Number System, Problems on Number

Let the numbers be x and (x + 16)
Then, x/3 - (x + 16)/7 = 4
⇔ 7x - 3(x +16) = 84
⇔ 4x = 84 + 48 = 132
⇔ x = 132/4 = 33
hence, the numbers are 33 and 49
Answer: 33 and 49

Let the number be X.
Then the sum of the number and 10 = X + 10.
and 1/21 of 5/7 of the number = 1/21 × 5/7 × X
From given,
We have X + 10 - 294 = 1/21 × 5/7 × X
X + 10 - 294 = 1/21 × 5/7 × X
⇒ X - 284 = 5X/147
⇒ 5X/147 - X = -284
⇒ 5X - 147X/147 = -284
⇒ -142X = 147 × (-284)
⇒ X = 147 × 2
⇒ X = 294
Hence the required number is 294.

Question: If you count from 1 to 100, how often will you encounter the number 5?

Solution:
১ - ১০ পর্যন্ত ৫ আছে = ১ টি
১১ - ২০ পর্যন্ত ৫ আছে = ১ টি
২১ - ৩০ পর্যন্ত ৫ আছে = ১ টি
৩১ - ৪০ পর্যন্ত ৫ আছে = ১ টি
৪১ - ৫০ পর্যন্ত ৫ আছে = ২ টি
৫১ - ৬০ পর্যন্ত ৫ আছে = ১০ টি
৬১ - ৭০ পর্যন্ত ৫ আছে = ১ টি
৭১ - ৮০ পর্যন্ত ৫ আছে = ১ টি
৮১ - ৯০ পর্যন্ত ৫ আছে = ১ টি
৯১ - ১০০ পর্যন্ত ৫ আছে = ১ টি

∴ মোট ৫ রয়েছে = ২০ টি

Let, these two numbers be 3x and 4x and their LCM = 12x
ATQ,
12x = 84
Or, x = 7
So, the greater number is = 4 × 7 = 28

Greatest number of four digits = 9999
LCM of 15, 25, 40 and 75 = 600
9999/600 = 16,
remainder = 399
Hence, greatest number of four digits which is divisible by 15, 25, 40 and 75
= 9999 - 399
= 9600

Numbers in which digit 5 is used between 1 & 100 are the following = 5 , 15 , 25 , 35 , 45 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 65 , 75 , 85 , 95 = 20 times

Question: A gardener planted trees in rows and columns such that the number of rows is seven more than the number of columns. If the total number of rows and columns is 87, find the number of trees.

Solution:
Let the number of columns = x.
Then, number of rows = x + 7

According to the question,
x + (x + 7) = 87
⇒ 2x + 7 = 87
⇒ 2x = 80
⇒ x = 40

Number of rows = x + 7 = 40 + 7 = 47

Total number of trees = rows × columns
= 47 × 40 = 1880

Question: In a class, 30 students play basketball, 20 students play volleyball, and 8 students play both. 12 students play neither basketball nor volleyball. What is the total number of students in the class?

​​Solution:
Number of students who play basketball, n(B) = 30
Number of students who play volleyball, n(V) = 20
Number of students who play both basketball and volleyball, n(B ∩ V) = 8
Number of students who play neither = 12

n(B ∪ V) = n(B) + n(V) - n(B ∩ V)
= 30 + 20 - 8 = 42

Total students in the class = students who play basketball or volleyball + students who play neither
n(U) = n(B ∪ V) + neither = 42 + 12 = 54

∴ Total 54 students in the class.

Given, 36 + 81 + 144
The series is = 6² + 9² + 12² + 15² + 18²
So, next term is 18² = 324

Let the six numbers be, x, (x + 1), (x + 2), (x + 3), (x + 4) and (x + 5)
Then,
⇔ x + (x + 1) + (x + 2) = 27
⇔ 3x + 3 = 27
Required sum :
= (x + 3) + (x + 4) + (x + 5)
= 3x + 12 
= (3x + 3) + 9
= 27 + 9
= 36

L.C.M of A and B is ''B'';
L.C.M of B and C is ''B';
⇒ L.C.M of A, B, C is ''B''
Answer : B

ধরি, সংখ্যাগুলো x - 3, x - 2, x - 1, x, x + 1, x + 2, x + 3, x + 4
শেষ চারটি সংখ্যার যোগফল = 4x + 10
প্রথম চারটি সংখ্যার যোগফল = 4x - 6
∴ ( - ) পার্থক্য = 16

9 × 7 = 63
Sum of decimal places = 5
∴ 0.09 × 0.007 = 0.00063

Let the integers be x and (x + 5)
Then,
⇔x(x+5)=500
⇔x²+5x−500=0
⇔(x+25)(x−20)=0
⇔x=20 
So, the numbers are 20 and 25

Question: Three numbers are in the ratio 1 : 2 : 3, and the sum of their cubes is 7776. The smallest number will be -

Solution:
Given,
The numbers be in the ratio 1 : 2 : 3
So, let:
Smallest number = x
Middle number = 2x
Largest number = 3x

ATQ,
x³ + 8x³ + 27x³ = 7776
⇒ 36x³ = 7776
⇒ x³ = 7776/36 = 216
∴ x = 6

So the smallest number is 6

Total number of students = (26 + 7) - 1 = 32

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.

Rational number is any number that can express in the form of qp of two integers, where 'q' cannot be zero.

i) First rational number between 3 and 4 can be calculated by finding average between them, which is
(3 + 4)/2 = 7/2 or 14/4
Now, we have three numbers i.e. 3, 7/2 and 4, so other remaining rational numbers can be calculated by taking average between 3 and 7/2, and between 7/2 and 4.

ii) Second rational number between 3 and 7/2 can be calculated by finding average between them.
(7/2 + 3) / 2 = {(7 + 6)/2} / 2 = 13/4

iii) The third rational number between 4 and 7/2 can be calculated by finding the average between them.
(7/2 + 4)/2 = {(7 + 8)/2}/2 = 15/4

iv) Similarly fourth rational number between 3 and 13/4 can be calculated by finding the average between them.
(13/4 + 3) / 2 = {(13 + 12)/4}/2 = 25/8

v) Similarly, fifth rational number between 4 and 13/4 can be calculated by finding average between them.
{(13/4) + 4}/2 = {(13 + 16)/4} / 2 = 29/8

Then the rational numbers between 3 and 4 are : 7/2 or 14/4, 13/4, 15/4, 25/8, 29/8

In this type of question, We need to find out the LCM of the given numbers.
LCM of 12, 15, 18 and 20;
12=2 × 2 × 3;
15=3 × 5;
18=2 × 3 × 3;
20=2 × 2 × 5

Hence, LCM = 2 × 2 × 3 × 5 × 3

Since, the soldiers are in the form of a solid square.
Hence, LCM must be a perfect square. To make the LCM a perfect square, We have to multiply it by 5,
hence,
The required number of soldiers = 2 × 2 × 3 × 3 × 5 × 5
= 900

Question: If the sum of three consecutive numbers is multiplied by 4, the result is 132. What is the largest number?

Solution:
ধরি,
তিনটি ধারাবাহিক সংখ্যা যথাক্রমে x, x + 1 এবং x + 2

প্রশ্নমতে,
⇒ {x + (x + 1) + (x + 2)} × 4 = 132
⇒ (3x + 3) × 4 = 132
⇒ 3x + 3 = 132 / 4
⇒ 3x + 3 = 33
⇒ 3x = 33 - 3
⇒ x = 30/3
∴ x = 10

বৃহত্তম সংখ্যাটি হলো = x + 2
= 10 + 2
= 12

Let the numbers are x, y, z
so, x²+ y² + z² = 532
Given,
x : y = 3 : 2 = 9 : 6
y : z = 3 : 2 = 6 : 4
Then x : y : z = 9 : 6 : 4
Now, Let the numbers are 9a, 6a, 4a
So, (9a)² + (6a)² + (4a)² = 532
Or, 81a² + 36a² + 16a² = 532
Or, 133a² = 532
Or, a² = 532/133 = 4
Or, a = 2
∴ the first number = 9×2 = 18

ধরি, শূণ্য স্থানের সংখ্যাটি x
প্রশ্নমতে, (x - 2763) ÷ 86 × 13 = 208
⇒ (x - 2763) = 208/13 × 86
⇒ x - 2763 = 1376
⇒ x = 1376 + 2763
⇒ x = 4139