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Chapter 4: Problem 7

$$ \frac{3}{14}-\frac{3}{7} $$

Short Answer

Expert verified
\( \frac{-3}{14} \)

Step by step solution

01

Find the Least Common Denominator

To subtract the fractions \( \frac{3}{14} \) and \( \frac{3}{7} \), we need to find a common denominator. The denominators are 14 and 7. The least common multiple of 14 and 7 is 14, so the least common denominator is 14.
02

Convert Fractions to the Common Denominator

The fraction \( \frac{3}{14} \) already has the denominator 14. For \( \frac{3}{7} \), we need to convert it to have the denominator 14. Multiply the numerator and the denominator by 2 to get \( \frac{6}{14} \).
03

Subtract the Fractions

Now, with the common denominator, subtract the numerators: \( \frac{3}{14} - \frac{6}{14} = \frac{3 - 6}{14} = \frac{-3}{14} \).

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Key Concepts

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

Understanding the Least Common Denominator
When subtracting fractions, we can't simply subtract the numerators unless the denominators are the same. This is where the concept of the least common denominator (LCD) comes into play. The LCD is the smallest number that both denominators can divide into without leaving a remainder.
For example, in the exercise with fractions \( \frac{3}{14} \) and \( \frac{3}{7} \), the denominators are 14 and 7. To find the LCD, first list the multiples of each denominator:
  • Multiples of 14: 14, 28, 42, ...
  • Multiples of 7: 7, 14, 21, 28, ...
The smallest multiple common in both lists is 14. So, the LCD is 14. Once we have the LCD, we can convert each fraction so they have the same denominator.
Converting Fractions to a Common Denominator
Once the least common denominator is identified, we need to convert each fraction so that they have this common denominator. This process involves adjusting the numerator to make sure the fraction's value remains unchanged.
For instance, with \( \frac{3}{7} \), we need to convert it to have a denominator of 14, which is the LCD. Since 14 is twice 7, we multiply both the numerator and the denominator of \( \frac{3}{7} \) by 2:
  • Numerator: \(3 \times 2 = 6\)
  • Denominator: \(7 \times 2 = 14\)
Thus, \( \frac{3}{7} \) is converted to \( \frac{6}{14} \). Now, both fractions have the common denominator, making them ready for subtraction.
Working with Negative Fractions
After converting fractions to a common denominator, subtraction can sometimes result in negative fractions. A negative fraction indicates a value that is less than zero, represented with a negative sign before the numerator.
For example, when you subtract \( \frac{6}{14} \) from \( \frac{3}{14} \):
  • Subtract the numerators: \(3 - 6 = -3\)
  • Keep the denominator the same: 14
This gives us \( \frac{-3}{14} \).
Negative fractions work similarly to positive fractions but represent values on the opposite side of zero on the number line. It's important to treat negative signs with care during calculations to ensure accurate results.

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Most popular questions from this chapter

Perform each indicated operation. If the result is an improper fraction, also write the improper fraction as a mixed number. $$2+\frac{2}{3}$$

Answer true or false for each statement. The sum of a negative fraction and a positive fraction is always a positive number.

$$\text { Evaluate each expression if } x=\frac{3}{4} \text { and } y=-\frac{4}{7}$$ $$4 x+y$$

Perform each indicated operation. If the result is an improper fraction, also write the improper fraction as a mixed number. $$9-\frac{5}{6}$$

Solve. See the Concept Checks in this section. Round the mixed number \(5 \frac{11}{12}\) to the nearest whole number
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