Question

Subtract the following fractions and mixed numbers. Reduce to lowest terms. \(2 \frac{1}{4}-1 \frac{1}{2}=\) ______

Short answer

\(2 \frac{1}{4}-1 \frac{1}{2} = \frac{3}{4}\)

Step-by-step solution

Convert Mixed Numbers to Improper Fractions

Firstly, convert the mixed numbers into improper fractions. For the number \(2 \frac{1}{4}\), the improper fraction is \(\frac{9}{4}\). This is found by multiplying the whole number 2 by the denominator 4, resulting in 8, and then adding the numerator 1. For the number \(1 \frac{1}{2}\), convert it to the improper fraction \(\frac{3}{2}\) by multiplying the whole number 1 by the denominator 2 and adding the numerator 1.

Find a Common Denominator

The fractions to be subtracted are \(\frac{9}{4}\) and \(\frac{3}{2}\). To subtract fractions, we need a common denominator. The denominators are 4 and 2, and the least common multiple is 4.

Adjust Fractions to Common Denominator

Convert \(\frac{3}{2}\) to have a denominator of 4 by multiplying both the numerator and the denominator by 2. Thus, \(\frac{3}{2} = \frac{6}{4}\). Now both fractions are \(\frac{9}{4}\) and \(\frac{6}{4}\).

Subtract the Fractions

Subtract \(\frac{6}{4}\) from \(\frac{9}{4}\): \[\frac{9}{4} - \frac{6}{4} = \frac{3}{4}\]

Simplify the Fraction (if necessary)

The fraction \(\frac{3}{4}\) is already in its simplest form, so no further simplification is necessary.

Key concepts

Mixed Numbers

When dealing with fractions, you might encounter mixed numbers, which consist of a whole number and a fraction combined. For example, in the exercise provided, there are two mixed numbers: \(2 \frac{1}{4}\) and \(1 \frac{1}{2}\). Understanding how to work with mixed numbers is essential.

• A mixed number can be broken down into its two parts: the whole number and the fractional component.
• To perform operations such as addition or subtraction, mixed numbers often need to be converted to improper fractions.

This conversion simplifies calculations, particularly when dealing with multiple terms. By working with improper fractions, you eliminate the complexity added by the mixed number format and focus purely on the fraction operation itself.

Improper Fractions

Improper fractions are an essential concept for simplifying calculations with mixed numbers. An improper fraction occurs when the numerator (top number) is greater than or equal to the denominator (bottom number).

• To convert a mixed number into an improper fraction, first multiply the whole number by the denominator.
• Then, add the result to the numerator of the fractional part.
• For example, to convert \(2 \frac{1}{4}\), you calculate \(2 \times 4 + 1 = 9\), resulting in the improper fraction \(\frac{9}{4}\).

This method simplifies subtraction and addition operations, providing a straightforward way to handle mixed numbers without juggling both whole and fractional parts at every step.

Common Denominator

When subtracting fractions, a common denominator is necessary. The common denominator is a shared multiple of the denominators of the fractions involved. It allows you to perform operations directly on the numerators.

• In the example provided, we are subtracting \(\frac{9}{4}\) and \(\frac{3}{2}\).
• The denominators are 4 and 2, so we seek the least common multiple to use as the common denominator, which is 4.
• Convert \(\frac{3}{2}\) to an equivalent fraction with a denominator of 4, yielding \(\frac{6}{4}\).

Now, with both fractions having the same denominator, you can easily subtract, resulting in \(\frac{9}{4} - \frac{6}{4} = \frac{3}{4}\). Having a common denominator streamlines operations, making calculations simpler and reducing error potential.