Comparing Fractions: Understanding Fraction Relationships

What are Fractions?

Fractions are numbers that represent parts of a whole. They consist of a numerator (the number above the line) and a denominator (the number below the line). For example, in the fraction \(\frac{3}{4}\), 3 is the numerator and 4 is the denominator.

Comparing Fractions

When comparing fractions, we're determining which fraction represents a larger or smaller part of a whole. There are several methods to compare fractions:

1. Common Denominator Method

To compare fractions with different denominators, we can convert them to equivalent fractions with a common denominator. The steps are:

1. Find the least common multiple (LCM) of the denominators.
2. Convert each fraction to an equivalent fraction with the LCM as the denominator.
3. Compare the numerators of the new fractions.

For example, to compare \(\frac{2}{3}\) and \(\frac{3}{4}\):

\[\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}\]

\[\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}\]

Now we can see that \(\frac{3}{4} > \frac{2}{3}\) because \(9 > 8\).

2. Cross Multiplication Method

For fractions \(\frac{a}{b}\) and \(\frac{c}{d}\), we can compare \(ad\) and \(bc\):

• If \(ad > bc\), then \(\frac{a}{b} > \frac{c}{d}\)
• If \(ad < bc\), then \(\frac{a}{b} < \frac{c}{d}\)
• If \(ad = bc\), then \(\frac{a}{b} = \frac{c}{d}\)

3. Decimal Conversion Method

We can convert fractions to decimals and compare the decimal values:

\[\frac{2}{3} \approx 0.6667\]

\[\frac{3}{4} = 0.75\]

0.75 > 0.6667, so \(\frac{3}{4} > \frac{2}{3}\)

Visual Representation

Fractions can be visually represented to aid in comparison:

This diagram visually shows that \(\frac{3}{4}\) (blue) is slightly larger than \(\frac{2}{3}\) (green).