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Chapter 1: Problem 453

Simplify. $$ -\frac{7}{24}+\frac{2}{24} $$

Short Answer

Expert verified
-\frac{5}{24}

Step by step solution

01

- Identify the common denominator

Both fractions \(-\frac{7}{24}\) and \(+\frac{2}{24}\) have the same denominator, which is 24. This means they can be directly added or subtracted from each other.
02

- Combine the numerators

Since the denominators are the same, combine the numerators by performing the subtraction: \(-7 + 2 = -5\).
03

- Form the new fraction

Place the result from Step 2 over the common denominator: \(-\frac{5}{24}\).
04

- Simplify the fraction if possible

Check if the fraction \(-\frac{5}{24}\) can be simplified further. In this case, 5 and 24 have no common factors other than 1, so the fraction is already in its simplest form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Denominator
When working with fractions, the common denominator is crucial for performing addition or subtraction. The common denominator is the same number in the denominator of each fraction involved.

In our example, we had the fractions \(-\frac{7}{24}\) and \(\frac{2}{24}\). Here, 24 is the common denominator. Because both fractions share this denominator, we can easily combine them. The common denominator eliminates the need for complex calculations and simplifies your work significantly. Always ensure that the denominators are the same before you combine any fractions. This can be achieved by finding the least common denominator if they are different initially.
Combining Numerators
Combining numerators is the next step once you have identified a common denominator. This involves adding or subtracting the top numbers, known as the numerators.

In our exercise, we combined the numerators of \(-\frac{7}{24}\) and \(\frac{2}{24}\). Since the denominators are both 24, you simply perform the operation with the numerators: \(-7 + 2\). This results in \(-5\).

So, your new fraction becomes \(-\frac{5}{24}\). Remember, the denominator stays the same while only the numerators are combined. This method ensures that your fraction remains accurate and easy to manage.
Simplification of Fractions
The last step is to simplify the fraction, if possible. Simplifying a fraction involves finding the greatest common factor (GCF) of the numerator and the denominator and dividing both by this number.

In our example, we have the fraction: \(-\frac{5}{24}\). To simplify, we check if there are any common factors between 5 and 24 other than 1. Since there aren't any in this case, the fraction \(-\frac{5}{24}\) is already in its simplest form.

Sometimes, simplifying a fraction can make it easier to understand or work with in further calculations. Always check if a fraction can be simplified to make your math tasks simpler.

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