Key Steps to Solve Rational Number Problems
Rational and Irrational numbers and worksheet on various operations on rational numbers:
Rational and Irrational numbers are one of the most important concepts for mathematics students.
Rational numbers are derived from the word ratio in mathematics. A rational number is the one that can be expressed as a fraction or quotient of two numbers, say p/q. Here, p is the numerator, and q is the non-zero denominator. Every integer in the number series is a rational number. For Example, 5 is also a rational number, represent-able as 5/1.
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Rational and irrational number worksheet contains all the operations and applications of these numbers. For the students learning the concepts, the worksheet helps in having the best practice materials compiled together at a single place. Furthermore, rational number worksheet separately and rational number worksheet with answers are also available to help the students get the right study material according to their needs and preferences.
Rational Coordinates and Curve:
There are other concepts also available related to rational numbers, like, a rational point is the one present on the number line with rational coordinates.
Rational Curve, on the other hand, is just a curve with parameters represented in terms of rational functions.
Rational Number Decimal Expansion and Worksheets:
Rational Numbers with decimal expansion either begin with a repetition of the sequence of digits with finite behaviour or would terminate after certain limited digits. Thus, Rational numbers might carry any form of repetition or termination of the decimals.
If any rational number represents in terms of decimal numbers like the fraction 10/3, its last digits will recur indefinitely. Like, 10/2 is equal to 3.333... and so on.
For the students to get a clear idea about how to represent numbers in the form of x/y, with y being a non-zero integer and x an integer, we precisely designed the ration number worksheet. Based on various operations on the numbers, worksheets are also available as adding and subtracting rational number worksheet and multiplying and dividing rational number worksheet.
Subtracting and Adding Rational Numbers Worksheet:
To perform addition or subtraction of rational numbers, say a/b and c/d, the below given procedure must be followed:
a/b + c/d = (a/b)*(d/d) + (c/d)*(b/b)
=>((ad)+(bc))/bd
Similarly the whole process goes for subtraction of the rational numbers.
Here are some problems on subtraction and addition of rational numbers worksheet:
1. Perform addition of the following:
½ +2/5
3/8+9/7
2. Perform removal of the following rational numbers:
5/2 – ½
¼ - 1/5
Dividing and Multiplying Rational Numbers Worksheet:
To perform multiplication of two numbers a/b and c/d, we must follow the below-mentioned set of steps:
a/b * c/d = (ac) /(bd)
And to divide two rational numbers a/b and c/d, the second number must be flipped first and then the multiplication is to be followed.
(a/b)/(c/d) = (a/b)*(d/c) = ad/bc
Here are some problems on multiplying and dividing rational numbers worksheet:
1. Perform multiplication and division operations as stated:
(1/2) / (3/4)
(3/4)*(1/2)
Fun Facts on Rational Numbers:
Pythagoras, an ancient Greek mathematician, believed that all the numbers available in the number line are rational. However, one of his intelligent students, Hippasus concluded using geometry that writing the square root of 2 is practically impossible as a fraction, thus it is an irrational number.
Still, the loyal followers of the prominent mathematician, Pythagoras, could not accept the concept of irrational numbers’ existence, and it is also known that the gods drowned Hippasus as a punishment.
FAQs on Rational Number Worksheet: Practice & Solutions
1. How should I use this Rational Number worksheet to effectively solve problems for my exams?
To best use this worksheet, first attempt to solve all the problems on your own. Focus on understanding the method behind each question, not just the final answer. If you get stuck, refer to the step-by-step solutions to identify where you went wrong. This practice helps reinforce concepts like properties of rational numbers and their representation, which are crucial for the CBSE 2025-26 exam pattern.
2. What is the correct method to check if two fractions, like 4/7 and 12/21, are equivalent rational numbers?
The most reliable method is to simplify both fractions to their standard form (where the numerator and denominator have no common factors other than 1).
- The fraction 4/7 is already in its simplest form.
- For 12/21, the greatest common divisor (GCD) of 12 and 21 is 3. Dividing both the numerator and denominator by 3 gives (12 ÷ 3) / (21 ÷ 3) = 4/7.
3. What is the step-by-step process for representing a negative rational number, like -5/4, on the number line?
To represent -5/4 on the number line, follow these steps:
- Step 1: Convert the improper fraction to a mixed fraction. -5/4 is equal to -1 1/4. This tells you the number lies between -1 and -2.
- Step 2: Draw a number line and mark the integers, especially -1 and -2.
- Step 3: The denominator is 4, so divide the segment between -1 and -2 into four equal parts.
- Step 4: The numerator is 1 (from 1/4). Starting from -1, move one part to the left. This point represents -5/4 on the number line.
4. How do you solve for the additive inverse and multiplicative inverse (reciprocal) of a rational number like -13/19?
To solve for the inverses, you apply two different rules:
- The additive inverse is the number that, when added to the original number, gives 0. To find it, you simply change the sign of the number. The additive inverse of -13/19 is +13/19.
- The multiplicative inverse (or reciprocal) is the number that, when multiplied by the original number, gives 1. To find it, you flip the numerator and denominator. The multiplicative inverse of -13/19 is -19/13.
5. How do you find five rational numbers between 1/4 and 1/2 using the correct NCERT method?
To find five rational numbers between 1/4 and 1/2, follow this CBSE-approved method:
- Step 1: Make the denominators equal. The LCM of 4 and 2 is 4. So, 1/2 becomes 2/4. Now we need numbers between 1/4 and 2/4.
- Step 2: To create a wider range, multiply the numerator and denominator of both fractions by a number greater than 5 (the number of rational numbers you need). Let's use 6.
- Step 3: 1/4 becomes (1×6)/(4×6) = 6/24. And 2/4 becomes (2×6)/(4×6) = 12/24.
- Step 4: Now, you can easily list five rational numbers between 6/24 and 12/24, such as 7/24, 8/24, 9/24, 10/24, and 11/24.
6. Why is zero (0) classified as a rational number, while 1/0 is not?
Zero (0) is a rational number because it can be written in the p/q form where 'q' is not zero (e.g., 0/1, 0/5, 0/100). This satisfies the definition. However, 1/0 is not a rational number—in fact, it is undefined. The fundamental rule of rational numbers is that the denominator 'q' can never be zero, because division by zero has no mathematical meaning.
7. How does the distributive property help in simplifying complex calculations with rational numbers? Provide an example.
The distributive property of multiplication over addition, a(b + c) = ab + ac, helps simplify problems by breaking them down. For example, to solve (2/5) × (-3/7 + 1/4), instead of adding the fractions in the bracket first, you can distribute:
- (2/5 × -3/7) + (2/5 × 1/4)
- = -6/35 + 2/20
- = -6/35 + 1/10
8. Can any terminating or repeating decimal always be expressed as a rational number? How?
Yes, absolutely. This is a core characteristic of rational numbers.
- A terminating decimal like 0.25 can be directly written as a fraction by placing it over a power of 10 (e.g., 25/100, which simplifies to 1/4).
- A repeating decimal like 0.777... can be converted to a fraction using an algebraic method. Let x = 0.777.... Then 10x = 7.777.... Subtracting the first equation from the second gives 9x = 7, so x = 7/9.
9. When comparing negative rational numbers like -7/5 and -8/5, why is the one with the 'smaller' numerator actually the larger number?
This is because of how negative numbers are placed on the number line. On the negative side, numbers closer to zero are larger. When you compare -7/5 and -8/5, you can think of their positions:
- -7/5 is equal to -1.4
- -8/5 is equal to -1.6
