Question

$$ \begin{aligned} &\frac{5}{4}-1 \frac{1}{2}\\\ &\begin{array}{|l|l|} \hline \text { Select... } & \nabla \\ \hline> \\ \hline< \\ \hline= \\ \hline \end{array} \end{aligned} $$

Short answer

\(<\)

Step-by-step solution

Convert Mixed Numbers to Improper Fractions

First, convert the mixed number 1 \(\frac{1}{2}\) to an improper fraction. This results in \(\frac{3}{2}\).

Find a Common Denominator

To subtract the fractions \(\frac{5}{4}\) and \(\frac{3}{2}\), find a common denominator. The least common denominator of 4 and 2 is 4.

Convert Fractions to Common Denominator

Convert \(\frac{3}{2}\) to \(\frac{6}{4}\) so both fractions have the same denominator. Now the subtraction problem is \(\frac{5}{4} - \frac{6}{4}\).

Perform the Subtraction

Subtract \(\frac{6}{4}\) from \(\frac{5}{4}\): \(\frac{5}{4} - \frac{6}{4} = -\frac{1}{4}\).

Compare the Result to Zero

Since \(-\frac{1}{4}\) is less than zero, \(\frac{5}{4} - 1\frac{1}{2}\) results in a negative fraction. Therefore, \(\frac{5}{4}\) is less than \(1\frac{1}{2}\).

Select the Correct Symbol

Based on our comparison, the correct symbol is '<'.

Key concepts

Improper Fractions

Fractions can be confusing, especially when they come in different forms. An improper fraction is when the numerator (top number) is larger than the denominator (bottom number). For example, \( \frac{5}{4} \) is an improper fraction because 5 is greater than 4.
To simplify operations involving mixed numbers (numbers that combine a whole number and a fraction), converting them to improper fractions can make subtraction easier. Mixed numbers like 1 \( \frac{1}{2} \) are converted to improper fractions by multiplying the whole number by the denominator and adding the numerator. For example: \[ 1 \frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{3}{2} \] This conversion helps us deal with only one type of fraction in our calculations.

Common Denominator

When adding or subtracting fractions, they need to have the same denominator. This is called finding a common denominator. Simply put, the denominators must match so that the fractions represent parts of the same whole.
For instance, to subtract \( \frac{5}{4} \) and \( \frac{3}{2} \), they must have the same denominator. The least common multiple of 4 and 2 is 4. Converting \( \frac{3}{2} \) to have the same denominator results in \[ \frac{3}{2} = \frac{3 \times 2}{2 \times 2} = \frac{6}{4} \] Now both fractions are ready for subtraction: \( \frac{5}{4} - \frac{6}{4} \).

Subtracting Fractions

Once you have fractions with a common denominator, subtracting them is straightforward. You simply subtract the numerators and keep the same denominator.
For example, with fractions \( \frac{5}{4} - \frac{6}{4} \), you subtract the numerators: \[ 5 - 6 = -1 \] So, \( \frac{5}{4} - \frac{6}{4} = \frac{-1}{4} \). The result is \( -\frac{1}{4} \). This indicates a negative fraction, meaning the result is less than zero.

Mixed Numbers

Mixed numbers are often easier for people to understand because they combine a whole number and a fraction, like 1 \( \frac{1}{2} \). Converting mixed numbers to improper fractions can simplify calculations, especially when performing operations like subtraction.
In our example, the mixed number 1 \( \frac{1}{2} \) was converted to the improper fraction \( \frac{3}{2} \). Converting helps align fractions for easy addition or subtraction. After solving, you can decide if converting the result back to a mixed number is useful, but the primary goal is accurate computation.
Remember, mastering mixed numbers and improper fractions enhances your overall understanding of fractions and their operations.