1. **Stating the problem:** Simplify the expression $$\frac{2a}{3b} - \frac{4a}{5b} + \frac{5a}{2b}$$ and explain why 10 is used as the number to expand it. 2. **Formula and rules:** When adding or subtracting fractions, we need a common denominator. The common denominator is usually the least common multiple (LCM) of the denominators. 3. **Identify denominators:** The denominators are $3b$, $5b$, and $2b$. Since $b$ is common in all, focus on the numbers: 3, 5, and 2. 4. **Find the LCM of 3, 5, and 2:** - Prime factors: 3, 5, and 2 are all prime. - LCM is their product: $$3 \times 5 \times 2 = 30$$ 5. **Why 10 is mentioned:** Sometimes, if only two denominators are considered, for example 2 and 5, their LCM is 10. But here, since 3 is also a denominator, the true LCM is 30. 6. **Rewrite each fraction with denominator $30b$:** - $$\frac{2a}{3b} = \frac{2a \times 10}{3b \times 10} = \frac{20a}{30b}$$ - $$\frac{4a}{5b} = \frac{4a \times 6}{5b \times 6} = \frac{24a}{30b}$$ - $$\frac{5a}{2b} = \frac{5a \times 15}{2b \times 15} = \frac{75a}{30b}$$ 7. **Combine the fractions:** $$\frac{20a}{30b} - \frac{24a}{30b} + \frac{75a}{30b} = \frac{20a - 24a + 75a}{30b} = \frac{71a}{30b}$$ 8. **Final answer:** $$\boxed{\frac{71a}{30b}}$$ **Summary:** 10 is used as a multiplier when finding a common denominator between 2 and 5, but since 3 is also a denominator here, the correct common denominator is 30, not 10.