⏱️ Rate, Time, Work, and Mixture Reference
🎯 Key Concepts
Rate
The amount of something that happens per unit of time.
Work
The amount of a task that can be completed.
Mixture
A combination of different substances with different properties.
Relative Rate
The rate at which two objects approach or separate from each other.
📐 Basic Rate Formulas
Distance, Rate, Time
$$D = R \times T$$
Usage: Find distance, rate, or time when two are known Example: If you travel 60 mph for 2 hours, distance = $60 \times 2 = 120$ miles
Work, Rate, Time
$$W = R \times T$$
Usage: Find work completed, rate, or time when two are known Example: If you work at 5 jobs/hour for 3 hours, work = $5 \times 3 = 15$ jobs
🔄 Relative Rates
Approaching Objects
$$R_{relative} = R_1 + R_2$$
Usage: When two objects move toward each other Example: Two cars at 40 mph and 60 mph approach at $40 + 60 = 100$ mph
Separating Objects
$$R_{relative} = R_1 + R_2$$
Usage: When two objects move away from each other Example: Two cars at 30 mph and 50 mph separate at $30 + 50 = 80$ mph
Same Direction
$$R_{relative} = |R_1 - R_2|$$
Usage: When two objects move in the same direction Example: Car at 60 mph passing car at 40 mph has relative speed of $60 - 40 = 20$ mph
🧪 Mixture Problems
Basic Mixture Formula
$$\text{Amount of substance} = \text{Concentration} \times \text{Total amount}$$
Usage: Find amount of pure substance in a mixture Example: 20% of 100 gallons = $0.20 \times 100 = 20$ gallons
Mixture Concentration
$$\text{Concentration} = \frac{\text{Amount of substance}}{\text{Total amount}}$$
Usage: Find concentration after mixing Example: Mix 10 gallons of 30% solution with 20 gallons of 50% solution:
- Total substance: $0.30 \times 10 + 0.50 \times 20 = 3 + 10 = 13$ gallons
- Total mixture: $10 + 20 = 30$ gallons
- Concentration: $\frac{13}{30} = 43.\overline{3}%$
🔢 Work Problems
Individual Work Rate
$$\text{Rate} = \frac{\text{Work completed}}{\text{Time taken}}$$
Usage: Find how fast someone works Example: If you complete 12 jobs in 4 hours, rate = $\frac{12}{4} = 3$ jobs/hour
Combined Work Rate
$$R_{combined} = R_1 + R_2 + R_3 + …$$
Usage: Find total rate when multiple people work together Example: If A works at 2 jobs/hour and B works at 3 jobs/hour, combined rate = $2 + 3 = 5$ jobs/hour
Time to Complete Work
$$T = \frac{W}{R}$$
Usage: Find time needed to complete work Example: If combined rate is 5 jobs/hour and there are 20 jobs, time = $\frac{20}{5} = 4$ hours
💡 Problem-Solving Strategies
- Identify what you’re looking for (distance, time, rate, work, concentration)
- 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
- Forgetting to convert units (hours to minutes, etc.)
- Miscounting work rates in combined problems
- Wrong concentration calculations in mixtures