Find three rational numbers between 3/6 and 3/4
In our daily lives, we use numbers. They are frequently referred to as numerals. We can’t count objects, date, time, money, or anything else without numbers. These numerals are sometimes used for measurement and other times for labeling. Numbers have features that allow them to conduct arithmetic operations on them. These figures are expressed both numerically and in words. For example, 3 is written as three, 33 is written as thirty-three, and so on. To learn further, students might practice writing the numbers from 1 to 100 in words.
There are various types of numbers that we learn in Math. Natural and whole numbers, odd and even numbers, rational and irrational numbers, and so on are all examples. In this article, we’ll go through all of the different varieties. Aside from that, the numbers are utilized in a variety of applications, including number series, arithmetic tables, and so on.
- A number is an arithmetic value that is used to represent and calculate a quantity. Numbers are represented by numerals, which are written symbols such as “2.”
- A number system is a method of writing numbers that uses logical digits or symbols to represent them.
Types of Numbers
The number system is a system for categorizing numbers into sets. In math, there are several different types of numbers:
- Natural Numbers: Natural numbers are positive integers from 1 to infinity that contain the positive integers 1 to infinity. The set of natural numbers is indicated by the letter “N,” and it consists of N = 1, 2, 3, 4, 5,…………
- Whole Numbers: Non-negative integers, often known as whole numbers, are non-negative integers that do not contain any fractional or decimal parts. It is symbolized by the letter “W,” and the set of whole numbers contains W = 0, 1, 2, 3, 4, 5,…………
- Integers: Integers are the set of all whole numbers, but they also include a set of negative natural numbers. Integers are represented by the letter “Z,” and the set of integers is Z = -3, -2, -1, 0, 1, 2, 3.
- Real Numbers: Real numbers are all positive and negative integers, fractional and decimal numbers that do not contain imaginary values. The letter “R” is used to signify it.
- Rational Numbers: Rational numbers are any numbers that may be expressed as a ratio of one number to another number. Any number that may be written in the form of p/q qualifies. The rational number is represented by the symbol “Q.”
- Irrational Numbers: Irrational numbers are numbers that cannot be expressed as a ratio of one to another and are denoted by the letter P.
- Complex Numbers: Complex numbers (C) are numbers that may be expressed in the form a+bi, where “a” and “b” are real numbers and I is an imaginary number.
Even after coining integers, one could not relax! 10 ÷ 5 is no doubt fine, giving the answer 2 but is 8 ÷ 5 comfortable? Numbers between numbers are needed. 8 ÷ 5 seen as 1.6, is a number between 1 and 2. But, where does (-3) ÷ 4 lie? Between 0 and -1. Thus, a ratio made by dividing an integer by another integer is called a rational number. The collection of all rational numbers is denoted by Q.
A Rational number is a number of the fractional form a/b, where a and b are integers and b ≠ 0.
Examples: 1/4 , 3/7 , (-11)/(-6)
- All-natural numbers, whole numbers, integers, and fractions are rational numbers.
- Every rational number can be represented on a number line.
- 0 is neither a positive nor a negative rational number.
Steps to find rational numbers between two given numbers
The numbers should be in the form of a/b and b≠0. Let us assume the numbers are a/b and c/d so to find the numbers first, we have to make b and d equal. So,
Step 1: Take the LCM of both denominators, let’s say L
Step 2: Multiply both the numbers with a number ‘x’ such that
a/b = a*x / b*x
= a*x / L i.e. A/L
Same to be done with c/d i.e. c*y/L i.e. C/L
Step 3: After making the denominator the same, all the numbers between these two new numbers can be considered as rational numbers between the given two numbers.
For example: A = 12/10, B = 16/15, find three rational numbers between these two numbers.
LCM of 10 and 15 = 2 * 3 * 5 = 30
A= 12 * 3 / 10 * 3 = 36 / 30
B = 16 * 2 / 15 * 2 = 32 / 30
So, 33/30, 34/30, 35/30 are the three rational numbers between A and B.
Give three rational numbers between 3/6 and 3/4
LCM of 6 and 4 = 2 * 2 * 3 = 12
Make the denominator same i.e.
3/6 = 3*2 / 6*2
3/4 = 3*3 / 4*3
So, now between 6/12 and 9/12 there are only two numbers i.e. 7/12 and 8/12
So, we will multiply the numerator and denominator of both numbers by 2.
6/12 = 6*2/12*2 = 12/24
9/12 = 9*2/12*2 = 18/24
So, three rational numbers between 12/24 and 18/24 are 13/24, 14/24, 15/24…..so on
Question 1: Give three rational numbers between 5/7 and 6/7.
Since the denominator of both numbers are already same. So to get more numbers between these two numbers, we will multiply the numerator and denominator of both numbers by any number let’s take 4.
5/7 = 5*4 / 7*4 = 20/28
6/7 = 6*4 / 7*4 = 24/28
So, three rational numbers between 20/28 and 24/28 are 21/28, 22/28, 23/28.
Question 2: Give five rational numbers between 2/3 and 5/4.
LCM of 3 and 4 = 3 * 4 i.e. 12
Make the denominator same i.e.
2/3 = 2*4 / 3*4 = 8/12
5/4 = 5*3 / 4*3 = 15/12
So, now between 8/12 and 15/12 there are 9/12, 10/12, 11/12, 12/12 and 13/12 are five rational numbers.