Bank Math Master

Question: If the sum of three consecutive odd integers is 117, what is the largest number?

Solution:
Let the three consecutive odd integers be:
x, x + 2, x + 4

Then:
x + (x + 2) + (x + 4) = 117
⇒ 3x + 6 = 117
⇒ 3x = 117 − 6
⇒ 3x = 111
⇒ x = 111 ÷ 3
⇒ x = 37

So the numbers are: 37, 39, 41 (Largest)

∴ The largest number is 41.

Question: The ages of five people are as follows 36, 48, 52, 40, and 44 years. What should be the age of a sixth person so that the average age of all six becomes 45 years?

Solution:
Let the age of the sixth person be x.
The average of six persons = Total age of 6 persons ÷ 6.

ATQ, 
(36 + 48 + 52 + 40 + 44 + x)/6 = 45
⇒ (220 + x)/6 = 45
⇒ 220 + x = 270
⇒ x = 270 - 220
⇒ x = 50

∴ The age of the sixth person should be 50 years.

Question: A group of 8 boxes has its average weight increased by 3 kg after replacing a 50 kg box with a new one. What is the weight of the new box?

Solution:
Let the weight of the new box be x kg.
Let the total weight of the original 8 boxes = W.
Original average = W/8.

After replacing the 50 kg box with a box weighing x kg, 
The total weight becomes = W - 50 + x. 
∴ The new average = (W - 50 + x)/8.

ATQ,
New average = Old average + 3
(W - 50 + x)/8 = W/8 + 3
⇒ W - 50 + x = W + 24
⇒ x = 24 + 50
⇒ x = 74

∴ The weight of the new box is 74 kg.

Question: The ratio of two numbers is 3 : 4 and their H.C.F is 6. Find their L.C.M.

Solution:
Let the two numbers be 3x and 4x.
∴ H.C.F = x = 6

∴ The two numbers are, 3 × 6 = 18 and 4 × 6 = 24

∴ Product of the two numbers = 18 × 24 = 432
And H.C.F = 6

We know,
L.C.M = (Product of two numbers)/H.C.F
= 432/6
= 72

∴ The L.C.M of the two numbers = 72.

Question: A sum of Tk. 4200 is divided among P, Q, and R such that P gets 3/5 of what Q gets and Q gets 1/4 of what R gets. Q’s share is:

Solution:
Let,
R’s share = Tk. x
Then,
Q’s share = Tk. x/4
P’s share = Tk. (3/5) × (x/4) = Tk. 3x/20

∴ P + Q + R = 4200
⇒ 3x/20 + x/4 + x = 4200
⇒ (3x + 5x + 20x)/20 = 4200
⇒ 28x/20 = 4200
⇒ 28x = 4200 × 20
⇒ 28x = 84000
⇒ x = 84000/28
⇒ x = 3000

∴ Q’s share = x/4 = 3000/4 = 750 Tk.

Question: The least number by which 150 must be multiplied to make it a perfect square is:

Solution:
Prime factorization of 150:
150 = 2 × 3 × 5 × 5
= 2¹ × 3¹ × 5²

Here, the powers of 2 and 3 are odd.
∴ To make it a perfect square, we need to multiply by 2 × 3 = 6.

Question: If √(7m) = 343, then the value of m is - 

Solution:
Given that, √(7m) = 343
⇒ √(7m) = 7³ (since, 343 = 7³)
⇒ {√(7m)}² = (7³)²
⇒ 7m = 7⁶
⇒ m = 7⁵
⇒ m = 16,807

∴ The value of m is 16,807.

Question:
 

Solution:

Question: A father is three times as old as his son. After 12 years, he will be twice as old as his son. Find the father’s present age.

Solution:
Let the son’s present age be x years.
Then, the father’s present age = 3x years.

After 12 years,
Son’s age = x + 12
Father’s age = 3x + 12

According to the question,
3x + 12 = 2(x + 12)
⇒ 3x + 12 = 2x + 24
⇒ 3x - 2x = 24 - 12
⇒ x = 12

∴ Father’s present age = 3 × 12 = 36 years.

Question: The average of ten numbers is 25. The average of seven of these numbers is 22. What is the average of the remaining three numbers?

Solution:
১০ টি সংখ্যার গড় = ২৫
১০ টি সংখ্যার সমষ্টি = ২৫ × ১০ = ২৫০

৭ টি সংখ্যার গড় = ২২
৭ টি সংখ্যার সমষ্টি = ২২ × ৭ = ১৫৪

∴ বাকী ৩ টি সংখ্যার সমষ্টি = ২৫০ - ১৫৪ = ৯৬
∴ ৩ টি সংখ্যার গড় = ৯৬/৩ = ৩২.

Question: Sanzida ate 3/4 of a pizza. Her brother Babu ate 1/2 of what was left. Then their friend Pavel ate 2/3 of what was still left. What fraction of the pizza remains uneaten?

Solution:
Sanzida ate = 3/4
∴ Remaining = 1 - 3/4 = 1/4

Babu ate = 1/2 × 1/4 = 1/8
∴ Remaining = 1/4 - 1/8 = (2 - 1)/8 = 1/8

Pavel ate = 2/3 × 1/8 = 2/24 = 1/12
∴ Remaining = 1/8 - 1/12 = (3 - 2)/24 = 1/24

∴ Fraction of pizza remaining uneaten = 1/24.

Question: (0.01 × 0.001 × 0.1 × 10⁶) is equal to:

Solution:
Given expression,
(0.01 × 0.001 × 0.1 × 10⁶)
= (1/100) × (1/1000) × (1/10) × 10⁶
= 10⁶/10⁶
= 10^{(6 - 6)}
= 10⁰
= 1

Question: Find the value of x if log_x 256 = 4.

Solution:
log_x 256 = 4
⇒ x⁴ = 256
⇒ (x²)² = 256
⇒ x² = 16
⇒ x = √16
⇒ x = 4

∴ The value of x is 4.

Question: If 0.24 ÷ q² = 6, then q equals:

Solution:
0.24 ÷ q² = 6
⇒ q² = 0.24/6
⇒ q² = 0.04
⇒ q = √0.04
⇒ q = 0.2

∴ The value of q is 0.2.

Question: Among 90 students, the average marks in science is 72. If the 60 girls scored an average of 75, determine the average score of the remaining 30 boys.

Solution:
Let, the average marks of the boys = k
Total marks of 90 students = 90 × 72 = 6480
Total marks of 60 girls = 60 × 75 = 4500

According to the question,
4500 + 30k = 6480
⇒ 30k = 6480 - 4500
⇒ 30k = 1980
⇒ k = 1980/30
⇒ k = 66

∴ Average marks of the remaining 30 boys = 66.

Question: Find the value of x, if 3^{(2x + 1)} = 243.

Solution:
3^{(2x + 1)} = 243
⇒ 3^{(2x + 1)} = 3⁵ (since 243 = 3⁵)
⇒ 2x + 1 = 5
⇒ 2x = 5 - 1
⇒ 2x = 4
⇒ x = 4/2
∴ x = 2

Question: How many prime numbers are there between 50 and 60?

Solution:
A prime number is a number that is divisible only by 1 and itself.

Numbers between 50 and 60 are:
51, 52, 53, 54, 55, 56, 57, 58, 59

Among these, the prime numbers are:
53, 59.

∴ Total number of prime numbers = 2.

Question: Among three numbers, the first number is twice the second and half of the third. If the average of the three numbers is 56, find the difference between the first and third numbers.

Solution:
Let, 
the second number be x.
Then first number = 2x, third number = 4x.

∴ 2x + x + 4x = 56 × 3
⇒ 7x = 168
⇒ x = 168/7
⇒ x = 24

Required difference:
= 4x - 2x
= 2x
= 2 × 24
= 48.

Question: If one-fourth of one-sixth of a number is 5, then what is 3/4 of the number?

Solution:
Let the number be x.
According to the question,
(1/4) of (1/6) of x = 5
⇒ (1/4) × (1/6) × x = 5
⇒ (1/24)x = 5
⇒ x = 5 × 24
∴ x = 120

Now,
(3/4) of x = (3/4) × 120
= 360/4
= 90

Question: The average age of all the students in a driving class is 22 years. The average age of the boys is 24 years and that of the girls is 18 years. If there are 15 girls in the class, find the number of boys.

Solution:
Let the number of boys in the class be x.

According to the question,
22 × (x + 15) = 24x + (18 × 15)
⇒ 22x + 330 = 24x + 270
⇒ 24x + 270 = 22x + 330
⇒ 24x - 22x = 330 - 270
⇒ 2x = 60
⇒ x = 30

∴ The number of boys in the class is 30.