Bank Math Master

Question: Four times the first of three consecutive even integers is 4 more than twice the third. The second integer is -

Solution:
Let the three integers be x, (x + 2) and (x + 4)

ATQ,
4x = 2(x + 4) + 4
⇒ 4x = 2x + 12
⇒ 2x = 12
∴ x = 6

∴ Second integer = x + 2
= 6 + 2
= 8

Question: The difference of two numbers is 20% of the larger number. If the smaller number is 12, the larger one is-

Solution:
Let,
The larger number = x

According to the question,
x - 12 = 20% of x
⇒ x - 12 = 20x/100
⇒ x - 12 = x/5
⇒ 5x - 60 = x
⇒ 5x - x = 60
⇒ 4x = 60
⇒ x = 60/4
∴ x = 15

So, the larger number is 15

Question: 


Solution:

Question: A is four years older than B, who is thrice as old as C. If the total of the ages of A, B, and C is 46, how old is A?

Solution:
Let
C's age be = a years
Then, B's age = 3a years
A's age = (3a + 4) years

∴ (3a + 4) + 3a + a = 32
⇒ 7a + 4 = 46
⇒ 7a = 42
⇒ a = 6

Hence, A's age = (3 × 6) + 4 = 22 years.

Question: If the average of 'm' numbers is √2n² and the average of 'n' numbers is √2m², what is the average of the combined (m + n) numbers?

Solution:
দেওয়া আছে,
 m সংখ্যার গড় = √2n²
∴ m সংখ্যার সমষ্টি = m × √2n²

n সংখ্যার গড় = √2m²
∴ n সংখ্যার সমষ্টি = n × √2m²

∴ মোট সমষ্টি = m + n = (m × √2n²) + (n × √2m²)
= √2mn² + √2m²n
= √2mn(m + n)

∴ তাদের গড় = মোট সমষ্টি/(m + n)
= √2mn(m + n)/(m + n)
= √2mn

Question: The average weight of three friends is 35 kg. None of the friends weighs less than 33 kg. What can be the maximum weight of any of the three friends?

Solution:
Here,
The average weight of three friends is 35 kg

∴ Total weight of three friends = (35 × 3) kg 
= 105 kg

Minimum weight of two friends (33 × 2) kg
= 66 kg

∴ The maximum weight of any three friends is (105 - 66) kg
= 39 kg

Question: 


Solution:

Question: The ratio of the ages of a father and his son is 5 : 2 respectively. Six years ago, the ratio of their ages was 3 : 1 respectively. What is the son's present age?

Solution:
ধরি, পিতার বর্তমান বয়স = 5x বছর
এবং পুত্রের বর্তমান বয়স = 2x বছর

ছয় বছর আগে,
পিতার বয়স = 5x − 6 বছর
পুত্রের বয়স = 2x − 6 বছর

শর্তমতে,
(5x − 6)/(2x − 6) = 3/1
⇒ 6x - 18 = 5x - 6
⇒ 6x - 5x = 18 - 6
⇒ x = 12

∴ পুত্রের বর্তমান বয়স = 2x = 2 × 12 = 24 বছর

Question: If e^x = 7, then x =?

Solution: 
Given, 
e^x = 7
⇒ ln(e^x) = ln 7
∴ x = ln 7;  [Formula: ln(e^x) = x]

Question: The difference between two integers is 4. If their product is 221, then the sum of the two numbers is?

Solution:
Let the two integers be x and y
The difference between the two integers is 4,
∴ x - y = 4 and their product is, xy = 221

We know,
⇒ (x + y)² = (x - y)² + 4xy
⇒ (x + y)² = 4² + (4 × 221)
⇒ (x + y)² = 16 + 884
⇒ (x + y)² = 900
⇒ x + y = √900
∴ x + y = 30

∴ The sum of the two numbers is 30

Question: The present ages of three cousins are in the ratio of 5 : 6 : 7. three years ago, their total age was 45 years. In two years, what will be the age of the youngest cousin?

Solution:
Present age ratio of three cousins is 5 : 6 : 7
Let their ages be 5x, 6x, and 7x, respectively

ATQ,
5x - 3 + 6x - 3 + 7x - 3 = 45
⇒ 18x - 9 = 45
⇒ 18x = 54
⇒ x = 3

∴ The present age of the youngest cousin is = 5x = 3 × 5 = 15 years.
In two years, his age will be = (15 + 2) = 17 years

Question: If m is an integer such that (- 2)^2m = 2^{9 - m}, then what is the value of m?

Solution: 
দেওয়া আছে,
(- 2)^2m = 2^{9 - m}
⇒ 2^2m = 2^{9 - m}
কোনো ঋণাত্মক সংখ্যার Power যদি জোড় সংখ্যা হয় তবে সংখ্যাটি ধনাত্মক হবে। আর যদি Power বিজোড় হয় তবে সংখ্যাটি ঋণাত্মক হবে। যেহেতু m একটি পূর্ণ সংখ্যা সেহেতু 2m একটি জোড় সংখ্যা। তাই (- 2)^2m সংখ্যাটি ধনাত্মক সংখ্যা হবে।

∴ 2m = 9 - m
⇒ 3m = 9
⇒ m = 9/3
⇒ m = 3

Question: Which number is to be added to the numerator and denominator of 7/17 to form 3/5?

Solution: 
ধরি, 
সংখ্যাটি x

প্রশ্নমতে,
(7 + x)/(17 + x) = 3/5
⇒ 5(7 + x) = 3(17 + x)
⇒ 35 + 5x = 51 + 3x
⇒ 5x - 3x = 51 - 35
⇒ 2x = 16
∴ x = 8

Question: If 2^a = 3^b and 2^{a + 2} - 3^{b + 1} = √3, then find the value of b.

Solution: 
Given, 
2^{a + 2} - 3^{b + 1} = √3
⇒ (2^a × 2²) - (3^b × 3¹) = √3
⇒ (3^b × 4) - (3^b × 3) = √3
⇒ 3^b(4 - 3) = √3
⇒ 3^b = 3^1/2
∴ b = 1/2

Question: The sum of the ages of a man and his son is 70 years. After 10 years, the ratio of their ages will be 2 : 1. What is the son's present age?

Solution:
ধরি,
বাবার বর্তমান বয়স x বছর
ছেলের বর্তমান বয়স y বছর

প্রশ্নমতে,
x + y = 70 ............ (1)

আবার, 10 বছর পর, 
(x + 10)/(y + 10) = 2/1
⇒ x + 10 = 2(y + 10)
⇒ x + 10 = 2y + 20
⇒ x - 2y = 10 ............. (2)

Equation, (1) - (2)
x + y - x + 2y = 70 - 10
⇒ 3y = 60
⇒ y = 20

∴ পুত্রের বর্তমান বয়স = 20 বছর

Question: log 2 + log 4 + log 8 + ............ Find the sum of the first 19th term-

Solution:
given that,
log2 + log4 +log8 + ............
= log 2¹ + log 2² + log 2³ + ............
= log 2 + 2 log 2 + 3 log 2 + ............
= (1 + 2 + 3 + .........) × log 2

The sum of the first 19 natural numbers is given by the formula:
Sum = n(n+1)/2
where n = 19
∴ Sum = 19(19 + 1)/2
= 19 × 10
= 190

So, the sum of the first 19 terms = 190 log 2

Question: In a group of five men, no two men have the same age. The oldest man is 50 years old, and the youngest is 30 years old. If X is the average age of the men in the group, which of the following best indicates all and only possible values of X? (All ages are in whole numbers)

Solution: 
Here, 
Oldest man = 50 years
Youngest man = 30 years
Average age = X

Since no two people are the same age, the minimum average = (30 + 31 + 32 + 33 + 50)/5
= 176/5
= 35.2 
and the maximum average = (30 + 47 + 48 + 49 + 50)/5
= 224/5
= 44.8

That is, the value of X is greater than 35 and less than 45.
i.e. 35 < X < 45 

Question: 


Solution:

Question: Which of the following numbers can be removed from the set S = {0, 2, 4, 5, 9} without changing the average of set S?

Solution: 
The average of the elements in the original set S is (0 + 2 + 4 + 5 + 9)/5
= 20/5
= 4

If we remove an element that equals the average, then the average of the new set will remain unchanged.
The new set after removing 4 is {0, 2, 5, 9}.

∴ The average of the elements is (0 + 2 + 5 + 9)/4 
= 16/4 
= 4

Question: The average age of the children in a tour group is 8 years, and that of the adults is 30 years. If the average age of the entire tour group is 15 years, find the ratio of children to adults in the group.

Solution:
Here,
Average age of children = 8 years
Average age of adults = 30 years
Average age of the entire group = 15 years

Let the number of children = m
and the number of adults = n

Then, the total number of people in the group is (m + n)

ATQ,
8m + 30n = 15(m + n)
⇒ 8m + 30n = 15m + 15n
⇒ 15m - 8m = 30n - 15n
⇒ 7m = 15n
⇒ m/n = 15/7
⇒ m : n = 15 : 7

∴ The ratio of children to adults in the group is 15 : 7