Problem 34 - Prealgebra

Question

Replace each \(\circ\) with \(<,>,\) or \(=\) to make a true sentence. $$\frac{5}{7} \circ \frac{10}{14}$$

Step-by-Step Solution

Answer

= (Both fractions are equal.)

Step 1: Simplify the Fractions

First, simplify both fractions to see if they can be expressed in simpler terms.For \(\frac{5}{7}\), the fraction is already in its simplest form as there are no common factors between 5 and 7.For \(\frac{10}{14}\), divide the numerator and the denominator by their greatest common divisor, which is 2.\[\frac{10}{14} = \frac{10 \div 2}{14 \div 2} = \frac{5}{7}\]

Step 2: Compare the Simplified Fractions

With both fractions simplified, we have \(\frac{5}{7}\) and \(\frac{5}{7}\).Since both fractions are identical, the relationship between them is equality, denoted by \(=\).

Step 3: Insert the Correct Symbol

Now, replace the placeholder \(\circ\) with the equal sign \(=\) to complete the comparison.This makes the true sentence: \[\frac{5}{7} = \frac{10}{14}\]

Key Concepts

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. When you simplify a fraction, you are looking for the greatest common divisor (GCD) of the numerator and the denominator. The fraction is then reduced by dividing both the numerator and the denominator by this GCD. For example, consider the fraction \(\frac{10}{14}\):

• First, identify the GCD of 10 and 14. In this case, it is 2.
• Then, divide both the numerator and the denominator by 2: \(\frac{10 \div 2}{14 \div 2} = \frac{5}{7}\).

By using this process, you've simplified \(\frac{10}{14}\) to \(\frac{5}{7}\). A simplified fraction is always equivalent to the original one, but it's easier to work with. Whenever you're working with fractions, it's good practice to simplify them whenever possible.

Comparing Fractions

Comparing fractions involves determining whether one fraction is larger, smaller, or equal to another. Simplifying fractions first, if possible, makes comparing them easier. For example, after simplifying \(\frac{10}{14}\) to \(\frac{5}{7}\), you can directly compare the two fractions:

• Check if the fractions are identical. If they simplify to the same numbers, they are equal.
• If the fractions are not the same, determine which is greater. You might need to find a common denominator to compare different fractions.

In the example provided, comparing \(\frac{5}{7}\) and \(\frac{5}{7}\) is straightforward because both are identical. Hence, they are equal, and you can conclude the relationship with an equals sign \(=\). When fractions are easier to compare, solving related problems becomes more straightforward.

Greatest Common Divisor

The greatest common divisor, or GCD, is a key concept when working with fractions. It's the largest number that divides both the numerator and the denominator without any remainder. Finding the GCD is crucial for simplifying fractions:

• List the factors of each number.
• Identify the largest factor that both numbers share.
• Use this factor to simplify the fraction by dividing both the numerator and the denominator by it.

For instance, to simplify the fraction \(\frac{10}{14}\):

• List the factors of 10: 1, 2, 5, 10.
• List the factors of 14: 1, 2, 7, 14.
• The largest common factor is 2, so the GCD is 2.

Using the GCD, you can simplify fractions, making them neater and more efficient to work with. Understanding the GCD not only helps in simplifying fractions but also in various mathematical operations, where the reduction of expressions is needed.