Multiply both sides by the LCD

Oct 13, 2025 by windailys.com 9 views

Introduction

Fractions can be a challenging aspect of mathematics, especially when it comes to solving equations. However, with the right techniques and strategies, you can master the art of solving equations with fractions. In this article, we will explore how to use the Least Common Denominator (LCD) to clear fractions and solve equations. We will also provide step-by-step solutions to three sample equations.

What is the Least Common Denominator (LCD)?

The Least Common Denominator (LCD) is the smallest multiple that two or more denominators have in common. It is used to clear fractions by multiplying both sides of the equation by the LCD. This process eliminates the fractions and makes it easier to solve the equation.

How to Find the LCD

To find the LCD, you need to list the multiples of each denominator and find the smallest multiple that they have in common. For example, if you have two fractions with denominators 3 and 5, the multiples of 3 are 3, 6, 9, 12, 15, and the multiples of 5 are 5, 10, 15. The smallest multiple that they have in common is 15, which is the LCD.

Step-by-Step Guide to Solving Equations with Fractions

Write down the equation: Start by writing down the equation that you want to solve.
Find the LCD: Find the LCD of the denominators in the equation.
Multiply both sides by the LCD: Multiply both sides of the equation by the LCD to clear the fractions.
Simplify the equation: Simplify the equation by combining like terms.
Solve for the variable: Solve for the variable by isolating it on one side of the equation.

Solving Equation Q1

Equation Q1

$\frac{1}{3} + \frac{4}{5} = \frac{x}{9}$

Step 1: Find the LCD

The denominators in the equation are 3 and 5. The multiples of 3 are 3, 6, 9, 12, 15, and the multiples of 5 are 5, 10, 15. The smallest multiple that they have in common is 15, which is the LCD.

Step 2: Multiply both sides by the LCD

Multiply both sides of the equation by 15 to clear the fractions.

$15(\frac{1}{3} + \frac{4}{5}) = 15(\frac{x}{9})$

Step 3: Simplify the equation

Simplify the equation by combining like terms.

$\frac{15}{3} + \frac{15 \times 4}{5} = \frac{15x}{9}$

$\frac{5}{1} + \frac{60}{5} = \frac{15x}{9}$

$5 + 12 = \frac{15x}{9}$

$17 = \frac{15x}{9}$

Step 4: Solve for the variable

Solve for the variable by isolating it on one side of the equation.

$17 = \frac{15x}{9}$

$17 \times 9 = 15x$

$153 = 15x$

$\frac{153}{15} = x$

$x = 10.2$

Solving Equation Q2

Equation Q2

$\frac{1}{3} - \frac{1}{2x} = \frac{1}{3}$

Step 1: Find the LCD

The denominators in the equation are 3 and 2x. The multiples of 3 are 3, 6, 9, 12, 15, and the multiples of 2x are 2x, 4x, 6x, 8x, 10x. The smallest multiple that they have in common is 6x, which is the LCD.

Step 2: Multiply both sides by the LCD

Multiply both sides of the equation by 6x to clear the fractions.

$6x(\frac{1}{3} - \frac{1}{2x}) = 6x(\frac{1}{3})$

Step 3: Simplify the equation

Simplify the equation by combining like terms.

$\frac{6x}{3} - \frac{6x}{2x} = \frac{6x}{3}$

$2x - 3 = \frac{6x}{3}$

$2x - 3 = 2x$

Step 4: Solve for the variable

Solve for the variable by isolating it on one side of the equation.

$2x - 3 = 2x$

$-3 = 0$

This equation has no solution.

Solving Equation Q3

Equation Q3

$\frac{2}{3} - \frac{1}{5} = \frac{x}{9}$

Step 1: Find the LCD

The denominators in the equation are 3 and 5. The multiples of 3 are 3, 6, 9, 12, 15, and the multiples of 5 are 5, 10, 15. The smallest multiple that they have in common is 15, which is the LCD.

Step 2: Multiply both sides by the LCD

Multiply both sides of the equation by 15 to clear the fractions.

$15(\frac{2}{3} - \frac{1}{5}) = 15(\frac{x}{9})$

Step 3: Simplify the equation

Simplify the equation by combining like terms.

$\frac{15 \times 2}{3} - \frac{15 \times 1}{5} = \frac{15x}{9}$

$\frac{30}{3} - \frac{15}{5} = \frac{15x}{9}$

$10 - 3 = \frac{15x}{9}$

$7 = \frac{15x}{9}$

Step 4: Solve for the variable

Solve for the variable by isolating it on one side of the equation.

$7 = \frac{15x}{9}$

$7 \times 9 = 15x$

$63 = 15x$

$\frac{63}{15} = x$

$x = 4.2$

Conclusion

Q: What is the Least Common Denominator (LCD)?

A: The Least Common Denominator (LCD) is the smallest multiple that two or more denominators have in common. It is used to clear fractions by multiplying both sides of the equation by the LCD.

Q: How do I find the LCD?

A: To find the LCD, you need to list the multiples of each denominator and find the smallest multiple that they have in common. For example, if you have two fractions with denominators 3 and 5, the multiples of 3 are 3, 6, 9, 12, 15, and the multiples of 5 are 5, 10, 15. The smallest multiple that they have in common is 15, which is the LCD.

Q: What is the difference between the LCD and the Greatest Common Denominator (GCD)?

A: The Greatest Common Denominator (GCD) is the largest multiple that two or more denominators have in common. The LCD, on the other hand, is the smallest multiple that two or more denominators have in common. While the GCD is used to simplify fractions, the LCD is used to clear fractions.

Q: Can I use the LCD to solve equations with fractions that have different signs?

A: Yes, you can use the LCD to solve equations with fractions that have different signs. When multiplying both sides of the equation by the LCD, make sure to distribute the negative sign to the terms on the other side of the equation.

Q: What if I have an equation with fractions that have different denominators and I'm not sure what the LCD is?

A: If you're not sure what the LCD is, you can try listing the multiples of each denominator and finding the smallest multiple that they have in common. Alternatively, you can use a calculator or online tool to find the LCD.

Q: Can I use the LCD to solve equations with fractions that have variables in the denominators?

A: Yes, you can use the LCD to solve equations with fractions that have variables in the denominators. When finding the LCD, make sure to include the variable in the list of multiples.

Q: What if I have an equation with fractions that have different denominators and I'm not sure how to multiply both sides by the LCD?

A: If you're not sure how to multiply both sides of the equation by the LCD, try breaking down the equation into smaller steps. For example, if you have an equation with two fractions, you can multiply both sides of the equation by the LCD of the first fraction and then multiply both sides of the equation by the LCD of the second fraction.

Q: Can I use the LCD to solve equations with fractions that have decimals?

A: Yes, you can use the LCD to solve equations with fractions that have decimals. When multiplying both sides of the equation by the LCD, make sure to convert the decimal to a fraction before multiplying.

Q: What if I have an equation with fractions that have different denominators and I'm not sure how to simplify the equation after multiplying both sides by the LCD?

A: If you're not sure how to simplify the equation after multiplying both sides by the LCD, try combining like terms and isolating the variable on one side of the equation.

Conclusion

Solving equations with fractions can be challenging, but with the right techniques and strategies, you can master the art of solving equations with fractions. By understanding the concept of the Least Common Denominator (LCD) and how to use it to clear fractions, you can solve equations with fractions that have different denominators. Remember to always follow the steps outlined in this article and practice regularly to become proficient in solving equations with fractions.