Simplify Rational Expressions Involving the Sum or Difference

In order to simplify rational expressions involving the sum or difference of three or more rational numbers, we may use the following steps:

Step I: Find the LCM of the denominator of all the numbers involved.

Step II: Write a rational number whose denominator is the LCM obtained in Step I and numerator is computed as follows:

Divide the LCM obtained in step I by the denominator of first rational number and get a quotient. Multiply the numerator of first rational number by this quotient. Repeat this procedure for all rational numbers. Retain the given signs of addition and subtraction between the given rational numbers and get an expression involving integers. Simplify this expression to get an integer as the numerator.

Step III: Reduce the rational number obtained in step II to the lowest form if it is not already so. This rational number so obtained is the required rational number.

How to simplify rational expressions involving the sum or difference of two or more rational numbers?

The following examples will illustrate the above procedure to simplify the expressions.

1. Simplify: -3/4 + 9/8 - (-5)/6

Solution:

We have,

-3/4 + 9/8 - (-5)/6 = -3/4 + 9/8 + 5/6, [Since, -(-5)/6 = 5/6]

Clearly, denominators of the three 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 4, 8 and 6.

The LCM of 4, 8 and 6 is 24.

Now, -3/4 = (-3) × 6/4 × 6 = -28/24,

9/8 = 9 × 3/8 × 3 = 27/24 and

5/6 = 5 × 4/6 × 4 = 20/24

Therefore, -3/4 + 9/8 - (-5)/6

            = -3/4 + 9/8 + 5/6

            = -28/24 + 27/24 + 20/24

            = (-28 + 27 + 20)/24

            = 19/24

Thus, -3/4 + 9/8 - (-5)/6 = 19/24

2. Simplify: 7/10 - (-7)/14 + 9/-5

Solution:

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

Clearly, denominators of 7/10 and (-7)/14 are positive.

The denominator of 9/-5 is negative.

The rational number 9/-4 with positive denominator is -9/5.

Therefore, 7/10 - (-7)/14 + 9/-5 = 7/10 - (-7)/14 + (-9)/5

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 10, 14 and 5.

The LCM of 10, 14 and 5 is 70.

Now, 7/10 = 7 × 7/10 × 7 = 49/70,

(-7)/14 = (-7) × 5/14 × 5 = (-35)/70 and

(-9)/5 = (-9) × 14/5 × 14 = (-126)/70

Therefore, 7/10 - (-7)/14 + 9/-5

            = 7/10 - (-7)/14 + (-9)/5

            = 49/70 - (-35)/70 + (-126)/70

            = 49/70 + 35/70 + (-126)/70, [Since, - (-35)/70 = 35/70]

            = [49 + 35 + (-126)]/70

            = -42/70

            = -3/5

Thus, 7/10 - (-7)/14 + 9/-5 = -3/5

● Rational Numbers

Introduction of Rational Numbers

What is Rational Numbers?

Is Every Rational Number a Natural Number?

Is Zero a Rational Number?

Is Every Rational Number an Integer?

Is Every Rational Number a Fraction?

Positive Rational Number

Negative Rational Number

Equivalent Rational Numbers

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

Addition of Rational Numbers

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

Product of Rational Numbers

Properties of Multiplication of Rational Numbers

Rational Expressions Involving Addition, Subtraction and Multiplication

Reciprocal of a Rational  Number

Division of Rational Numbers

Rational Expressions Involving Division

Properties of Division of Rational Numbers

Rational Numbers between Two Rational Numbers

To Find Rational Numbers

8th Grade Math Practice

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