Problem 132 Determine which of the two given... [FREE SOLUTION]

91影视

Prealgebra

David Arnold

$Math Studyset 91影视 Explanations$ Math

2020 Edition

Chapter 4: Problem 132

Determine which of the two given statements is true. $\frac{3}{2}<\frac{7}{9}$ or $\frac{3}{2}>\frac{7}{9}$.

Short Answer

Expert verified

The true statement is $ \frac{3}{2} > \frac{7}{9} $.

Step by step solution

Write the Inequalities

First, let's write the two statements clearly: 1. $ \frac{3}{2} < \frac{7}{9} $2. $ \frac{3}{2} > \frac{7}{9} $

Find a Common Denominator

To compare the fractions, we need to have a common denominator. The denominators here are 2 and 9. The least common multiple of 2 and 9 is 18.

Convert to Equivalent Fractions

Convert both fractions so they have the common denominator of 18. \[ \frac{3}{2} = \frac{3 \times 9}{2 \times 9} = \frac{27}{18} \]\[ \frac{7}{9} = \frac{7 \times 2}{9 \times 2} = \frac{14}{18} \]

Compare the Fractions

Now, compare the two equivalent fractions: $ \frac{27}{18} $ and $ \frac{14}{18} $. Since 27 is greater than 14, $ \frac{27}{18} > \frac{14}{18} $.

Determine the True Statement

Since $ \frac{3}{2} = \frac{27}{18} $ and $ \frac{7}{9} = \frac{14}{18} $, we determine that the correct inequality is $ \frac{3}{2} > \frac{7}{9} $.

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Key Concepts

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

Understanding Common Denominators

When you want to compare two fractions, having a common denominator makes it easier to understand which fraction is larger. A **common denominator** is simply a shared bottom number for both fractions. This practice ensures that you are comparing apples to apples. If the denominators are different, you can't directly compare the numerators.

**Finding a common denominator**: To find one, look for the least common multiple (LCM) of the two denominators. This is the smallest number that both denominators can divide without leaving a remainder.
In our example, the fractions are $\frac{3}{2}$ and $\frac{7}{9}$, thus 2 and 9 are the denominators. The LCM of 2 and 9 is 18.
Now, change each fraction so that they both have this LCM as their new denominator.This step helps you align the fractions and makes their comparison more straightforward.

With a common denominator, you simply compare the numerators. The fraction with the larger numerator is the greater fraction.

Exploring Fraction Inequalities

Fractions can be compared, much like whole numbers, using **inequalities**. These are the less than (<), greater than (>), and equal to (=) symbols. When comparing fractions, the goal is to determine which fraction represents a larger or smaller value.

The key to comparing fractions using inequalities is to first convert them to have a common denominator. Only then can accurate comparisons be made.
Once both fractions have the same denominator, the inequality involves comparing the numerators. A greater numerator indicates a larger fraction because the context of the same total parts (denominator) is given.
For example, converting $\frac{3}{2}$ and $\frac{7}{9}$ to $\frac{27}{18}$ and $\frac{14}{18}$, respectively, lets us see that 27 (from $\frac{27}{18}$) is greater than 14 (from $\frac{14}{18}$).

This comparison thus illustrates that $\frac{3}{2}$ is greater than $\frac{7}{9}$ since $\frac{27}{18}$ > $\frac{14}{18}$.

Equivalent Fractions Simplified

**Equivalent fractions** are different expressions for the same value. They represent the same portion of a whole, even if they look different because their numerators and denominators have been altered.

To create equivalent fractions, either multiply or divide both the numerator and the denominator by the same non-zero number.
For example, the fraction $\frac{3}{2}$ can be rewritten as $\frac{27}{18}$ by multiplying both the numerator and the denominator by 9. Similarly, $\frac{7}{9}$ becomes $\frac{14}{18}$ by multiplying both parts by 2.
These transformations do not change the fraction's value but make comparisons feasible by providing a common baseline for analysis.

In essence, finding equivalent fractions is a useful skill to compare fractions efficiently. By using a common denominator, it becomes easy to determine which fraction is larger or smaller.

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91影视

Determine which of the two given statements is true. \(\frac{3}{2}<\frac{7}{9}\) or \(\frac{3}{2}>\frac{7}{9}\).

Short Answer

Step by step solution

Write the Inequalities

Find a Common Denominator

Convert to Equivalent Fractions

Compare the Fractions

Determine the True Statement

Key Concepts

Understanding Common Denominators

Exploring Fraction Inequalities

Equivalent Fractions Simplified

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