/*! 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 47 Simplify the following. $$\lef... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 47

Simplify the following. $$\left(\frac{3}{4}-1\right)\left(\frac{1}{8}+\frac{1}{2}\right)$$

Short Answer

Expert verified
The simplified expression is \(-\frac{5}{32}\).

Step by step solution

01

Simplify the First Bracket

First, we take the expression inside the first bracket: \( \frac{3}{4} - 1 \). To subtract 1 from \( \frac{3}{4} \), convert 1 to a fraction with a denominator of 4: \( 1 = \frac{4}{4} \). Now subtract: \( \frac{3}{4} - \frac{4}{4} = \frac{3 - 4}{4} = -\frac{1}{4} \). Thus, the first bracket simplifies to \( -\frac{1}{4} \).
02

Simplify the Second Bracket

Next, simplify the expression in the second bracket: \( \frac{1}{8} + \frac{1}{2} \). To add these fractions, they need to have the same denominator. Convert \( \frac{1}{2} \) to have a denominator of 8: \( \frac{1}{2} = \frac{4}{8} \). Now add: \( \frac{1}{8} + \frac{4}{8} = \frac{1 + 4}{8} = \frac{5}{8} \). So, the second bracket simplifies to \( \frac{5}{8} \).
03

Multiply the Simplified Brackets

Now multiply the results from the two simplified brackets obtained in Steps 1 and 2. Thus, multiply \( -\frac{1}{4} \) by \( \frac{5}{8} \): Multiply the numerators: \( -1 \times 5 = -5 \).Multiply the denominators: \( 4 \times 8 = 32 \).Combine to get the product: \( -\frac{5}{32} \). The expression simplifies to \( -\frac{5}{32} \).

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.

Subtracting Fractions
Subtracting fractions might seem tricky, but with a systematic approach, it can be done easily. To subtract one fraction from another, you need a common denominator. This means both fractions must be rewritten so that the part below the fraction line is the same for both.
For example, consider the problem \( \frac{3}{4} - 1 \). Here, the number 1 can be converted into a fraction \( \frac{4}{4} \) so that it has the same denominator as \( \frac{3}{4} \).
Then, subtraction becomes straightforward:
  • Write both fractions with a common denominator: \( \frac{3}{4} \) and \( \frac{4}{4} \).
  • Ignore the denominators for a moment and subtract the numerators: \( 3 - 4 \).
  • Write the result over the common denominator: \( \frac{-1}{4} \).
The answer is \( -\frac{1}{4} \), and that's how you subtract fractions!
Adding Fractions
Adding fractions is very similar to subtracting them, just with a plus sign instead of a minus. The key is ensuring that both fractions have the same denominator, which allows you to add the numerators directly. This is called finding a 'common denominator.'
For instance, to add \( \frac{1}{8} + \frac{1}{2} \), you should first convert \( \frac{1}{2} \) to a fraction with a denominator of 8. Why? Because both fractions need that shared base.
  • First, change \( \frac{1}{2} \) to \( \frac{4}{8} \) to match the denominator of 8.
  • Now, simply add the numerators: \( 1 + 4 \).
  • Place the result over the denominator of 8: \( \frac{5}{8} \).
That's it! So, \( \frac{1}{8} + \frac{1}{2} = \frac{5}{8} \). Easy peasy!
Multiplying Fractions
Multiplying fractions is often considered the simplest of operations among fractions because it doesn't require a common denominator. You only need to focus on the numerators and denominators separately.
Take the result from the previous operations, \( -\frac{1}{4} \) and \( \frac{5}{8} \) as an example:
  • Multiply the top numbers: \( -1 \times 5 = -5 \).
  • Multiply the bottom numbers: \( 4 \times 8 = 32 \).
  • Write the result as a fraction: \( -\frac{5}{32} \).
That's it! You've just multiplied two fractions. The negative sign carries through since multiplying a negative by a positive gives a negative result. So, the final answer is \( -\frac{5}{32} \). Simple, right? Just remember, no need for common denominators when multiplying!

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

Solve. See the Concept Checks in this section. Round the mixed number \(7 \frac{1}{8}\) to the nearest whole number.

Solve. In your own words, describe how to divide mixed numbers.

Answer true or false for each statement.What operation should be performed first to simplify $$\frac{1}{5} \cdot \frac{5}{2}-\left(\frac{2}{3}+\frac{4}{5}\right)^{2} ?$$ Explain your answer.

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

Perform each indicated operation. If the result is an improper fraction, also write the improper fraction as a mixed number. $$4-\frac{1}{5}$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

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.