### Key Concepts

- Add and subtract rational numbers with different signs.
- Use properties of operations to add and subtract.
- Find distances on a number line.

## Introduction:

- In this chapter, we will learn to add and subtract rational numbers with different signs.
- Addition and subtraction of rational numbers with same denominator.
- Addition and subtraction of rational numbers with different denominators.
- Practice properties of operations to solve addition and subtraction problems.
- Solve the questions to find the distance between any two points on a number line.
- Finding the absolute value of their difference.

**1.7 ****Add and subtract rational numbers**

**What are Rational Numbers? **

The numbers that are expressed in the form of p/q , where *p* and *q* are integers (*q* ≠ 0) are called rational numbers. Rational numbers consist of natural, whole numbers, fractions and integers.

**1. Positive Rational Numbers:**

Positive rational numbers are the numbers whose both the numerator and denominator are positive.

Example: 6/7 , 13/11 etc.,

**2. Negative Rational Numbers:**

Negative rational numbers are the numbers whose one of the numerator or denominator is negative.

Example: -2 / 6 , 36/-3 etc.,

**Zero** is neither a positive nor a negative rational number.

**Rational Numbers in Standard Form:**

A rational number is said to be in the standard form, when the denominator is a positive integer and there is no common factor between the numerator and denominator other than 1.

If any rational number is not in the standard form, then we can reduce it to its standard form or the lowest form by dividing its numerator and denominator by their HCF.

**Example:**

Write the standard form of 12/18.

**Solution: **

12/18 ÷6/6=2/3

∴∴2/3 is the standard or simplest form of 12/18.

**1.7.1 Add and subtract rational numbers with different signs**

**Addition:**

Addition of rational numbers are categorised into three types:

- Addition of two rational numbers with the same denominator

- Addition of two Rational Numbers with different denominator

- Additive Inverse

**1. Addition of two rational numbers with the same denominator:**

Two rational numbers can be added by adding their numerators, keeping the denominator same.

**Example 1:**

Add 1/5 and 2/5.

**Solution:**

On the number line, we have to move right from 0 to 1/5 units and then move 2/5 units more to the right.

If we have to add two rational numbers whose denominators are same, then we simply add their numerators and the denominator remain the same.

**2. Addition of two rational numbers with different denominator:**

As in the case of fractions, we first find the L.C.M. of the two denominators. Then we find the rational numbers equivalent to the given rational numbers with this L.C.M. as the denominator.

**Example 2:**

Add 2/5 and 3/7.

**Solution:**

To add the two rational numbers, first, we need to take the L.C.M. of denominators then find the equivalent rational numbers.

L.C.M. of 5 and 7 is 35.

**3. Additive Inverse**

Like integers, the additive inverse of rational numbers is also the same.

The additive inverse of the rational number p/q is -p/q.

This shows that the additive inverse of 3/7 is -(3/7).

This shows that,

**Subtraction:**

If we have to subtract two rational numbers, then we have to add additive inverse of the rational number that can be subtracted to the other rational number.

a − b = a + (−b)a − b = a + (−b)

**Example 1:**

Subtract 4/21 from 8/21.

**Solution:**

In the first method, we will simply subtract the numerator and the denominator remains the same.

In the second method, we will add the additive inverse of the second number to the first number.

**1.7.2 Use Properties of Operations to Add and Subtract**

The properties of rational numbers are classified as:

- Closure Property
- Commutative Property

- Associative Property
- Identity Property
- Inverse Property

**Closure Property:**

This property states that when any two rational numbers are added, subtracted, multiplied or divided, the result is also a rational number.

x/y ±m/n=xn ±ym / yn

**Example:**

Addition:7/6 + 2/5 – 47/30

Subtraction: 5/1 – 1/3 – 1/2

**Commutative Property:**

This property states that two rational numbers can be added without concern of their order.

This property does not hold true for subtraction of rational numbers.

x / y+m / n=m / n+x / y

**Example:**

**Associative Property:**

Adding any three rational numbers in such a way, that they can be rearranged.

This property does not hold true for subtraction of rational numbers.

x / y+(m / n+p / q)=(x / y+m / n)+pq

**Example: **

1/2(3+8/12)-(2-1/4) + 2/3 -17/12- 17/12.

**Additive Identity:**

Zero is the additive identity of any rational number. When we add zero to any rational number, the result we get is the number itself.

x / y+0=x / y

**Example:**

1/ 2+0=1 / 2

**Additive Inverse:**

For any rational number

x/y, there exists−x / y such that the addition of both the numbers gives 0.

−x/ y is the additive inverse of x/y.

xy+(−xy)=0xy+−xy=0

**Example:**

The additive inverse of 1/3 is −1/3.Hence 1 / 3+(−1 / 3)=0

**1.7.3 Find distances on a number line**

**Rational Numbers on the Number Line:**

Representation of whole numbers, natural numbers and integers on a number line is done as follows:

Rational Numbers can also be represented on a number line like integers i.e., positive rational numbers are on the right to 0 and negative rational numbers are on the left of 0.

Representation of rational numbers can be done on a number line as follows:

**Basic rules on representing rational numbers on the number line:**

- If the rational number fraction is proper then, it lies between 0 and 1.

- If the rational number fraction is improper then, first convert it to mixed fraction and then the given rational number lies between the whole numbers.

**Example:**

Represent 4/7 on the number line.

**Step 1:**

Draw a number line

**Step 2:**

As the number 4/7 is a positive number so it will be on right side of zero.

**Step 3:**

After zero mark,

1/7,2/7 ,3/7, 4/7, ,5/7 ,6/7 , ( ,7/7 = 1).

## Exercise:

- Simplify the expression: 5/6 +(-2/5)-(-2/10)
- What should be subtracted from(3/4 – 2/3) to get 1/6 ?
- Add: -7/27 and 11/18
- Add: -3 and 3/5
- Subtract: -8/9 from 3/5
- Subtract: 3/4 from 1/3
- The roots of a plant reach 3 3/4 inches below the ground. How many inches is the plant above the ground?
- Simplify the expression ( 13.2) + 8.1.
- When Sam simplified the expression -2.6 +(-5.4), he got 2.8. What mistake did Sam likely make?
- Find the sum or difference of the expression: 3.2 – (–5.7).

### What have we learnt:

- Learn how to add and subtract rational numbers with different signs
- Add and subtract rational numbers with same denominator
- Add and subtract rational numbers with different denominators
- Use properties of operations to add and subtract
- Find the distance between any two points on a number line

Find the absolute value of their difference