8. Word Problems That Hide the Equation

The hardest GED® algebra is usually a word problem where you must build the equation yourself. Name the unknown, translate each phrase, then solve.

Example 1 - reverse percent:
After a 20 percent discount, a jacket costs 68 dollars. Original price?
A 20 percent discount means you pay 80 percent: \(0.8p=68\Rightarrow p=85\) dollars.

Example 2 - mixture as algebra:
How many liters of pure water must be added to 6 liters of 50 percent salt solution to make it 30 percent salt? The salt amount stays fixed at \(0.5(6)=3\) liters.
\[ \frac{3}{6+w}=0.30. \]
Cross-multiply:
\[ 3=0.30(6+w)=1.8+0.3w. \]
So \(0.3w=1.2\), \(w=4\) liters.

Example 3 - consecutive with a condition:
The sum of three consecutive even integers is 10 more than twice the smallest. Find the smallest.
\[ n+(n+2)+(n+4)=2n+10. \]
Left: \(3n+6\). So \(3n+6=2n+10\Rightarrow n=4\).

Common mistake: translating "20 percent off" as multiplying by 0.20 instead of 0.80. "Off" means what remains.