We will learn the subtraction of rational number with different denominator. To find the difference of two rational numbers which do not have the same denominator, we follow the following steps:

Step I: Let us obtain the rational numbers and see whether their denominators are positive or not. If the denominator of one (or both) of the numerators is negative, re-arrange it so that the denominators become positive.

Step II: Obtain the denominators of the rational numbers in step I.

Step III: Find the lowest common multiple of the denominators of the two given rational numbers.

Step IV: Express both the rational numbers in step I so that the lowest common multiple of the denominators becomes their common denominator.

Step V: Write a rational number whose numerator is equal to the difference of the numerators of rational numbers obtained in step IV and denominators is the lowest common multiple obtained in step III.

Step VI: The rational number obtained in step V is the required difference (simplify if required).Following examples will illustrate the above procedure.

**1.** Subtract 9 from 4/5

**Solution:**

We have, 9 = 9/1

Clearly, denominators of the two rational numbers are positive. We now re-write them so that they have a common denominator equal to the LCM of the denominators.

In this case the denominators are 1 and 5.

The LCM of 1 and 5 is 5.

We have, 9 = 9/1 = 9 × 5/1 × 5 = 45/5

Therefore, 4/5 - 9

= 4/5 - 9/1

= 4/5 - 45/5

= (4 - 45)/5

= -41/5

Therefore, 4/5 - 9 = -41/5

**2.** Find the difference of: -3/4 - 5/6

**Solution: **

The denominators of the given rational numbers are 4 and 6 respectively.

LCM of 4 and 6 = (2 × 2 × 3) = 12.

Now, -3/4 = (-3) × 3/4 × 3 = -9/12

and 5/6 = 5 × 2/6 × 2 = 10/12

Therefore, -3/4 - 5/6

= -9/12 - 10/12

= (-9 - 10)/12

= -19/12

Therefore, -3/4 - 5/6 = -19/12

**3.** Simplify: 3/-15 - 7/-12

**Solution:**

First we write each of the given numbers with positive denominator.

3/-15 = 3 × (-1)/(-15) × (-1) = -3/15, [Multiplying the numerator and denominator by -1]

⇒ 3/-15 = -3/15

7/-12 = 7 × (-1)/(-12) × (-1) = -7/12, [Multiplying the numerator and denominator by -1]

⇒ 7/-12 = -7/12

Therefore, 3/-15 - 7/-12 = -3/15 - (-7)/12

Now, we find the LCM of 15 and 12.

The LCM of 15 and 12 = 60

Rewriting -3/15 in the form in which it has denominator 60, we get

-3/15 = -3 × 4/15 × 4 = -12/60

Rewriting -7/12 in the form in which it has denominator 60, we get

-7/12 = -7 × 5/12 × 5 = -35/60

Therefore, 3/-15 - 7/-12

= -3/15 - (-7)/12

= -12/60 - (-35)/60

= (-12) - (-35)/60

= -12 + 35/60

= 23/60

Thus, 3/-15 - 7/-12 = 23/60.

**4.** Simplify: 11/-18 - 5/12

**Solution:**

First we write each one of the given rational numbers with positive denominator.

Clearly, denominator of 5/12 is positive.

The denominator of 11/-18 is negative.

The rational number 11/-18 with positive denominator is -11/18.

Therefore, 11/-18 - 5/12 = -11/18 - 5/12

The LCM of 18 and 12 is 36.

Rewriting -11/18 in forms having the same denominator 36, we get

-11/18 = (-11) × 2/18 × 2, [Multiplying the numerator and denominator by 2]

⇒ -11/18 = -22/36

Rewriting 5/12 in forms having the same denominator 66, we get

5/12 = 5 × 3/12 × 3, [Multiplying the numerator and denominator by 3]

⇒ 5/12 = 15/36

Therefore, 11/-18 - 5/12

= -11/18 - 5/12

= -22/36 - 15/36

= -22 - 15/36

= -37/36

Therefore, 11/-18 - 5/12 = -37/36

If a/b and c/d are two rational numbers such that b and d do not have a common factor other than 1, i.e., HCF of b and d is 1, then

a/b - c/d = a × d - c × b/b × d

For example, 5/18 - 3/13 = 5 × 13 - 3 × 18/18 × 13 = 65 - 54/234 = 11/234

and -2/11 - 3/14 = (-2) × 14 - (3 × 11)/11 × 14 = -28 - 33/154 = -61/154

● **Rational Numbers**

Introduction of Rational Numbers

Is Every Rational Number a Natural Number?

Is Every Rational Number an Integer?

Is Every Rational Number a Fraction?

Equivalent form of Rational Numbers

Rational Number in Different Forms

Properties of Rational Numbers

Lowest form of a Rational Number

Standard form of a Rational Number

Equality of Rational Numbers using Standard Form

Equality of Rational Numbers with Common Denominator

Equality of Rational Numbers using Cross Multiplication

Comparison of Rational Numbers

Rational Numbers in Ascending Order

Rational Numbers in Descending Order

Representation of Rational Numbers on the Number Line

Rational Numbers on the Number Line

Addition of Rational Number with Same Denominator

Addition of Rational Number with Different Denominator

Properties of Addition of Rational Numbers

Subtraction of Rational Number with Same Denominator

Subtraction of Rational Number with Different Denominator

Subtraction of Rational Numbers

Properties of Subtraction of Rational Numbers

Rational Expressions Involving Addition and Subtraction

Simplify Rational Expressions Involving the Sum or Difference

Multiplication of Rational Numbers

Properties of Multiplication of Rational Numbers

Rational Expressions Involving Addition, Subtraction and Multiplication

Reciprocal of a Rational Number

Rational Expressions Involving Division

Properties of Division of Rational Numbers

Rational Numbers between Two Rational Numbers

**8th Grade Math Practice**

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● **Rational Numbers - Worksheets**

Worksheet on Equivalent Rational Numbers

Worksheet on Lowest form of a Rational Number

Worksheet on Standard form of a Rational Number

Worksheet on Equality of Rational Numbers

Worksheet on Comparison of Rational Numbers

Worksheet on Representation of Rational Number on a Number Line

Worksheet on Adding Rational Numbers

Worksheet on Properties of Addition of Rational Numbers

Worksheet on Subtracting Rational Numbers

Worksheet on Addition and
Subtraction of Rational Number

Worksheet on Rational Expressions Involving Sum and Difference

Worksheet on Multiplication of Rational Number

Worksheet on Properties of Multiplication of Rational Numbers

Worksheet on Division of Rational Numbers

Worksheet on Properties of Division of Rational Numbers

Worksheet on Finding Rational Numbers between Two Rational Numbers

Worksheet on Word Problems on Rational Numbers

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