Lcd Fractions 3047F4

1. **State the problem:** We need to solve the equation $$\frac{2}{5}x + \frac{2}{3} = \frac{7}{4}$$ by clearing the fractions using the least common denominator (LCD). 2. **Find the LCD:** The denominators are 5, 3, and 4. The prime factors are: - 5 is prime - 3 is prime - 4 = 2^2 The LCD is the product of the highest powers of all primes: $$\text{LCD} = 5 \times 3 \times 2^2 = 5 \times 3 \times 4 = 60$$ 3. **Number of terms to multiply:** There are 3 terms with denominators: $$\frac{2}{5}x$$, $$\frac{2}{3}$$, and $$\frac{7}{4}$$. Each term must be multiplied by the LCD to clear fractions. 4. **Multiply each term by the LCD:** $$60 \times \frac{2}{5}x + 60 \times \frac{2}{3} = 60 \times \frac{7}{4}$$ Calculate each: $$60 \times \frac{2}{5}x = \cancel{\frac{60}{5}} \times 2x = 12 \times 2x = 24x$$ $$60 \times \frac{2}{3} = \cancel{\frac{60}{3}} \times 2 = 20 \times 2 = 40$$ $$60 \times \frac{7}{4} = \cancel{\frac{60}{4}} \times 7 = 15 \times 7 = 105$$ 5. **Rewrite the equation without fractions:** $$24x + 40 = 105$$ 6. **Solve for $$x$$:** Subtract 40 from both sides: $$24x + 40 - 40 = 105 - 40$$ $$24x = 65$$ Divide both sides by 24: $$x = \frac{65}{24}$$ This fraction cannot be simplified further. **Final answer:** $$x = \frac{65}{24}$$