When coefficients contain decimals, calculations tend to become complicated. Therefore, use the method of multiplying both sides by 10, 100, or another power of 10 to eliminate the decimals. Although an equation can be solved with decimals, converting them to integers reduces mistakes.
How to Solve Equations Containing Decimals
① Multiply both sides equally by 10, 100, 1000… to change the decimals into integers
② Then solve the equation using the same procedure as an ordinary linear equation
The important point is to multiply every term by the same number. Do not multiply only some of the terms.
Example ①
Solve the following equation: 0.3x + 0.5 = 1.4
Answer:
① Multiply both sides by 10: 3x + 5 = 14
② Transpose: 3x = 14 − 5 → 3x = 9
③ Divide both sides by 3: x = 3
Solve the following equation: 0.3x + 0.5 = 1.4
Answer:
① Multiply both sides by 10: 3x + 5 = 14
② Transpose: 3x = 14 − 5 → 3x = 9
③ Divide both sides by 3: x = 3
Example ②
Solve the following equation: 0.04x − 0.1 = 0.06
Answer:
① Multiply both sides by 100, according to the smallest decimal place: 4x − 10 = 6
② Transpose: 4x = 6 + 10 → 4x = 16
③ Divide both sides by 4: x = 4
Solve the following equation: 0.04x − 0.1 = 0.06
Answer:
① Multiply both sides by 100, according to the smallest decimal place: 4x − 10 = 6
② Transpose: 4x = 6 + 10 → 4x = 16
③ Divide both sides by 4: x = 4
When Integers and Decimals Are Mixed
Do not forget the integer terms; multiply them by 10, 100, or the same factor as well. For example, in 0.5x = 2 → 5x = 20, the integer term must also be multiplied by the same factor.
Example ③
Solve the following equation: 0.2x + 3 = 0.5x − 1.5
Answer:
① Multiply both sides by 10: 2x + 30 = 5x − 15
② Transpose: 2x − 5x = −15 − 30 → −3x = −45
③ Divide both sides by −3: x = 15
Solve the following equation: 0.2x + 3 = 0.5x − 1.5
Answer:
① Multiply both sides by 10: 2x + 30 = 5x − 15
② Transpose: 2x − 5x = −15 − 30 → −3x = −45
③ Divide both sides by −3: x = 15
Key Points
- For equations containing decimals, multiply both sides by 10, 100, … to convert them to integers
- Multiply “every term.” Do not forget to multiply integer terms by the same number
- Choose whether to multiply by 10 or 100 according to the smallest decimal place
- Once the terms are integers, use the ordinary solving procedure
Practice Problems
- Solve the following: 0.5x − 1.2 = 0.3
- Solve the following: 0.07x + 0.4 = 0.5
- Solve the following: 0.4(x − 2) = 0.2x + 0.4
Answers and Explanations
- Answer: x = 3
Explanation: Multiply both sides by 10: 5x − 12 = 3 → 5x = 15 → x = 3. - Answer: x = 10/7
Explanation: Multiply both sides by 100: 7x + 40 = 50 → 7x = 10 → x = 10/7. - Answer: x = 6
Explanation: Multiply both sides by 10: 4(x−2) = 2x + 4 → 4x − 8 = 2x + 4 → 2x = 12 → x = 6.