Adding and Subtracting of Rational Numbers

Adding and Subtracting of Rational Numbers

Rational numbers are those numbers that could be presented in the form a/b where a and b are integers and b is not 0. They consist of the positive, negative, and zero numbers. Studying the addition and subtraction of rational numbers is a significant aspect of mathematics as it enables us to comprehend the way of interactions and changes of numbers.

Just as whole numbers, and integers, addition and subtraction of rational numbers have some rules. However, when fractions or negative numbers are involved, we must be careful with signs and denominators. 

Understanding these operations makes it easier to solve math problems related to measurements, algebra, and even real-life situations like handling money or temperature changes. Students will learn about adding rational numbers, subtracting rational numbers, examples and more below.

Adding Rational Numbers

In addition to rational numbers, one should consider the signs of the numbers and the presence or absence of similar or different denominators. The outcome will be determined by whether the numbers are positive, negative, or in one case, positive and negative.

Rules for Adding Rational Numbers

Follow these basic rules when adding rational numbers:

1. If the signs are the same

• Add their absolute values.

• Keep the common sign.

Example: (+6) + (+3) = +9 and (-4) + (-5) = -9

2. If the signs are different

• Subtract the smaller absolute value from the larger.

• Keep the sign of the number with the larger absolute value.

Example: (+7) + (-5) = +2 and (-8) + (+3) = -5

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3. Adding fractions with the same denominator

• Add the numerators directly.

• Keep the same denominator.

Example: 3/8 + 2/8 = 5/8

4. Adding fractions with different denominators

• Find the least common denominator (LCD).

• Convert both fractions so the denominators match.

• Add the numerators.

Example: 1/3 + 1/4 = 4/12 + 3/12 = 7/12

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5. Simplify the result

• Reduce the final fraction to its simplest form.

Key Points to Remember

• Always check the sign of each number before adding.

• Convert mixed numbers or decimals into fractions when needed.

• Simplify the result after addition.

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Subtracting Rational Numbers

The process of subtracting rational numbers is very similar to addition. In fact, subtraction can be thought of as adding the opposite. This means whenever a number is subtracted, its opposite sign is added.

Rules for Subtracting Rational Numbers

1. Change the subtraction sign to addition

• Replace subtraction with addition and change the sign of the second number.

Example: 5 - 3 = 5 + (-3) and (-7) - (-2) = -7 + 2

2. Apply the rules of addition

• After changing to addition, follow the same rules used for adding rational numbers.

Example: (-4) - (+6) = -4 + (-6) = -10

3. Subtraction of fractions with the same denominator

• Subtract the numerators directly and keep the same denominator.

Example: 7/9 - 3/9 = 4/9

4. Subtraction of fractions with different denominators

• Find the least common denominator (LCD), make the denominators the same, and then subtract.

Example: 3/4 - 1/2 = 3/4 - 2/4 = 1/4

5. Simplify the result

• Reduce the final answer to its simplest form.

Important Notes

• Subtraction can always be turned into addition to the opposite.

• Be careful with negative signs, as double negatives become positive.

• Always check if denominators are the same before subtracting fractions.

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Adding Rational Numbers Examples

Examples help in understanding the rules of adding rational numbers more clearly. Here are some step-by-step examples:

• Example 1: Add 3/5 and 2/5  = (3 + 2)/5 = 5/5 = 1

• Example 2: Add -4/7 and -2/7  = (-4 + -2)/7 = -6/7

• Example 3: Add 5/8 and -3/8  = (5 + -3)/8 = 2/8 = 1/4

• Example 4: Add -2.5 and 1.2 = -2.5 + 1.2 = -1.3

Observation

In all these adding rational numbers examples, note that:

• When signs are the same, numbers are added.

• When signs are different, subtraction is performed.

• Denominators must be the same to add fractions.

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Subtracting Rational Numbers Examples

The following subtracting rational numbers examples show how subtraction works step by step:

• Example 1: Subtract 5/6 - 1/6  = (5 - 1)/6 = 4/6 = 2/3

• Example 2: Subtract 3/4 - (-1/4) = 3/4 + 1/4 = 4/4 = 1

• Example 3: Subtract -5/9 - 2/9 = (-5 - 2)/9 = -7/9

• Example 4: Subtract -3.6 - (-1.4) = -3.6 + 1.4 = -2.2

Observation

In all subtracting rational numbers examples, subtraction becomes easier when the problem is changed into an additional problem.

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