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 rewrite 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 rewrite 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
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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Worksheet on Finding Rational Numbers between Two Rational Numbers
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