Comparing Fractions: Exploring Methods and Applications

1. Common Denominator Method

Tip: Find a common denominator for both fractions, then compare the numerators. For example, to compare $\frac{1}{3}$  and $\frac{2}{5}$ , find a common denominator (15 in this case), then compare $\frac{5}{15}$ and $\frac{6}{15}$ . Since $\frac{6}{15}$  is greater than $\frac{5}{15}$ , $\frac{2}{5}$  is greater than $\frac{1}{3}$ .

2. Cross Multiplication Method

Tip: Multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa. Then compare the results. For example, to compare $\frac{1}{3}$  and $\frac{2}{5}$ , cross multiply to get 5 × 1 = 5 and 3 × 2 = 6. Since 6 is greater than 5, $\frac{2}{5}$  is greater than $\frac{1}{3}$ .

3. Visualize on a Number Line

Tip: Represent each fraction on a number line. This can help visualize which fraction is greater. For example, if you represent $\frac{1}{3}$  and $\frac{2}{5}$  on a number line from 0 to 1, you’ll see that $\frac{2}{5}$  is closer to 1 than $\frac{1}{3}$ , making it greater.

4. Convert to Decimals

Tip: Convert both fractions to decimals and then compare them. For example, $\frac{1}{3}$  ≈ 0.333 and $\frac{2}{5}$  ≈ 0.4. Since 0.4 is greater than 0.333, $\frac{2}{5}$  is greater than $\frac{1}{3}$ .

5. Benchmark Fractions

Tip: Know some benchmark fractions (like 1/2, 1/3, 1/4, etc.) and use them as references. For example, if you’re comparing $\frac{1}{3}$  and $\frac{3}{4}$  you know that $\frac{1}{3}$  is smaller than $\frac{1}{2}$ , and $\frac{3}{4}$  is greater than $\frac{1}{2}$ , so $\frac{3}{4}$  must be greater than $\frac{1}{3}$ .