Equations with fractions and decimals

Multiply x/3 + 1/2 = 5/6 by 6, the least common denominator, and it becomes 2x + 3 = 5, with no fractions at all.

Clearing the fractions first

An equation full of fractions can be solved with the steps you know, but every addition then needs a common denominator, and the arithmetic piles up. A cleaner route is to get rid of the fractions at the very start. Find the least common denominator (LCD) of every fraction in the equation, and multiply both sides of the equation by it. The Multiplication Property of Equality says that keeps the sides equal, and each denominator divides into the LCD exactly, so every fraction turns into a whole number.

Take x/3 + 1/2 = 5/6. The denominators are 3, 2 and 6, and the LCD is 6. Multiply every term on both sides by 6: 6 × x/3 = 2x, 6 × 1/2 = 3, and 6 × 5/6 = 5. The equation becomes 2x + 3 = 5. Subtract 3 to get 2x = 2, and divide by 2 to get x = 1. Check in the original: 1/3 + 1/2 = 2/6 + 3/6 = 5/6. The key detail is to multiply every term, including the ones that are already whole numbers. Forgetting one term is the usual mistake.

One more: (3/4)x − 2 = 7. The only denominator is 4, so multiply both sides by 4: 3x − 8 = 28. Note that the 2 became 8 and the 7 became 28, because they were multiplied too. Add 8 to get 3x = 36, and divide by 3 to get x = 12. Check: three quarters of 12 is 9, and 9 − 2 = 7. When a fraction sits in front of parentheses, as in (1/2)(x + 4) = 5, you can either distribute first or multiply both sides by 2 straight away, giving x + 4 = 10 and x = 6.

Decimals: multiply by 10, 100 or 1000

A decimal is a fraction whose denominator is a power of ten, so the same idea works. Look for the decimal with the most digits after the point and multiply both sides by 10 for one digit, 100 for two, 1000 for three. In 0.4x + 1.2 = 3.6, every decimal has one digit after the point, so multiply by 10: 4x + 12 = 36. Then 4x = 24 and x = 6. In 0.25n + 0.5 = 2, the 0.25 has two digits, so multiply by 100: 25n + 50 = 200, 25n = 150, n = 6. Money problems often look like this, because cents are hundredths of a dollar.