Question

$$ \begin{aligned} &\frac{7}{4}-\frac{13}{8}\\\ &\begin{array}{|l|l|} \hline \text { Select... } & \boldsymbol{\nabla} \\ \hline> \\ \hline< \\ \hline= \\ \hline \end{array} \end{aligned} $$

Short answer

>

Step-by-step solution

Find a common denominator

The two fractions \(\frac{7}{4}\) and \(\frac{13}{8}\) must be compared by finding a common denominator. The least common denominator (LCD) of 4 and 8 is 8.

Convert the fractions

Convert \(\frac{7}{4}\) to have the denominator 8. \(\frac{7}{4} = \frac{7 \times 2}{4 \times 2} = \frac{14}{8}\). The fraction \(\frac{13}{8}\) already has the denominator 8, so no conversion is necessary.

Compare the numerators

Now that both fractions have the same denominator, compare their numerators. \(\frac{14}{8}\) has a numerator of 14 and \(\frac{13}{8}\) has a numerator of 13.

Determine the relationship

Since 14 is greater than 13, \(\frac{14}{8} > \frac{13}{8}\). Therefore, \(\frac{7}{4} > \frac{13}{8}\).

Key concepts

Common Denominator

When comparing fractions, it's crucial to have a common denominator. This makes both fractions comparable because it puts them on the same scale. Think of the denominator as the 'whole' or base unit. If these units aren't the same, you can't directly compare the fractions.
For instance, in our problem, we are comparing \(\frac{7}{4}\) and \(\frac{13}{8}\). Since these fractions have different denominators, we need to find a common one to proceed.
Common denominators simplify the comparison process and are foundational in fraction arithmetic operations, such as addition, subtraction, and comparison.

Least Common Denominator (LCD)

The least common denominator (LCD) is the smallest number that is a multiple of both denominators. It is smaller and more manageable than any other common multiple. To find the LCD:

• List the multiples of each denominator
• Identify the smallest multiple that appears in both lists

For 4 and 8, the multiples are:
\(4: 4, 8, 12, 16, ...\)
\(8: 8, 16, 24, ...\)
The smallest common multiple is 8, making it the LCD. Here, we use 8 as the common denominator to convert both fractions.

Fraction Conversion

Once the LCD is found, you need to convert each fraction to have this common denominator. This process is called fraction conversion. If you change a fraction's denominator, you must adjust its numerator to keep the value the same.
For our example, convert \(\frac{7}{4}\) to have 8 as the denominator:

• Multiply the numerator and denominator by the same number so that the denominator matches the LCD
• \(\frac{7 \times 2}{4 \times 2} = \frac{14}{8}\)

The conversion ensures both denominators are the same, allowing an accurate comparison. Since \(\frac{13}{8}\) already has 8 as the denominator, no conversion is needed for it.

Numerator Comparison

With both fractions converted to the same denominator, you can now compare their numerators. This comparison tells you which fraction is larger or smaller.
In our problem, compare the numerators of \(\frac{14}{8}\) and \(\frac{13}{8}\):

• \(14 > 13\)

Since their denominators are the same, we only need to look at the numerators. Therefore, \(\frac{14}{8} > \frac{13}{8}\).
Translating this back to the original fractions, \(\frac{7}{4} > \frac{13}{8}\). The process is complete once you identify the relationship between the numerators.