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Chapter 2: Problem 16

Perform the operations. $$\frac{6}{7} \cdot \frac{5}{8}$$

Short Answer

Expert verified
The result of \(\frac{6}{7} \cdot \frac{5}{8}\) is \(\frac{15}{28}\).

Step by step solution

01

Multiplication of Numerators

First, multiply the numerators of both fractions, which are \(6\) and \(5\). The multiplication of \(6\) and \(5\) gives \(30\). So, this results in a numerator of \(30\).
02

Multiplication of Denominators

Next, multiply the denominators of both fractions, which are \(7\) and \(8\). The multiplication of \(7\) and \(8\) gives \(56\). So, this results in a denominator of \(56\).
03

Form the Resultant Fraction

Combine the resultant numerator and denominator to form a fraction. This gives \(\frac{30}{56}\).
04

Simplification of the Fraction

The fraction \(\frac{30}{56}\) can be simplified by dividing the numerator and denominator by their greatest common factor, which is \(2\). After simplification, the fraction becomes \(\frac{15}{28}\).

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Key Concepts

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

Numerator
The numerator is the top number in a fraction. It tells us how many parts of the whole we have. For example, in the fraction \(\frac{6}{7}\), the number \(6\) is the numerator. Think of it as slicing a pizza. If there are 7 slices total and you have 6, then you have \(\frac{6}{7}\) of the pizza. So, when multiplying fractions, our first task is to multiply the numerators, like multiplying the parts of each given fraction. This process creates a new top number for our resulting fraction.
Denominator
The denominator is the bottom number of a fraction. It indicates the total number of equal parts that make up a whole. For instance, in the fraction \(\frac{6}{7}\), \(7\) is the denominator, meaning the whole is divided into 7 equal parts. When we multiply fractions, we multiply the denominators, creating our new bottom number. For example, \(\frac{6}{7} \times \frac{5}{8}\) involves multiplying the denominators: \(7\) and \(8\). This gives a new denominator of \(56\), redistributing the whole into a larger number of parts, keeping our fraction consistent.
Simplification
Simplification is the process of making a fraction as simple as possible, meaning the numerator and the denominator don't have any common factors other than 1. After multiplying fractions, like getting \(\frac{30}{56}\), we simplify it to make it easier to understand and use. To simplify \(\frac{30}{56}\), we need to find a common number that can divide both the numerator and the denominator evenly. Simplification doesn鈥檛 change the value of the fraction, just its appearance, making it more manageable.
Greatest Common Factor
The greatest common factor (GCF) is the largest number that can evenly divide both the numerator and denominator in a fraction. It's key to simplifying fractions. To find the GCF of \(30\) and \(56\), list the factors of each number:
  • Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
  • Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
The greatest number that both lists share is \(2\). By dividing \(30\) and \(56\) by their GCF \(2\), we simplify \(\frac{30}{56}\) to \(\frac{15}{28}\), making the fraction easier to work with or compare.

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