Bank Math Master

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 be 42, the how old is A?

Solution: 
Let,
C is = x years old
B is 2x years old
A is (2x + 2) years old

Now,
2x + 2x + 2 + x = 42
⇒ 5x + 2 = 42
⇒ 5x = 40
∴ x = 8
A is (2 × 8) + 2 years old
= 16 + 2 years old
= 18 years old

Question: The average of the first five multiples of 11 is-

Solution: 
The  first five multiples of 11: (11 × 1), (11 × 2), (11 × 3), (11 × 4), (11 × 5)
their sum = (11 × 1) + (11 × 2) + (11 × 3) + (11 × 4) + (11 × 5)
= 11 (1 + 2 + 3 + 4 + 5)
= 11 × 15

∴ average = (11 × 15)/5
= 33

প্রশ্ন: The average of 4 consecutive numbers is 10.5 . The largest of these numbers is:

সমাধান:
৪ টি ক্রমিক সংখ্যার গড় ১০.৫
৪ টি সংখ্যার সমষ্টি = ১০.৫ × ৪ = ৪২

ধরি, সংখ্যাগুলি হল a, a + ১, a + ২, a + ৩
প্রশ্নমতে,
a + a + ১ + a + ২ + a + ৩ = ৪২
⇒ ৪a + ৬ = ৪২
⇒ ৪a = ৪২ - ৬
⇒ ৪a = ৩৬
⇒ a = ৯

∴ বড় সংখ্যাটি হল a + ৩ 
= ৯ + ৩
= ১২

Question: The average age of three boys is 15 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 15 years.
sum of three boys = (15 × 3)
= 45 years

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

3x + 5x + 7x = 45 
⇒ 15x = 45
∴ x = 3

age of the eldest boy is (7 × 3) years 
= 21 years 

Question: A father said to his son, 'I was as old as you are at the present at the time of your birth'. If the father's age is 48 years now, the son's age three years back was-

Solution:
let, at present son is x years old
so, at time of his birth the father was x years old

∴ at present the age of father is = x + x year
= 2x year

2x = 48
⇒ x = 48/2
= 24 year

∴ the son's age five years back was = 24 - 3
= 21 year

Question: 4log√5 + 3log2 - (1/4)log10000 =?

Solution: 
4log√5 + 3log2 - (1/4)log10000
= log(√5)⁴ + 3log2 - (1/4)log10⁴
= log5² + 3log2 - (4/4) log10
= 2log5 + 3log2 - log10
= 2log5 + 3log2 - log (2 × 5)
= 2log5 + 3log2 - log2 - log5
= log5 + 2log2
= log5 + log2²
= log (5 × 4)
= log20

প্রশ্ন:

সমাধান: 

Question: If 3^{m+ n} = 81, 81^{m - n} = 3, then what is the value of n? 

Solution: 
3^{m + n} = 81
⇒ 3^{m + n} = 3⁴
⇒ m + n = 4

81^{m - n} = 3
⇒ 3^{4(m - n)} = 3
⇒ 4(m - n) = 1
⇒ m - n = 1/4 

m + n - m + n = 4 - (1/4)
⇒ 2n = 15/4
∴ n = 15/8

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

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

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

Question: If log(p/q) + log(q/p) = log(p + q), then-

Solution: 
log(p/q) + log(q/p) = log(p + q)
⇒ log {(p/q) × (q/p)} = log(p + q)
⇒ log1 = log(p + q)
∴ p + q = 1

Question: Out of 10 persons, 9 persons spent Tk. 20 each for their meals. The tenth one spent Tk. 9 more than the average expenditure of all the ten. The total money spent by all of them was-

Solution:
৯ জনের প্রতি জন খরচ করে ২০ টাকা
৯ জন মোট খরচ করে = (৯ × ২০) টাকা
= ১৮০ টাকা  

ধরি, দশম জন খরচ করে x টাকা 
দশজন মোট খরচ করে ১৮০ + x টাকা
গড় = (১৮০ + x)/১০ টাকা 

প্রশ্নমতে, 
{(১৮০ + x)/১০} + ৯ = x
⇒ {(১৮০ + x)/১০} = x - ৯ 
⇒ ১৮০ + x = ১০x - ৯০ 
⇒ ১০x - x = ১৮০ + ৯০ 
⇒ ১০x - x = ২৭০ 
⇒ ৯x = ২৭০ 
∴ x = ৩০ 

দশজন মোট খরচ করে = ১৮০ + ৩০ টাকা 
= ২১০ টাকা

Question: Sadman is younger than Miraj by 7 years. If their ages are in the respective ratio of 7 : 8, how old is Sadman?

Solution:
Let
Miraj 's age be x years.
Sadman's age = (x - 7) years.

Now
(x - 7)/x = 7/8
8x - 56 = 7x
8x - 7x = 56
x = 56

Sadman's age = (56 - 7) years.
= 49 years

Question:

Solution: 

Question: If  log₁₀(2^m + m - 4) = m(1 - log₁₀5), then m =?

Solution: 
log₁₀(2^m + m - 4) = m(1 - log₁₀5)
⇒ log₁₀(2^m + m - 4) = m (log₁₀10 - log₁₀5)
⇒ log₁₀(2^m + m - 4) = m log₁₀(10/5)
⇒ log₁₀(2^m + m - 4) = m log₁₀2
⇒ log₁₀(2^m + m - 4) =  log₁₀2^m
⇒ 2^m + m - 4 = 2^m
⇒ m = 4

Question: If 382A is divisible by 9, find the value of the smallest natural number A?

Solution: 
একটি সংখ্যা ৯ দ্বারা বিভাজ্য হবে, যদি সংখ্যাটির অংকগুলোর যোগফল ৯ দ্বারা বিভাজ্য হয়।

৩ + ৮ + ২ = ১৩। ১৩ এর সাথে ৫ যোগ করলে ১৮ হয় যা ৯ দ্বারা বিভাজ্য। 
অতএব, A অঙ্কটি হবে ৫। 

Queston: What is the geometric average of 4 and 16? 

Solution: 
geometric average of 4 and 16 = √(4 × 16)
= √64
= 8

Question: If 4xy = 4, what is the value of  log_yx -

Solution: 
 4xy = 4
⇒ xy = 1
∴ x = 1/y

log_yx
=log_y(1/y)
= log_yy^-1
= -1 log_yy
= -1