Problem 40 - Prealgebra

Question

Replace each \(\circ\) with \(<,>,\) or \(=\) to make a true statement. $$\frac{3}{5} \diamond \frac{4}{7}$$

Step-by-Step Solution

Answer

\(\frac{3}{5} > \frac{4}{7}\)

Step 1: Understand the Problem

We need to compare two fractions: \( \frac{3}{5} \) and \( \frac{4}{7} \). We'll replace the symbol \( \diamond \) with \( <, >, \) or \( = \) to make this a true statement.

Step 2: Cross-Multiplication

When comparing two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), we can use cross-multiplication. Multiply \( a \) by \( d \) and \( b \) by \( c \). This gives: \( 3 \times 7 \) and \( 5 \times 4 \).

Step 3: Perform the Multiplications

Calculate the products: \( 3 \times 7 = 21 \) and \( 5 \times 4 = 20 \).

Step 4: Compare the Products

Compare the results of the two multiplications: 21 and 20. Since 21 is greater than 20, \( \frac{3}{5} > \frac{4}{7} \).

Key Concepts

Cross-Multiplication

Cross-multiplication is a handy technique used to compare two fractions, without needing to convert them to a common denominator. It involves multiplying the numerator of one fraction by the denominator of the other fraction. Here's how it works step-by-step:

• Consider two fractions, \( \frac{a}{b} \) and \( \frac{c}{d} \).
• Multiply the numerator of the first fraction (\( a \)) by the denominator of the second fraction (\( d \)).
• Similarly, multiply the numerator of the second fraction (\( c \)) by the denominator of the first fraction (\( b \)).

This gives us two products: \( ad \) and \( bc \). By comparing these two products, we can determine which fraction is greater without converting them into decimals. For example, in our problem, \( 3 \times 7 \) results in 21 and \( 5 \times 4 \) results in 20. Since 21 is greater than 20, \( \frac{3}{5} \) is greater than \( \frac{4}{7} \). Cross-multiplication simplifies the process and reduces potential errors in calculation.

Fraction Comparison

When comparing fractions, the goal is to determine which fraction represents a larger or smaller value. There are different methods to accomplish this: converting to decimal, finding a common denominator, or using cross-multiplication. Cross-multiplication is often the simplest method because:

• It avoids the need for common denominators.
• It allows for direct calculation with simple multiplication.
• It's quick, minimizing the steps needed to reach a solution.

In our example, using cross-multiplication helped us determine that \( \frac{3}{5} \) is greater than \( \frac{4}{7} \), since 21 (the result of \( 3 \times 7 \)) is larger than 20 (the result of \( 5 \times 4 \)). This technique is particularly useful with academic exercises where efficiency and accuracy are important. Remember, correctly performing the multiplications is key to accurate comparison.

Inequalities in Math

Inequalities express relationships of greater than, less than, or equal to between two values or expressions. In mathematics, it's crucial to determine and understand these relationships as they form the basis for solving equations and representing real-world situations, like comparing sizes, speeds, or quantities. When it comes to fractions, inequalities help us:

• Understand which fraction is larger or smaller.
• Represent solutions to practical problems.
• Analyze data in statistics or probability.

For instance, inequality symbols in our example were used to fill in for \( \diamond \), demonstrating that \( \frac{3}{5} \) is greater than \( \frac{4}{7} \) by placing \( > \). Recognizing inequalities allows us to make informed comparisons and decisions in both academic settings and daily life. Always be diligent in ensuring calculations are correct when assessing inequalities, as they must truly reflect the relationships represented.