/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 59 Perform each indicated operation... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 59

Perform each indicated operation. $$ \frac{2}{6}-\frac{3}{4} $$

Short Answer

Expert verified
The result of the operation \(\frac{2}{6}-\frac{3}{4}\) is \(-\frac{5}{12}\).

Step by step solution

01

Find a Common Denominator

To subtract fractions, they must have the same denominator. The denominators here are 6 and 4. The least common multiple of 6 and 4 is 12. Thus, the common denominator will be 12.
02

Convert Fractions to Have the Same Denominator

Convert \(\frac{2}{6}\) and \(\frac{3}{4}\) so that they both have the denominator of 12. Convert \(\frac{2}{6}\):\[\frac{2}{6} = \frac{2 \times 2}{6 \times 2} = \frac{4}{12}\]Convert \(\frac{3}{4}\):\[\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}\]
03

Subtract the Fractions

Now that both fractions have the same denominator, subtract them:\[\frac{4}{12} - \frac{9}{12} = \frac{4-9}{12} = -\frac{5}{12}\]
04

Simplify the Result

The fraction \(-\frac{5}{12}\) is already in its simplest form since 5 and 12 have no common factors other than 1.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Least Common Multiple
When subtracting fractions, the first step is to find a common denominator. This requires determining the least common multiple (LCM) of the denominators. For the fractions \( \frac{2}{6} \) and \( \frac{3}{4} \), their denominators are 6 and 4, respectively. The LCM is the smallest number that both denominators divide into evenly.
The prime factorizations of these numbers can help. For 6, it is 2 and 3. For 4, it is 2 and 2. The LCM is 12, which combines the highest power of each prime (2 x 2 x 3 = 12). Once you have this, you can set both fractions over this common denominator.
Common Denominator
Using a common denominator helps align the fractions so they can be subtracted easily. Since we've determined that the LCM of 6 and 4 is 12, this number becomes our common denominator.
To convert, adjust the numerators accordingly. For \( \frac{2}{6} \), multiply the numerator and denominator by 2 to get \( \frac{4}{12} \). For \( \frac{3}{4} \), multiply by 3 to reach \( \frac{9}{12} \). Now both fractions are prepared for direct subtraction.
Simplifying Fractions
Once you subtract fractions, it's essential to simplify your result, which means reducing it to its simplest form. After subtracting, the result of \( \frac{4}{12} - \frac{9}{12} \) is \( -\frac{5}{12} \).
In this case, the fraction \( -\frac{5}{12} \) is already simplified because 5 is a prime number and has no common factors with 12 other than 1. Simplifying helps to ensure the fraction is easy to understand and use in further calculations.
Fraction Conversion
To work with fractions in operations like subtraction, conversion to the same denominator is crucial. You change each fraction to equivalent fractions with a shared denominator.
This involves scaling up both the numerator and denominator so they match without altering the value of the fraction. By converting \( \frac{2}{6} \) and \( \frac{3}{4} \) to have the denominator of 12, we created \( \frac{4}{12} \) and \( \frac{9}{12} \). It's a simple but essential technique that forms the basis for adding, subtracting, and comparing fractions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Then list four equivalent forms for each rational expression. $$ -\frac{x+11}{x-4} $$

Does \(\frac{(x-3)(x+3)}{x-3}\) have the same value as \(x+3\) for all real numbers? Explain why or why not.

Perform each indicated operation. $$ \frac{13}{20} \div \frac{2}{9} $$

Simplify each expression. Each exercise contains a four-term polynomial that should be factored by grouping. $$ \frac{5 x+15-x y-3 y}{2 x+6} $$

Perform the indicated operations. $$ \frac{5 x}{7}+\frac{9 x}{7} $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.