6. Fraction and Decimal Equations

Fractions and decimals are still solved with the same balance rule. The cleanest strategy is often to clear the fraction or decimal early.

Example 1:
\[ \frac{x}{3}+4=10. \]
Subtract 4:
\[ \frac{x}{3}=6. \]
Multiply by 3:
\[ x=18. \]

Example 2:
\[ \frac{2}{5}x = 8. \]
Divide by \(\frac{2}{5}\), which is the same as multiplying by \(\frac{5}{2}\):
\[ x = 8\cdot \frac{5}{2}=20. \]

Example 3:
\[ \frac{3x+1}{2}=11. \]
Multiply both sides by 2:
\[ 3x+1=22. \]
Subtract 1:
\[ 3x=21. \]
Divide by 3:
\[ x=7. \]

Example 4:
\[ 0.2x+4=10. \]
Subtract 4:
\[ 0.2x=6. \]
Divide by \(0.2\):
\[ x=30. \]

Common mistake: multiplying only part of a side when clearing fractions. If you multiply by 2, every term on both sides must be multiplied by 2.