🎯 Rate, Time, Work, and Mixture Problem Types
📊 Problem Pattern Catalog
Type 1: Basic Rate Problems
Pattern: Find distance, rate, or time using $D = R \times T$ Key Formula: $D = R \times T$, $R = \frac{D}{T}$, $T = \frac{D}{R}$
Worked Example:
A car travels 240 miles in 4 hours. What is its average speed?
Solution: $R = \frac{D}{T} = \frac{240}{4} = 60$ mph
Type 2: Work Problems
Pattern: Find work completed, rate, or time using $W = R \times T$ Key Formula: $W = R \times T$, $R = \frac{W}{T}$, $T = \frac{W}{R}$
Worked Example:
If 3 workers can complete 12 jobs in 4 hours, how many jobs can 5 workers complete in 6 hours?
Solution: First find individual rate: $R = \frac{W}{T} = \frac{12}{4} = 3$ jobs/hour per worker
With 5 workers: $R_{total} = 5 \times 3 = 15$ jobs/hour
In 6 hours: $W = R \times T = 15 \times 6 = 90$ jobs
Type 3: Relative Rate Problems
Pattern: Find when two objects meet or how fast they approach/separate Key Concept: Add rates for approaching/separating, subtract for same direction
Worked Example:
Two cars start 200 miles apart and drive toward each other. One travels 50 mph, the other 60 mph. How long until they meet?
Solution: Relative rate = $50 + 60 = 110$ mph (approaching)
Time = $\frac{D}{R} = \frac{200}{110} = \frac{20}{11} \approx 1.82$ hours
Type 4: Mixture Problems
Pattern: Find concentration or amount after mixing different solutions Key Formula: Amount = Concentration × Total
Worked Example:
Mix 20 gallons of 30% acid solution with 30 gallons of 50% acid solution. What is the final concentration?
Solution: Amount of acid in first solution: $0.30 \times 20 = 6$ gallons
Amount of acid in second solution: $0.50 \times 30 = 15$ gallons
Total acid: $6 + 15 = 21$ gallons
Total solution: $20 + 30 = 50$ gallons
Final concentration: $\frac{21}{50} = 0.42 = 42%$
Type 5: Combined Work Problems
Pattern: Find time when multiple people work together Key Concept: Add individual rates to get combined rate
Worked Example:
Alice can paint a room in 6 hours, Bob can paint it in 4 hours. How long will it take them to paint it together?
Solution: Alice’s rate: $\frac{1}{6}$ room/hour
Bob’s rate: $\frac{1}{4}$ room/hour
Combined rate: $\frac{1}{6} + \frac{1}{4} = \frac{2}{12} + \frac{3}{12} = \frac{5}{12}$ room/hour
Time together: $\frac{1}{\frac{5}{12}} = \frac{12}{5} = 2.4$ hours
🔍 Problem-Solving Strategy
- Identify the type of problem (rate, work, mixture)
- Write down what you know about each quantity
- Use the appropriate formula for the situation
- Set up equations if multiple unknowns
- Check your answer with estimation
⚠️ Common Mistakes
- Mixing up rate and time in formulas
- Wrong relative rate for same direction vs. approaching
- Forgetting to convert units (hours to minutes, etc.)
- Miscounting work rates in combined problems
- Wrong concentration calculations in mixtures