Rational numbers are a fundamental concept in mathematics. They are defined as any number that can be expressed as the quotient or fraction
In this article, we will discuss what rational numbers are, explore their formulas, and provide examples of rational numbers.
Table of Content
What are Rational Numbers?
Rational numbers are the numbers that can be expressed in the form of a fraction, where both the numerator and denominator are integers ex. 1,2,0.5,
1.1\overline{1}
In other words, a rational number can be expressed in the form of p/q, where q ≠ 0 and p & q are any integer.
Important Formulas on Rational Numbers
Some key formulas and concepts of rational numbers are discussed below:
Basic Formula
A rational number is written in the form of p/q
where,
q ≠ 0 and p & q are any integer.
Arithmetic Operations
Let us discuss about how to perform arithmetic operations on rational numbers.
- Addition: When we add two rational numbers p/q and s/t, we have to make the denominator same.
p/q + s/t = (pt+qs)/qt
- Subtraction: When we subtract two rational numbers a/b and c/d, we also have to make the denominator same.
a/b - c/d = (ad-bc)/bd
- Multiplication: In case of multiplication of two rational numbers a/b and c/d, the numerators and denominators are multiplied respectively.
a/b × c/d = ac/bd
- Division: If p/q is divided by s/t, then the result will be,
p/q ÷ s/t = pt/qs
Multiplicative Inverse of a Rational Number
If a rational number is represented in the form p/q, which is a fraction, then the multiplicative inverse of that rational number will be the reciprocal of that fraction.
For example: The multiplicative inverse of 5/6 will be 6/5.
Practice Questions on Rational Numbers - Solved
The following Practice Questions on Rational Numbers are various types of word problems based on the concepts of rational numbers.
1. From a rope 11 m long, two pieces of lengths 13/5 m and 33/10 m are cut off. What is the length of the remaining rope?
Total length of the two pieces = ( 13/5 + 33/10 )
= 59/10 m.
Length of the rope = 11m.
Length of the remaining rope = (11 - 59/10)
= (110 - 59)/10
= 51/10 m.
2. A car is moving at an average speed of 196/3 km/hr. How much distance will it cover in 15/2 hours?
Avg. speed of the car is = 196/3 km/hr. i.e., the car travels 196/3 km in 1 hr.
1hr = 196/3km
15/2hrs = 196/3 × 15/2
= 98 × 5
= 490 km
∴ The car will travel 490km in 15/2 hrs.
3. A truck is traveling at a speed of 7 miles per hour. How long will it take the truck to travel 4½ miles.
The truck goes 7 miles in one hour.
It will go 4½ miles in (1/7 × 4½)
= 1/7 × 9/2
= 9/14 hr.
∴ It will take 9/14 hrs for the truck to travel 4½ miles.
4. How many 0.26L glasses can be filled by 20.8L water bottle?
We are splitting 20.8L water into a glasses of 0.26L.
Now,
(20.8 ÷ 0.26) = 2080 ÷ 26 = 80
∴ Total 80 glasses can be filled.
5. A cord of length 143/2 m has been cut into 26 pieces of equal length. What is the length of each piece?
The length of the cord is = 143/2 m. which has been cut into 26 pieces of equal length.
The length of each piece = 143/2 ÷ 26
= 11/4 m.
6. Determine whether the given numbers are rational or irrational.
(a) 1.33 (b) 0.1 (c) 0 (d) √5
- 1.33 is a rational number as it can be represented as 133/100.
- 0.1 is a rational number as it can be represented as 1/10.
- 0 is a rational number as it can be represented as 0/1.
- √5 is an irrational number as it can not be represented as p/q.
7. Which of the following is irrational or rational:
1/2, 13, -4, √3, and π.
Here in this problem, 1/2, 13 and -4 are rational numbers because they can be expressed in p/q format. While √3 and π are not rational as they can not be expressed by p/q format.
8. What are the five rational numbers between 0 to 1?
The five rational numbers between 0 to 1 are, 0.1, 0.2, 0.3, 0.4, 0.5.
9. Simplify the equation (1/2 + 3/4 – 4/5)
Lets take LCM of the denominators LCM(2, 4, 5) = 20
Equivalent fraction of 1/2, 3/4 and 4/5 with denominator 20 are:
1/2 = (1×10)/(2×10) = 10/20
3/4 = (3×5)/(4×5) = 15/20
4/5 = (4×4)/(5×4) = 16/20
Now, simplify
10/20 + 15/20 - 16/20
= (10 + 15 - 16)/20
= 9/20
10. Find out the subtraction of the rational number 7/8 from 5/6.
Lets take LCM of the denominator LCM(8, 6) = 24
Equivalent fraction of 5/6 and 7/8 with denominator 24 are:
5/6 = (5×4)/(6×4) = 20/24
7/8 = (7×3)/(8×3) = 21/24
Now, simplify
(20/24 - 21/24) = -1/24
11. Multiply 2/5 and 5/6.
- multiply the numerator
2×5 = 10
- multiply the denominator
5×6 = 30
- Now Simplifying the fractions we get (10/30 )= 1/3
12. Find the rational number that should be added to 2/9 to get -1
Let the required number = x
then,
x + 2/9 = -1
x = -1 - 2/9
x = (-9-2)/9
x = -11/9
Therefore, -11/9 should be added to 2/9 to get -1.
13. Verify the distributive property of rational numbers
x × (y + z) = (x × y) + (x × z), where x = -3, y = 9/2, z = -7/9
Here in this problem, we have to equalize LHS = RHS
LHS = x × (y + z)
Putting the given values we get,
= -3 × (9/2 + -7/9)
= -3 × {(81-14)/18}
= -3 × 67/18
= - 67/6.........................(i)
RHS = (x × y) + (x × z)
= (-3 × 9/2) + (-3 × -7/9)
= -27/2 + 7/3
= (-81+14)/6
= - 67/6......................(ii)
From equation (i) and (ii), LHS = RHS
Hence the distributive property is verified.
14. Find the reciprocal of the rational number
a) 113/15
b) 90/173
a) The reciprocal of 113/15 = 15/113
b) The reciprocal of 90/173 = 173/90.
15. Arrange the following rational numbers in ascending order:
-7/10, -5/8, -3/4
The given rational numbers are -7/10, -5/8, -3/4. We will take LCM of the denominators to convert these numbers to equivalent rational numbers.
So, the LCM of (10, 8, 4) = 40
The equivalent rational numbers are-
-7/10 = -28/40
-5/8 = -25/40
-3/4 = -30/40
Now, from the above numbers, we can easily find that:
-30<-28<-25
or, -3/4<-7/10<-5/8
Therefore the ascending order is -3/4<-7/10<-5/8 .
Practice Questions on Rational Numbers - Unsolved
This worksheet on Rational Numbers contain some practice questions on Rational numbers which you can solve to test your understanding of the concept.
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