/*! 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 239 In the following exercises, simp... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 239

In the following exercises, simplify. $$\frac{8 \cdot 9-7 \cdot 6}{5 \cdot 6-9 \cdot 2}$$

Short Answer

Expert verified
\(\frac{5}{2}\)

Step by step solution

01

Simplify the Numerator

Calculate the expression in the numerator: The numerator is \(8 \times 9 - 7 \times 6\). First, calculate \(8 \times 9\): \(8 \times 9 = 72\). Next, calculate \(7 \times 6\): \(7 \times 6 = 42\). Subtract the results: \(72 - 42 = 30\). So, the numerator simplifies to 30.
02

Simplify the Denominator

Calculate the expression in the denominator: The denominator is \(5 \times 6 - 9 \times 2\). First, calculate \(5 \times 6\): \(5 \times 6 = 30\). Next, calculate \(9 \times 2\): \(9 \times 2 = 18\). Subtract the results: \(30 - 18 = 12\). So, the denominator simplifies to 12.
03

Simplify the Fraction

The simplified fraction is: \(\frac{30}{12}\). Both the numerator and denominator can be divided by their greatest common divisor, which is 6. So, \(\frac{30/6}{12/6} = \frac{5}{2}\).

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.

Numerator
The numerator is the top part of a fraction. It represents how many parts of a whole are considered. For example, in the fraction \( \frac{3}{4} \), the numerator is 3.
It tells us that 3 out of 4 equal parts are in use.

In our exercise, the numerator begins as \(8 \times 9 - 7 \times 6\). Here鈥檚 how it's simplified:
  • Calculate \(8 \times 9\): This equals 72.
  • Calculate \(7 \times 6\): This equals 42.
  • Subtract the results: \(72 - 42 = 30\).
So, the simplified numerator is 30. Understanding the numerator helps us see how many parts or units we are dealing with in any fraction.
Denominator
The denominator is the bottom part of a fraction. It shows the total number of equal parts the whole is divided into.
For example, in the fraction \( \frac{3}{4} \), the denominator is 4. It means the whole is divided into 4 equal parts.

In our exercise, the denominator starts as \(5 \times 6 - 9 \times 2\). Here鈥檚 how this part is simplified:
  • Calculate \(5 \times 6\): This equals 30.
  • Calculate \(9 \times 2\): This equals 18.
  • Subtract the results: \(30 - 18 = 12\).
So, the simplified denominator is 12. Grasping the denominator helps us understand the fraction's overall context or the 'whole' in the division.
Greatest Common Divisor
The greatest common divisor (GCD) is the largest number that can divide both the numerator and the denominator without leaving a remainder. Using the GCD helps us simplify fractions to their lowest terms.

Here鈥檚 how we found the GCD in our exercise:
  • The simplified fraction was \( \frac{30}{12} \).
  • We need the largest number that divides both 30 and 12.
  • Listing the factors of each:
    • Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30.
    • Factors of 12: 1, 2, 3, 4, 6, 12.
  • The common factors are 1, 2, 3, and 6, where 6 is the largest.

So, the GCD is 6. We simplified our fraction by dividing both the numerator and denominator by 6:
\( \frac{30/6}{12/6} = \frac{5}{2} \). Understanding the GCD ensures we can reduce fractions to their simplest form easily and accurately.

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

In the following exercises, perform the indicated operations and simplify. $$\frac{11}{12}-\frac{2}{3}$$

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. 8\(u^{2} v^{3}\) when \(u=-\frac{3}{4}\) and \(v=-\frac{1}{2}\)

In the following exercises, find the least common denominator. $$\frac{1}{3} \text { and } \frac{4}{5}$$

In the following exercises, find the least common denominator. $$\frac{1}{3} \text { and } \frac{1}{12}$$

In the following exercises, simplify. $$ -\frac{22}{25}+\frac{9}{40} $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

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.