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Chapter 8: Problem 49

Give the result as a fraction in simplest form. $$ \frac{2}{11}-\frac{x}{11} $$

Short Answer

Expert verified
\( \frac{2-x}{11} \)

Step by step solution

01

Understand the Problem

We need to subtract two fractions that have the same denominator. The expression given is \( \frac{2}{11} - \frac{x}{11} \). Both fractions share the denominator 11.
02

Subtract the Numerators

Since the denominators are the same, we can directly subtract the numerators. The expression becomes \( \frac{2-x}{11} \).
03

Simplify the Fraction

The fraction \( \frac{2-x}{11} \) is already in its simplest form since the numerator and the denominator have no common divisors other than 1. Therefore, no further simplification is needed.

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Key Concepts

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

Simplifying Fractions
Simplifying fractions is an essential step in working with fractions. The goal is to reduce a fraction to its simplest form. A fraction is considered simplified if the numerator and denominator share no common factors other than 1.

To simplify fractions:
  • Find the greatest common factor (GCF) of the numerator and the denominator.
  • Divide both the numerator and the denominator by the GCF.
  • The fraction after dividing by the GCF is your simplified fraction.
For instance, if you have the fraction \( \frac{4}{8} \), the GCF of 4 and 8 is 4. Dividing both by 4 gives you \( \frac{1}{2} \), which is the simplified fraction.

In the solution provided for your exercise, the fraction \( \frac{2-x}{11} \) is already in simplest form because 2 and 11 have no common factors other than 1. This is why no further simplification was needed.
Common Denominators
Fractions must have common denominators to be subtracted directly. This means both fractions should have the same bottom number, or denominator, before you proceed with operations like addition or subtraction. Having a common denominator simplifies the process.

For example:
  • If you're subtracting \( \frac{1}{4} \) from \( \frac{3}{4} \), you notice both have the common denominator 4.
  • This allows you to directly subtract the numerators.
  • The result will be \( \frac{2}{4} \). Then, you simplify this to \( \frac{1}{2} \).

In your exercise, \( \frac{2}{11} - \frac{x}{11} \) already has a common denominator of 11, so you can move directly to subtracting the numerators without any additional steps to create a common denominator.
Numerator Subtraction
Subtracting fractions with the same denominator involves subtracting the numerators directly. This process is relatively straightforward:
  • Keep the denominator the same.
  • Subtract the second numerator from the first.
For example, given \( \frac{5}{13} - \frac{3}{13} \):
  • Subtract 3 from 5 to get 2.
  • Your answer is \( \frac{2}{13} \).

In your exercise, subtract \( x \) from 2 in the expression \( \frac{2}{11} - \frac{x}{11} \):
  • The result is \( \frac{2-x}{11} \).
  • The denominator remains 11, and the numerator becomes \( 2-x \).
So, focus on only adjusting the numerators when denominators are already matched. This keeps fraction subtraction efficient and clear.

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