⇒ x = 2

{³√(5²).³√5} / √5
= ³√(5².5) / √5
= ³√(5¹⁺²) / √5
= ³√5³ / √5
= 5/√5
= (√5.√5)/√5
= √5

(a^x/a^y)^z . (a^y/a^z)^x . (a^z/a^x)^y 
= (a^xz/a^yz) . (a^xy/a^xz) . (a^yz/a^xy)
=  a^{xz - yz + xy - xz + yz - xy}
= a⁰
= 1

Question: If 2^{2x + 1} = 128 then, what is the value of x?

Solution:
2^{2x + 1} = 128
⇒ 2^{2x + 1} = 2⁷
⇒ 2x + 1 = 7
⇒ 2x = 6
⇒ x = 3

Question: Find the value of x; 2^{x - 4} = 4a^{x - 6} [a > 0, a ≠ 2]

Solution:
2^{x - 4} = 4a^{x - 6}
⇒ (2^{x - 4})/4 = a^{x - 6}
⇒ (2^{x - 4})/2² = a^{x - 6}
⇒ 2^{x - 4 - 2} = a^{x - 6}
⇒ 2^{x - 6} = a^{x- 6}
⇒ (2/a)^{x - 6} = 1
⇒ (2/a)^{x - 6} = (2/a)⁰
⇒ x - 6 = 0
⇒ x = 6

Question: 4^{x + 2} = 2^{2x + 1} + 14, Find the value of x.

Solution:
4^{x + 2} = 2^{2x + 1} + 14
⇒ 4².4^x  - 2^2x.2¹ = 14
⇒ 16.4^x - 4^x.2 = 14
⇒ 4^x(16 - 2) = 14
⇒ 4^x.14 = 14
⇒ 4^x= 1
⇒ 4^x = 4⁰
⇒ x = 0

3log2 + log5
= log2³ + log5
= log8 + log5
= log (8 × 5)
= log40

Question: The difference between the greatest and the least four-digit numbers that begins with 3 and ends with 5 is -

Solution:
The greatest four-digit number that begins with 3 and ends with 5 = 3995
The least four-digit number that begins with 3 and ends with 5 = 3005

∴ Required difference = 3995 - 3005 = 990

Question: Which among 2^1/2, 3^1/3, 4^1/4, 6^1/6, and 12^1/12 is the largest?

Solution:
2^1/2, 3^1/3, 4^1/4, 6^1/6, and 12^1/12
= 2^{(1/2) × 12}, 3^{(1/3) × 12}, 4^{(1/4) × 12}, 6^{(1/6) × 12}, and 12^{(1/12) × 12}
= 2⁶, 3⁴, 4³, 6² and 12¹
= 64, 81, 64, 36, and 12

Here, 3^1/3 is the largest number.

Question: Three numbers are in the ratio 1 : 2 : 3 and their HCF 12. The sum of three numbers is -

Solution:
Let the three numbers are x, 2x, 3x respectively 
Their HCF = x

ATQ,
x = 12

So, the three numbers are 12, 24, and 36 respectively 
The average of the three number = (12 + 24 + 36)/3 = 24

Question: The average of 75 numbers is 35. If each number is increased by 5, the average of new numbers is -

Solution:
Sum of 75 numbers = 75 × 35 = 2625

Total increase = 5 × 75 = 375
Increased sum = 2625 + 375 = 3000

So, the new average = 3000/75 = 40

Question: Rifat got married 8 years ago. His present age is 6/5 times his age at the time of his marriage. What is the present age of Rifat?
Solution:
Let, Rifatj's age 8 years ago be x years
His present age is = x + 8 years

ATQ,
x + 8 = 6x/5
⇒ 5x + 40 = 6x
⇒ x = 40

His present age is = 40 + 8 = 48 years