Number System, Problems on Number

Question: If (n - 1) is an odd number, what are the two other odd numbers nearest to it? 

Solution: 
n - 1 is an odd number 
previous odd number = n - 1 - 2 = n - 3
next odd number = n - 1 + 2 = n + 1

Question:  A cricketer whose bowling average is 12.4 runs per wicket takes 5 wickets for 26 runs and thereby decreases his average by 0.4. The number of wickets taken by him till the last match was -

Solution: 
Let the cricketer takes x wickets before last match 

Total run = 12.4x + 26 
New average = (12.4x + 26) / (x + 5)

ATQ,
(12.4x + 26) / (x + 5) = 12.4 - 0.4 
⇒ (12.4x + 26) = 12 (x + 5)
⇒ 12.4x - 12x = 60 - 26
⇒ 0.4x = 34
⇒ x = 34/0.4
= 85 

The number of wickets taken by him till the last match was = 85 wickets 

Question: If 35% of a certain number is 84, then find the number-

Solution:
Let the number be x.
Then,
35% of x = 84
⇒ (35/100) × x = 84
⇒ 7x/20 = 84
⇒ x = (20 × 84)/7
⇒ x = 1680/7
∴ x = 240

4 x 162 = 648. Sum of decimal places = 6.
So,
0.04 x 0.0162
= 0.000648
= 6.48 x 10^-4

Question: What is the value of (255 - 55) ÷ 4 × 15 - 504 ÷ 3 = ?

Solution:
(255 - 55) ÷ 4 × 15 - 504 ÷ 3
= 200 ÷ 4 × 15 - 504 ÷ 3
= 50 × 15 - 168
= 750 - 168
= 582

Question: The sum of two integers is 48 and their product is 432. What is the smaller number?

Solution:
Let the two integers be x and y.
Then we get, 
x + y = 48
y = 48 - x .....(1)

And, xy = 432
⇒ x(48 - x) = 432 ; [From (1)] 
⇒ 48x - x² = 432
⇒ x² - 48x + 432 = 0
⇒ x² - 36x - 12x + 432 = 0
⇒ x(x - 36) - 12(x - 36) = 0
⇒ (x - 36)(x - 12) = 0
Now, x - 36 = 0
∴ x = 36
Or, x - 12 = 0
∴ x = 12
Therefore the two numbers are 12 and 36.

∴ The smaller number is 12.

Question: Three numbers which are co-prime to each other are such that the product of the first two is 551 and that of the last two is 1073. The sum of the three numbers is-

Solution: 
Let the three numbers be a, b, and c.

Given that, 
a × b = 551 = 19 × 29 ...… (1) ; [both 19 and 29 are prime numbers]
∴ a = 19, b = 29

And,
b × c = 1073 = 29 × 37 ...… (2) ; [both 29 and 37 are prime numbers]
∴ b = 29, c = 37

∴ a : b : c = 19 : 29 : 37
So the three numbers are 19, 29, 37

∴ sum = 19 + 29 + 37 = 85

প্রশ্ন: যদি m একটি বিজোড় স্বাভাবিক সংখ্যা হয়, নিচের কোনটি অবশ্যই একটি জোড় স্বাভাবিক সংখ্যা হবে?

সমাধান:
ধরি, m = ৩ (বিজোড় সংখ্যা)

ক) m + ১ = ৩ + ১ = ৪ (জোড়)
খ) m² = ৩² = ৯ (বিজোড়)
গ) ৩m = ৩ × ৩ = ৯ (বিজোড়)
ঘ) m² + m + ১ = ৯ + ৩ + ১ = ১৩ (বিজোড়)

Question: A student got twice as many sums wrong as he got right. If he attempted 42 sums in all, how many did he solve correctly?
Solution:
Let he has solved correctly X no. of sums.
Therefore incorrect no. of sums = 2X
Now,
X + 2X = 42
⇒ 3X = 42
⇒ X = 14

∴ 14 sums he has done correctly.

The sum of all prime numbers from 1 to 20
= (2 + 3 + 5 + 7 + 11 + 13 + 17 + 19)
= 77

Question: What is the highest power of 5 in the prime factorization of 625?

Solution:
Given that, Highest power of 5 in the prime factorization of 625
Now, The prime factorization of 625 = 5 × 5 × 5 × 5 = 5⁴
Highest power of 5 = 4

∴ The highest power of 5 is 4.

Question: Let N be the smallest positive integer that is divisible by both 20 and 30. How many distinct prime factors does N have?

Solution:
এখানে, N হলো 20 এবং 30 দ্বারা বিভাজ্য ক্ষুদ্রতম সংখ্যা।
সুতরাং, N হবে 20 এবং 30 এর ল.সা.গু।

এখন, 20 = 2 × 2 × 5 = 2² × 5¹
এবং 30 = 2 × 3 × 5 = 2¹ × 3¹ × 5¹

LCM(20, 30) = 2² × 3¹ × 5¹ = 60
অতএব, N = 60

60 এর মৌলিক উৎপাদক = 2² × 3 × 5

স্বতন্ত্র মৌলিক উৎপাদকগুলি হলো 2, 3 এবং 5।

∴ N এর স্বতন্ত্র মৌলিক উৎপাদকের সংখ্যা হলো 3টি।

Question: What should come in place of both n in the equation (n/√162) = (√128/n)?

Solution: 
Here,
n/√162 = √128/n
⇒ n² = √(128 × 162)
⇒ n² = √(64 × 2 × 18 × 9)
⇒ n² = √(64 × 36 × 9)
⇒ n² = √(8² × 6² × 3²)
⇒ n² = 8 × 6 × 3
⇒ n² = 144
⇒ n = √144
∴ n = 12

Question: Which one of the following is a rational number?

Solution:
ক) √3 × √5 = √15 ........ irrational
খ) √11 × √2 = √22 ........ irrational
গ) √3 × √27 = √81 = 9 ........ rational
ঘ) √6 × √16 = √96 ........ irrational
ঙ) √7 × √13 = √91 ........ irrational

Answer: গ) √3 × √27 = 9 is a rational number

Choose n to be 0.
Then (n -2)/2
= (0 -2)/2
= -1 which is an integer.
So, eliminate
next, √n = √0 = 0.
Eliminate.
Next, 2/(n +1) = 2/1 = 2
eliminate, 
Next, √1/(n² + 2)
= √1/2
= 1/√2 which is not an integer
So, the Answer is: √1/(n² + 2)

a + b + c = 150.

Since, we have to find out the most possible smallest value,
We assume a = 1, b = 2 and c = 147.

So, the median is 2.

First, identify the number that is multiple of 3 more than 100.
That type of number is 102.
So, 102//3 = 34.
Second, we have to identify the number that is multiple of 3 but nearest less than 198.
Now, 198/3 = 66.
Answer is (66 - 34) + 1 = 33

Question: Half of the people on a bus get off at each stop after the first and no one gets on after the first stop. If only 4 person gets off at stop number 4, how many people got on at the first stop?

Solution:
stop 4 এ যাত্রী ছিলো 4 জন
stop 3 এ যাত্রী ছিলো 8 জন
stop 2 এ যাত্রী ছিলো 16 জন
stop 1 এ যাত্রী ছিলো 32 জন

প্রথম stopageএ কেউ নামেনি। তার পরের প্রতিটিতে অর্ধেক করে নেমে গিয়েছে।
প্রথম স্টেশনে যাত্রী ছিল 32 জন।

Question: If n is an even integer, which of the following must be an odd integer?

Solution:
Let,
n = 2

ক) n + 2 = 2 + 2 = 4  ; even
খ) n² = 2² = 4   ; even
গ) 2n +1 = (2 × 2) + 1 = 5   ; odd
ঘ) n² + n = 2² + 2 = 6   ; even

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: If x and y are odd numbers, which number is even?

Solution:
Let x = 1 and y = 3 (both are odd numbers)

a) x × y = (1 × 3) = 3 ............. Odd

b) x + y + 1 = (1 + 3 + 1) = 5 ......... Odd

c) 3x + 4 = (3 × 1 + 4) = 3 + 4 = 7 ......... Odd 

d) 2x + 2y = (2 × 1 + 2 × 3) = 2 + 6 = 8 .......... Even 

Question: Among students in a class, 30 are basketball players, 20 are volleyball players, and 8 play both games. If 12 students play neither game, how many students are in the class altogether?

Solution:
Let the number of students who play basketball = 30
Number of students who play volleyball = 20
Number of students who play both basketball and volleyball = 8
Number of students who play neither = 12

First, calculate the number of students who play basketball or volleyball:
n(B ∪ V) = n(B) + n(V) − n(B ∩ V)
n(B ∪ V) = 30 + 20 − 8 = 42

Now, add the students who play neither sport to get total students:
Total students = n(B ∪ V) + neither

Total students = 42 + 12 = 54

Given,
5 - [4 - {3 - (3 - 3 - 6)}]
= 5 - [4 - {3 - (-6)}]
= 5 - [4 - {3 +6}]
= 5 - [4 - {9}]
= 5 - [4 - 9]
= 5 - [-5]
= 5 + 5
= 10

Question: If one-fifth of one-sixth of a number is 10, then what is 2/3 of that number?

Solution:
Let the number be x.
So one-sixth of the number = x/6

Now, one-fifth of one-sixth of the number = (1/5) × (x/6) = x/30

According to the question:
⇒ x/30 = 10
⇒ x = 10 × 30
∴ x = 300

∴ Now calculate 2/3 of the number = (2/3) × 300
= 2 × 100
= 200

Let, The numbers are x & y,
therefore, x - y = 11 ---- (1) and
1/5 (x + y) = 9
or, x + y = 45 ------ (2)
Adding two equation we got,
2x = 56 or, x = 28
Putting the value of x in equation 1,
We get, y = 17

Question: The daily rate for a hotel room that sleeps 4 people is Tk. 390 for one person and X taka for each additional person. If 3 people take the room for one day and each pays Tk 210 for the room, then what is the value of X?

Solution:
The daily rate for 1 person is Tk. 390
For each additional person daily rate  X taka
The total cost for 3 people is = 390 + 2X

If 3 people take the room for one day and each pays Tk 210 for the room
Total cost = 210 × 3 = Tk. 630 

According to the question
390 + 2X = 630 
⇒ 2X = 630 - 390 
⇒ 2X =240
∴ X = 120