As a comprehensive review of methods for solving equations, we will learn about equations containing fractions and proportions. Learn how to clear denominators and the important property of proportions.
Equations Containing Fractions (Clearing Denominators)
When an equation contains fractions, the basic method is to multiply both sides by the least common multiple of the denominators to clear the denominators and change the equation into one with integer coefficients.
Example: How to solve x/2 + 1 = 4
Multiply both sides by 2 → 2×(x/2) + 2×1 = 2×4 → x + 2 = 8 → x = 6
Solve the following equation: x/3 + 1/2 = 5/6
Answer:
The least common multiple of the denominators 3, 2, and 6 is 6. Multiply both sides by 6.
6×(x/3) + 6×(1/2) = 6×(5/6)
2x + 3 = 5
2x = 2 → x = 1
Solve the following equation: (x+1)/2 = (2x − 3)/5
Answer:
The least common multiple of the denominators 2 and 5 is 10. Multiply both sides by 10.
10×(x+1)/2 = 10×(2x−3)/5
5(x+1) = 2(2x−3)
5x + 5 = 4x − 6
5x − 4x = −6 − 5 → x = −11
Note: Do not forget the parentheses in the numerator! When (x+1) is the numerator, the expression to which 5 is distributed is all of x+1.
Proportions and Their Property
An expression connecting equal ratios, such as a : b = c : d, is called a proportion. A proportion has the following property.
a : b = c : d ⇔ ad = bc (outer × outer ＝ inner × inner)
Using this property, a proportion containing a letter can also be solved as an equation.
Solve the following proportion: x : 6 = 8 : 12
Answer:
Using outer × outer ＝ inner × inner,
12x = 6 × 8 = 48
x = 4
- For equations containing fractions, multiply both sides by the least common multiple of the denominators to clear them
- When the numerator is a polynomial, be sure to add parentheses after clearing the denominator
- Use a:b = c:d ⇔ ad = bc to change a proportion into an equation
- After solving, substitute the answer into the original expression to check it
Practice Problems
- Solve the following: x/4 − x/6 = 1
- Solve the following: (2x − 1)/3 = (x + 2)/4
- Solve the following proportion: (x + 1) : 5 = 4 : 2
Answers and Explanations
- Answer: x = 12
Explanation: Multiply both sides by 12. 3x − 2x = 12 → x = 12. - Answer: x = 2
Explanation: Multiply both sides by 12. 4(2x−1) = 3(x+2) → 8x − 4 = 3x + 6 → 5x = 10 → x = 2. - Answer: x = 9
Explanation: 2(x+1) = 5×4 → 2x + 2 = 20 → 2x = 18 → x = 9.