/*! 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 22 Add and simplify. $$ \frac{2... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 22

Add and simplify. $$ \frac{27}{28}-\frac{5}{28}-\frac{28}{28} $$

Short Answer

Expert verified
The simplified result is \(\frac{-3}{14}\).

Step by step solution

01

Identify Common Denominator

Observe that all fractions have the same denominator of 28. This means we can easily add and subtract the numerators without changing the denominator.
02

Add/Subtract Numerators

Keeping the common denominator of 28, add the numerators: \[\frac{27}{28} - \frac{5}{28} - \frac{28}{28} = \frac{27 - 5 - 28}{28}.\]
03

Perform the Arithmetic in the Numerator

Calculate the result of the numerator:\[27 - 5 = 22,\] then,\[22 - 28 = -6.\] So, the fraction simplifies to:\[\frac{-6}{28}.\]
04

Simplify the Fraction

To simplify \(\frac{-6}{28}\), find the greatest common divisor (GCD) of 6 and 28, which is 2. Divide both numerator and denominator by 2:\[\frac{-6 \div 2}{28 \div 2} = \frac{-3}{14}.\]

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.

Common Denominator
When dealing with fractions, especially in addition or subtraction, having a common denominator is crucial. The denominator is the bottom number of a fraction and represents the total number of equal parts. When fractions share the same denominator, you can directly add or subtract the numerators, or the top numbers. In the exercise, fractions like \( \frac{27}{28} \), \( \frac{5}{28} \), and \( \frac{28}{28} \) all have the denominator 28. This simplifies the process greatly because you don't need to find a new common base for comparison. This means you focus only on manipulating the numerators while keeping the denominator constant.
Simplifying Fractions
Simplifying fractions is an essential skill to make an answer cleaner and easier to understand. A fraction is considered simplified when the numerator and the denominator have no common factors other than 1. After performing the arithmetic on the numerators, we ended up with \( \frac{-6}{28} \). This fraction isn't in its simplest form because both the numerator and the denominator can be divided by the same number. Simplifying \( \frac{-6}{28} \) involves dividing both the top and bottom by their greatest common divisor, resulting in \( \frac{-3}{14} \). This makes the fraction more concise and understandable.
Greatest Common Divisor
The greatest common divisor (GCD) of two numbers is the largest positive integer that divides each of the numbers without being left with a remainder. In fraction simplification, the GCD helps to reduce fractions to their simplest form. To simplify \( \frac{-6}{28} \) from the exercise, we first find the GCD of 6 and 28. List the divisors of each number:
  • The divisors of 6: 1, 2, 3, 6
  • The divisors of 28: 1, 2, 4, 7, 14, 28
From these lists, the GCD is 2, as it's the largest number that appears in both sets of divisors. By dividing both -6 and 28 by the GCD, 2, the fraction simplifies to \( \frac{-3}{14} \). This step ensures that the fraction is in its simplest, most reduced form.

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

Perform the indicated operations. $$ -4 \frac{2}{5} \cdot 2 \frac{3}{10} $$

Write each improper fraction as a mixed number or a whole number. See Example 21. $$\frac{47}{15}$$

Solve. A student asked you to check his work below. Is it correct? If not, where is the error? \(3 \frac{2}{3} \cdot 1 \frac{1}{7} \stackrel{?}{=} 3 \frac{2}{21}\)

Solve. In your own words, explain how to multiply a. fractions b. mixed numbers

Answer true or false for each statement.A student is to evaluate \(x-y\) when \(x=\frac{1}{5}\) and \(y=-\frac{1}{7} .\) This student is asking you if he should evaluate \(\frac{1}{5}-\frac{1}{7} .\) What do you tell this student and why?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Pure 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.