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Chapter 6: Problem 78

Perform each indicated operation. See Section 1.3 $$ \frac{4}{10}-\frac{7}{10} $$

Short Answer

Expert verified
The result of \( \frac{4}{10} - \frac{7}{10} \) is \( \frac{-3}{10} \).

Step by step solution

01

Identify the Operation

The exercise requires us to perform a subtraction of two fractions: \( \frac{4}{10} \) and \( \frac{7}{10} \). Both fractions have the same denominator, which means they are 'like' fractions.
02

Subtract the Numerators

Since the fractions have the same denominator, we subtract the numerators directly: \( 4 - 7 = -3 \).
03

Write the Resulting Fraction

The resulting fraction from the subtraction is \( \frac{-3}{10} \). This fraction cannot be simplified further since 3 and 10 have no common factors other than 1.

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Key Concepts

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

Like Fractions
Like fractions are ones that share the same denominator. In simpler terms, the bottom number of each fraction is the same. This is important because having like denominators makes the process of subtracting or adding fractions much simpler. When you encounter fractions such as \( \frac{4}{10} \) and \( \frac{7}{10} \), you don't have to worry about changing the denominators. This is because they already match. As a result, you can jump straight into adding or subtracting the numerators.
Numerators
The numerator is the top number in a fraction. It represents how many parts of the whole you have. For instance, in \( \frac{4}{10} \), 4 is the numerator, which means you have 4 parts out of 10 in total.When it comes to subtraction, you only work with the numerators if the fractions are like fractions. So for \( \frac{4}{10} - \frac{7}{10} \), you subtract the numerators: 4 - 7. This gives you \(-3\). The new numerator \(-3\) is placed over the common denominator 10, resulting in \( \frac{-3}{10} \). This operation shows why it's crucial to be comfortable with negative numbers in fractions.
Common Factors
To simplify a fraction, it is important to identify common factors between the numerator and the denominator. A common factor is a number that divides exactly into two numbers.In the case of \( \frac{3}{10} \) or \( \frac{-3}{10} \), the numerator and the denominator have only 1 as a common factor.Since there are no other common factors, you can鈥檛 simplify the fractions further. To find common factors, the following steps are useful:
  • List the factors of the numerator.
  • List the factors of the denominator.
  • Identify the greatest common factor (GCF).
When fractions have no common factors (other than 1), they are already in their simplest form.
Simplification of Fractions
The simplification of fractions involves reducing them to their simplest form, where the numerator and the denominator share no common factors other than 1.To simplify a fraction:- Identify the common factors of the numerator and the denominator.- Divide both by their greatest common factor.In cases where there are no factors aside from 1, like with \( \frac{-3}{10} \), the fraction is already simplified, and no further action is required. Simplification is important because it makes fractions easier to understand and work with, both in calculations and in comparative operations.

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