Mathematical Reasoning › Worked Example: Solving a Multi-Step Algebra Problem

To make these concepts concrete, let us trace through how a well-trained model approaches a multi-step algebra problem and where chain-of-thought reasoning helps.

Problem: "A store sells two types of coffee: Regular at $8 per pound and Premium at $14 per pound. A merchant wants to create a 20-pound blend that costs $10 per pound. How many pounds of each type should be used?"

Without chain-of-thought: A model might produce "8 pounds of Premium and 12 pounds of Regular" or attempt a direct answer without showing work. Even if the answer is correct, there is no way to detect errors in intermediate reasoning.

With chain-of-thought:

Step 1: Let $r$ = pounds of Regular and $p$ = pounds of Premium.

Step 2: Total weight constraint: $r + p = 20$.

Step 3: Total cost constraint: $8r + 14p = 10 \times 20 = 200$.

Step 4: From the first equation, $r = 20 - p$.

Step 5: Substituting: $8(20 - p) + 14p = 200$.

Step 6: Expanding: $160 - 8p + 14p = 200$.

Step 7: Simplifying: $6p = 40$, so $p = 6.67$ pounds.

Step 8: Therefore $r = 20 - 6.67 = 13.33$ pounds.

Step 9: Verify: $8 \times 13.33 + 14 \times 6.67 = 106.67 + 93.33 = 200$. Correct.

The chain-of-thought solution is longer but serves several purposes. Each step is checkable independently. The substitution in step 4 is an explicit mathematical move that a process reward model can evaluate. The verification in step 9 closes the loop by confirming the answer satisfies the original constraints. This structure is exactly what process reward models learn to evaluate: they assign high scores to steps like step 4 (correct substitution) and step 9 (verification) and low scores to steps that make unjustified leaps.

Notice also what the chain-of-thought reveals about the problem structure. By explicitly setting up variables and equations, the model commits to a representation that either works or does not. A model that jumps to the answer "8 and 12" cannot easily be checked; a model that writes down the two equations can be verified at the equation-setting-up stage, before any solving occurs. This early verification is particularly valuable for catching errors in problem representation, which are the hardest class of errors to detect in final-answer-only evaluation.