# Subtracting Rational Numbers by Adding the Inverse

## Subtracting Rational Numbers by Adding the Inverse

In this text, we're tackling a handy mathematical technique Subtracting Rational Numbers by Adding the Inverse. This method can simplify your calculations and save you time. Don't let the terminology intimidate you — we're going to explore this topic step by step, ensuring you grasp the concept and learn how to apply it effectively.

## Subtracting by Adding the Inverse – Explanation

When we talk about subtracting rational numbers by adding the inverse, we're essentially discussing a more efficient way to perform subtraction.

A rational number is any number that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, where $q \neq 0$. This includes integers, fractions, mixed numbers, and finite and repeating decimals.

The inverse of a number refers to its additive inverse, which is the number that, when added to the original number, results in zero. For example, the additive inverse of $7$ is $-7$ because $7 + (-7) = 0$.

Subtracting a negative rational number is particularly interesting because you effectively add the opposite of a negative, which is a positive. For instance, subtracting $-3$ is the same as adding $3$. This makes calculations easier because adding is often more straightforward than subtracting, especially when dealing with negatives.

What is the additive inverse of $-\frac{3}{4}$?
If you have $5$ and you want to subtract $2$, what inverse number would you add to $5$?
Is the additive inverse of a positive decimal also a positive decimal?

## Subtracting Rational Numbers by Adding the Inverse – Example

Let's go through an example to see this process in action.

Subtract $\frac{2}{3}$ from $3$ using the inverse.

• Identify the number to subtract: We want to subtract $\frac{2}{3}$ from $3$.
• Find the inverse: The inverse of $\frac{2}{3}$ is $-\frac{2}{3}$.
• Add the inverse: Perform the addition $3 + (-\frac{2}{3})$.

Illustration request: A number line with $3$ on it and then showing a move of $-\frac{2}{3}$ to the left, landing at $2\frac{1}{3}$.

• Result: $3 + (-\frac{2}{3}) = 2\frac{1}{3}$.

Let’s now take a look at subtracting a negative rational number.

Subtract $-\frac{4}{5}$ from $2$ using the inverse.

• Identify the number to subtract: We want to subtract $-\frac{4}{5}$ from $2$.
• Find the inverse: The inverse of $-\frac{4}{5}$ is $\frac{4}{5}$ because the additive inverse of a negative number is its positive counterpart.
• Add the inverse: Perform the addition $2 + \frac{4}{5}$.

Illustration request: A number line with $2$ on it and then showing a move of $\frac{4}{5}$ to the right, landing at $2\frac{4}{5}$.

• Result: $2 + \frac{4}{5} = 2\frac{4}{5}$.

## Subtracting Rational Numbers by Adding the Inverse – Guided Practice

Now, let's do a guided practice together.

Subtract $\frac{5}{6}$ from $4$ by adding the inverse.
Subtract $-1.2$ from $4.5$ by adding the inverse.

Now apply this to your independent practice!

## Subtracting Rational Numbers by Adding the Inverse – Summary

Key Learnings from this Text:

• To subtract a rational number, add its additive inverse to the other number.
• The additive inverse of a number is what, when added to the original number, equals zero.
• This method can simplify the process, especially with fractions and negative numbers.

Keep practicing this technique, and you'll soon master subtracting rational numbers with ease. Don't forget to check out other interactive practice problems and learning materials on our platform!

## Subtracting Rational Numbers by Adding the Inverse – Frequently Asked Questions

What is the purpose of adding the inverse when subtracting rational numbers?
Can this method be used for all rational numbers?
How do you find the inverse of a fraction?
Is subtracting by adding the inverse the same as the original subtraction?
Does the order of numbers affect the result when subtracting by adding the inverse?
Can this method be applied to mixed numbers?
What are some real-world applications of subtracting rational numbers by adding the inverse?
How can this method help with understanding algebraic expressions?
Is it necessary to use a common denominator when subtracting fractions by adding the inverse?
How do you teach subtracting rational numbers by adding the inverse to students who are visual learners?

## Subtracting Rational Numbers by Adding the Inverse exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the learning text Subtracting Rational Numbers by Adding the Inverse.
• ### Define the keywords.

Hints

$3$ and $-3$ are additive inverses because $3 + -3 = 0$

Rational numbers include integers, fractions and decimals.

Examples of rational numbers include -

Integers: $5, 7, -4, 126$

Fractions: $\frac{3}{5}, \frac{9}{2}, 5\frac{7}{8}$

Decimals: $4.2, 7.0451, 0.\dot{1}\dot{7}$

Solution
• An additive inverse is a pair of numbers that, when added together, equal zero.
• An integer is a whole number.
• A rational number is any number that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, where $q \neq 0$.
• A fraction is a part of a whole.
• ### Find the maths sentence that uses the additive inverse.

Hints

Additive inverses cancel each other out. The additive inverse of $11$ is $-11$.

Subtracting a number is the same as adding its inverse. For example $6 - 2 = 6 + -2$.

Example: $8 - -3$

• Identify the number to subtract: We want to subtract $-3$ from $8$.
• Find the inverse: The inverse of $-3$ is $+3$.
• Add the inverse: The equivalent math sentence is $8 + 3$.

The equals sign does not always mean find the answer, here $=$ means the two statements are equivalent to each other.

Solution

$3 - 2 = 3 + -2$

$5 - -4 = 5 + 4$

$5 - 4 =5 + -4$

$6.1 - -2 =6.1 + 2$

• ### Choose the correct answer for each calculation.

Hints

First,

• Identify the number to subtract.

Then,

• Find the inverse.

Finally,

Solution

$4 -2\frac{1}{2} = 4 + -2\frac{1}{2} =$ $\bf{1\frac{1}{2}}$

$9 - -3 = 9 + 3 =$ $\bf{12}$

$8 - -1.6 = 8 + 1.6 =$ $\bf{9.6}$

• ### Solve the subtractions.

Hints

To subtract a rational number, add its additive inverse to the other number.

The additive inverse of a number is what, when added to the original number, equals zero.

An example:

$8 - -2\frac{2}{5} = 8 + 2\frac{2}{5} = 10\frac{2}{5}$

Solution

$3 - 1.5 = 3 + -1.5 = \bf{1.5}$

$7 - -3\frac{1}{4} = 7 + 3\frac{1}{4} = \bf{10\frac{1}{4}}$

$9 - 5\frac{1}{4} = 9 + -5\frac{1}{4} = \bf{3\frac{3}{4}}$

$4 - - 0.5 = 4 + 0.5 = \bf{4.5}$

• ### What is $5 - -2$?

Hints

Find the additive inverse of $-2$.

Now add the additive inverse to $5$ to find the answer.

A similar example:

$8 - - 6 = 8 + 6 = 14$

Solution

$5 - -2 = 5 + 2 = 7$

• ### Solve the calculations and place in order from smallest to largest.

Hints

Solve each subtraction first using additive inverses.

Write the solutions down and order them from the smallest answer to the largest answer.

Solution

The correct order is:

$\bf{6- 4.9, 10 - 5\frac{1}{2}, 5 - -1.2, 8 - -1\frac{3}{5}}$

because

$6- 4.9 = 6 + -4.9 = 1.1$

$10 - 5\frac{1}{2} = 10 + -5\frac{1}{2} = 4\frac{1}{2}$

$5 - -1.2 = 5 + 1.2 = 6.2$

$8 - -1\frac{3}{5} = 8 + 1\frac{3}{5} = 9\frac{3}{5}$

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Subtracting Rational Numbers by Adding the Inverse
CCSS.MATH.CONTENT.7.NS.A.1.C