Apply the seven-step problem-solving strategy, the total-value model, and the known-total technique to a ticket mixture problem involving a given total number of tickets sold at two distinct price points.

Problem: A whale-watching ship had $40$ paying passengers on board. The total revenue collected from tickets was $$1{,}196$. Full-fare passengers paid $$32$ each and reduced-fare passengers paid $$26$ each. How many full-fare passengers and how many reduced-fare passengers were on the ship?

1. Read the problem and identify the types involved: full-fare passengers (worth $$32$ each) and reduced-fare passengers (worth $$26$ each). The total number of passengers is $40$ and the total revenue is $$1{,}196$.
2. Identify what to find: the number of full-fare passengers and the number of reduced-fare passengers.
3. Name the unknowns using a single variable and the known-total technique. Let $f$ = the number of full-fare passengers. Since the total number of passengers is $40$, the number of reduced-fare passengers is $40 - f$. Organize in a table:

Type	Number	Value ($)	Total Value ($)
Full-fare	$f$	$32$	$32f$
Reduced-fare	$40 - f$	$26$	$26(40 - f)$
Total	$40$		$1{,}196$

4. Translate into an equation. The value of full-fare tickets plus the value of reduced-fare tickets equals the total value: $32f + 26(40 - f) = 1{,}196$

5. Solve the equation:

• Distribute $26$: $32f + 1{,}040 - 26f = 1{,}196$
• Combine like terms: $6f + 1{,}040 = 1{,}196$
• Subtract $1{,}040$ from both sides: $6f = 156$
• Divide both sides by $6$: $f = 26$

Find the number of reduced-fare passengers: $40 - 26 = 14$.

6. Check the mathematical result against the stated scenario: There were $26$ full-fare passengers who paid $$32$ each and $14$ reduced-fare passengers who paid $$26$ each. $26 \cdot 32 = 832$ $14 \cdot 26 = 364$ Combining these yields: $832 + 364 = 1{,}196$ $\checkmark$

7. Answer: There were $26$ full-fare passengers and $14$ reduced-fare passengers.

This core example demonstrates the typical ticket mixture problem setup. By recognizing the known total ($40$ passengers) and assigning expressions like $f$ and $40 - f$, we seamlessly create expressions for the total value without needing multiple variables.