⏱️ Rate, Time, Work, and Mixture Formulas
🎯 Basic Rate Formulas
Distance, Rate, Time
$$D = R \times T$$
Usage: Find distance when rate and time are known Micro-example: If rate is 60 mph and time is 2 hours, distance = $60 \times 2 = 120$ miles
Rate Formula
$$R = \frac{D}{T}$$
Usage: Find rate when distance and time are known Micro-example: If distance is 150 miles and time is 3 hours, rate = $\frac{150}{3} = 50$ mph
Time Formula
$$T = \frac{D}{R}$$
Usage: Find time when distance and rate are known Micro-example: If distance is 200 miles and rate is 40 mph, time = $\frac{200}{40} = 5$ hours
🔄 Relative Rate Formulas
Approaching Objects
$$R_{relative} = R_1 + R_2$$
Usage: When two objects move toward each other Micro-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 Micro-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 Micro-example: Car at 60 mph passing car at 40 mph has relative speed of $60 - 40 = 20$ mph
🔧 Work Formulas
Work, Rate, Time
$$W = R \times T$$
Usage: Find work completed when rate and time are known Micro-example: If rate is 5 jobs/hour and time is 3 hours, work = $5 \times 3 = 15$ jobs
Work Rate
$$R = \frac{W}{T}$$
Usage: Find rate when work and time are known Micro-example: If 12 jobs are completed in 4 hours, rate = $\frac{12}{4} = 3$ jobs/hour
Work Time
$$T = \frac{W}{R}$$
Usage: Find time when work and rate are known Micro-example: If 20 jobs need to be done at 4 jobs/hour, time = $\frac{20}{4} = 5$ hours
Combined Work Rate
$$R_{combined} = R_1 + R_2 + R_3 + …$$
Usage: Find total rate when multiple people work together Micro-example: If A works at 2 jobs/hour and B works at 3 jobs/hour, combined rate = $2 + 3 = 5$ jobs/hour
🧪 Mixture Formulas
Basic Mixture
$$\text{Amount} = \text{Concentration} \times \text{Total}$$
Usage: Find amount of pure substance in a mixture Micro-example: 25% of 80 gallons = $0.25 \times 80 = 20$ gallons
Concentration
$$\text{Concentration} = \frac{\text{Amount}}{\text{Total}}$$
Usage: Find concentration after mixing Micro-example: If 15 gallons of pure substance are in 50 gallons total, concentration = $\frac{15}{50} = 0.30 = 30%$
Two-Part Mixture
$$C_1 \times A_1 + C_2 \times A_2 = C_{final} \times (A_1 + A_2)$$
Usage: Find final concentration when mixing two solutions Micro-example: Mix 10 gallons of 30% solution with 20 gallons of 50% solution:
- $0.30 \times 10 + 0.50 \times 20 = 3 + 10 = 13$ gallons of substance
- Final concentration = $\frac{13}{30} = 43.\overline{3}%$
📊 Average Rate
Weighted Average Rate
$$R_{avg} = \frac{D_1 + D_2 + D_3 + …}{T_1 + T_2 + T_3 + …}$$
Usage: Find average rate over multiple segments Micro-example: Drive 60 miles at 40 mph, then 40 miles at 60 mph:
- Total distance = 100 miles
- Total time = $\frac{60}{40} + \frac{40}{60} = 1.5 + 0.67 = 2.17$ hours
- Average rate = $\frac{100}{2.17} \approx 46.1$ mph
💡 Quick Reference
| Problem Type | Formula | Usage |
|---|---|---|
| Distance | $D = R \times T$ | Find distance |
| Rate | $R = \frac{D}{T}$ | Find rate |
| Time | $T = \frac{D}{R}$ | Find time |
| Approaching | $R_{relative} = R_1 + R_2$ | Objects moving toward each other |
| Same Direction | $R_{relative} = | R_1 - R_2 |
| Work | $W = R \times T$ | Find work completed |
| Combined Work | $R_{combined} = R_1 + R_2$ | Multiple people working together |
| Mixture | Amount = Conc. × Total | Find amount of substance |
| Two-Part Mix | $C_1A_1 + C_2A_2 = C_{final}(A_1 + A_2)$ | Mix two solutions |
⚠️ Common Pitfalls
- Rate vs Time: Don’t mix up which variable to solve for
- Relative Rates: Add for approaching/separating, subtract for same direction
- Work Rates: Add rates when working together
- Mixture Math: Convert percentages to decimals first
- Average Rate: Use weighted average, not simple average