Question

# To find: The least common denominator of the fractions. \frac{2}{5x} + \frac{3x}{4} \frac{x}{2y} + \frac{y}{3x}

Factors and multiples
To find:
The least common denominator of the fractions.
1) $$\displaystyle{\frac{{{2}}}{{{5}{x}}}}+{\frac{{{3}{x}}}{{{4}}}}$$
2) $$\displaystyle{\frac{{{x}}}{{{2}{y}}}}+{\frac{{{y}}}{{{3}{x}}}}$$

2021-08-04
The least common denominator is defined as the least common multiple of the denominators of the set of fractions.
Example 1:
$$\displaystyle{\frac{{{2}}}{{{5}{x}}}}+{\frac{{{3}{x}}}{{{4}}}}$$
Now write each denominator as a product of prime factors,
$$\displaystyle{5}{x}={5}\times{x}$$
$$\displaystyle{4}={2}\times{2}$$
Now, form the product of all prime factors,
$$\displaystyle{L}.{C}.{D}.={5}\times{x}\times{2}\times{2}={20}{x}$$
Example 2:
$$\displaystyle{\frac{{{x}}}{{{2}{y}}}}+{\frac{{{y}}}{{{3}{x}}}}$$
Now write each denominator as a product of prime factors,
$$\displaystyle{2}{y}={2}\times{y}$$
$$\displaystyle{3}{x}={3}\times{x}$$
Now, form the product of all prime factors,
$$\displaystyle{L}.{C}.{D}.={2}\times{y}\times{3}\times{x}={6}{x}{y}$$
The Final Statement:
The least common denominator is defined as the least common multiple of the denominators of the set of fractions.