What Rational Number Should Be Subtracted From -9/25 To Get 7/10?

Jun 24, 2025 by ADMIN 66 views

Subtracting Rational Numbers A Comprehensive Guide to Solving -9/25 to Get 7/10

In the realm of mathematics, rational numbers play a crucial role, forming the building blocks for various numerical operations and problem-solving scenarios. One such scenario involves understanding the concept of subtraction within the domain of rational numbers. This article delves into the intricacies of subtracting rational numbers, focusing on a specific problem: determining the rational number that should be subtracted from -9/25 to obtain 7/10. By exploring this problem, we aim to provide a comprehensive guide to understanding the underlying principles and techniques involved in subtracting rational numbers.

Understanding Rational Numbers

To effectively address the problem at hand, it's essential to have a solid grasp of what rational numbers are. In essence, a rational number is any number that can be expressed as a fraction p/q, where p and q are integers, and q is not equal to zero. This definition encompasses a wide range of numbers, including integers, fractions, and terminating or repeating decimals. For instance, -9/25 and 7/10, the numbers involved in our problem, are both rational numbers as they fit the p/q format.

The Subtraction of Rational Numbers

Subtracting rational numbers involves finding the difference between two rational numbers. This process can be visualized as moving along a number line. When we subtract a number, we move to the left on the number line, and when we add a number, we move to the right. To subtract rational numbers, we need to ensure that they have a common denominator. This common denominator serves as the reference point for comparing and subtracting the numerators.

Finding the Common Denominator

The first step in subtracting rational numbers is to find the least common multiple (LCM) of the denominators. The LCM is the smallest number that is a multiple of both denominators. In our problem, the denominators are 25 and 10. To find the LCM, we can list the multiples of each number until we find a common one:

Multiples of 25: 25, 50, 75, 100, ... Multiples of 10: 10, 20, 30, 40, 50, ...

The LCM of 25 and 10 is 50. This means that we need to convert both fractions to have a denominator of 50.

Converting Fractions to a Common Denominator

To convert -9/25 to a fraction with a denominator of 50, we multiply both the numerator and denominator by 2:

(-9/25) * (2/2) = -18/50

Similarly, to convert 7/10 to a fraction with a denominator of 50, we multiply both the numerator and denominator by 5:

(7/10) * (5/5) = 35/50

Now, we have both fractions with a common denominator: -18/50 and 35/50.

Setting Up the Equation

Let's represent the unknown rational number that we need to subtract from -9/25 as 'x'. According to the problem statement, subtracting 'x' from -9/25 should result in 7/10. We can express this relationship as an equation:

-9/25 - x = 7/10

Our goal is to solve for 'x', which will give us the rational number that needs to be subtracted.

Solving for x

To solve for 'x', we need to isolate it on one side of the equation. We can do this by adding x to both sides and subtracting 7/10 from both sides:

-9/25 - x + x = 7/10 + x

-9/25 = 7/10 + x

Now, subtract 7/10 from both sides:

-9/25 - 7/10 = 7/10 + x - 7/10

-9/25 - 7/10 = x

We already converted the fractions to have a common denominator of 50, so we can substitute those values:

-18/50 - 35/50 = x

Now, subtract the numerators:

(-18 - 35)/50 = x

-53/50 = x

Therefore, the rational number that should be subtracted from -9/25 to get 7/10 is -53/50.

To ensure that our solution is correct, we can substitute -53/50 back into the original equation and verify that it holds true:

-9/25 - (-53/50) = 7/10

Convert -9/25 to -18/50:

-18/50 - (-53/50) = 7/10

Subtracting a negative number is the same as adding its positive counterpart:

-18/50 + 53/50 = 7/10

Combine the fractions:

(-18 + 53)/50 = 7/10

35/50 = 7/10

Simplify the fraction:

7/10 = 7/10

The equation holds true, confirming that our solution of -53/50 is correct.

While we have solved this problem using a step-by-step approach, there are alternative methods that can be employed. One such method involves using the concept of additive inverses. The additive inverse of a number is the number that, when added to the original number, results in zero. For example, the additive inverse of 7/10 is -7/10.

To solve the problem using additive inverses, we can rewrite the equation as:

-9/25 - x = 7/10

Add the additive inverse of 7/10 to both sides:

-9/25 - x + (-7/10) = 7/10 + (-7/10)

-9/25 - x - 7/10 = 0

Now, we can isolate 'x' by adding x to both sides:

-9/25 - 7/10 = x

This leads us back to the same equation we solved earlier, and we can proceed with the same steps to find the solution of x = -53/50.

When subtracting rational numbers, there are several common mistakes that students often make. One mistake is failing to find a common denominator before subtracting the fractions. This can lead to incorrect results as the fractions are not being compared on the same scale. Another mistake is mishandling negative signs. Remember that subtracting a negative number is the same as adding its positive counterpart.

The concept of subtracting rational numbers has numerous real-world applications. For instance, in financial calculations, subtracting rational numbers can be used to determine the difference between two amounts or to calculate the remaining balance after a payment. In measurement, subtracting rational numbers can be used to find the difference in length, weight, or volume. Understanding how to subtract rational numbers is a valuable skill in various practical scenarios.

In this article, we have explored the process of subtracting rational numbers, focusing on the specific problem of finding the rational number that should be subtracted from -9/25 to get 7/10. We have walked through the steps of finding a common denominator, setting up the equation, solving for the unknown variable, and verifying the solution. We have also discussed alternative approaches and common mistakes to avoid. By mastering the concepts and techniques presented in this article, you will be well-equipped to tackle a wide range of problems involving the subtraction of rational numbers.

Subtracting rational numbers involves finding the difference between two rational numbers.
To subtract rational numbers, they must have a common denominator.
The least common multiple (LCM) is used to find the common denominator.
The additive inverse can be used as an alternative method for solving subtraction problems.
Common mistakes include failing to find a common denominator and mishandling negative signs.
Subtracting rational numbers has various real-world applications in finance, measurement, and other fields.

Q1: What is a rational number? A rational number is a number that can be expressed as a fraction p/q, where p and q are integers, and q is not equal to zero.

Q2: Why is a common denominator needed when subtracting rational numbers? A common denominator is needed to ensure that the fractions are being compared on the same scale, allowing for accurate subtraction.

Q3: What is the least common multiple (LCM)? The LCM is the smallest number that is a multiple of both denominators.

Q4: What is the additive inverse? The additive inverse of a number is the number that, when added to the original number, results in zero.

Q5: What are some real-world applications of subtracting rational numbers? Subtracting rational numbers has applications in finance, measurement, and various other fields.

Rational numbers, subtraction, common denominator, least common multiple, LCM, additive inverse, equation, solving for x, real-world applications, verification, fractions, numerators, denominators, negative signs, mathematical operations.