Comparing fractions

In this lesson, we will learn how to compare fractions. Comparing fractions helps us understand which fraction is greater, less than, or equal to another.

Key Concepts

1. Common Denominator: To compare fractions, we can find a common denominator. This means finding a number that both denominators can divide into.
2. Cross-Multiplication: Another method to compare fractions is cross-multiplication, where we multiply the numerator of one fraction by the denominator of the other.

Methods to Compare fractions

1. Finding a Common Denominator
To compare $\frac{a}{b}$ and $\frac{c}{d}$, convert them to have a common denominator: $\frac{a}{b} = \frac{a \times d}{b \times d} \text{ and } \frac{c}{d} = \frac{c \times b}{d \times b}$

Then compare the numerators:

• If $a \times d > c \times b$, then $\frac{a}{b} > \frac{c}{d}$.
• If $a \times d < c \times b$, then $\frac{a}{b} < \frac{c}{d}$.

2. Cross-Multiplication
For fractions $\frac{a}{b}$ and $\frac{c}{d}$:

• Calculate $a \times d$ and $c \times b$.
• Compare the products:
  • If $a \times d > c \times b$, then $\frac{a}{b} > \frac{c}{d}$.
  • If $a \times d < c \times b$, then $\frac{a}{b} < \frac{c}{d}$.

Example

Let’s compare $\frac{2}{3}$ and $\frac{3}{4}$ using cross-multiplication:

1. Calculate the cross products:

   • $2 \times 4 = 8$
   • $3 \times 3 = 9$

2. Compare the results:

   • Since $8 < 9$, we conclude that $\frac{2}{3} < \frac{3}{4}$.

Key Questions

• What is a common denominator?
• How do you use cross-multiplication to compare fractions?
• Can you give an example of comparing fractions using both methods?

Conclusion

Understanding how to compare fractions is essential for many arithmetic operations. Practice comparing different fractions to strengthen your skills!