⏱️ Rate, Time, Work, and Mixture Topics

🎯 Core Subtopics

Basic Rate Problems

  • Formula: $D = R \times T$
  • Solving for each variable: $R = \frac{D}{T}$, $T = \frac{D}{R}$
  • Micro-example: If distance is 120 miles and time is 2 hours, rate = $\frac{120}{2} = 60$ mph
  • Trap: Mixing up which variable to solve for

Work Problems

  • Formula: $W = R \times T$
  • Individual rates: Rate = Work/Time
  • Combined rates: $R_{total} = R_1 + R_2 + R_3 + …$
  • Micro-example: If A works at 3 jobs/hour and B works at 2 jobs/hour, together they work at 5 jobs/hour
  • Trap: Forgetting to add rates when working together

Relative Rate Problems

  • Approaching: $R_{relative} = R_1 + R_2$
  • Same direction: $R_{relative} = |R_1 - R_2|$
  • Micro-example: Two cars at 40 mph and 60 mph approaching have relative speed of 100 mph
  • Trap: Using wrong formula for same direction vs. approaching

Mixture Problems

  • Basic formula: Amount = Concentration × Total
  • Concentration formula: Concentration = Amount/Total
  • Micro-example: 25% of 80 gallons = $0.25 \times 80 = 20$ gallons
  • Trap: Forgetting to convert percentages to decimals

Percent Mixture Problems

  • Two-part mixtures: $C_1 \times A_1 + C_2 \times A_2 = C_{final} \times (A_1 + A_2)$
  • Micro-example: Mix 10 gallons of 30% solution with 20 gallons of 50% solution:
    • Total substance: $0.30 \times 10 + 0.50 \times 20 = 13$ gallons
    • Final concentration: $\frac{13}{30} = 43.\overline{3}%$
  • Trap: Not accounting for total volume in final concentration

Time and Distance with Multiple Stops

  • Total distance: Sum of all segments
  • Total time: Sum of all time segments
  • Average rate: Total distance/Total time
  • 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
  • Trap: Using simple average instead of weighted average

🚨 Common Traps

  1. Rate vs Time: Confusing which variable to solve for
  2. Relative Rates: Using wrong formula for same direction
  3. Work Rates: Forgetting to add rates when working together
  4. Mixture Math: Not converting percentages to decimals
  5. Average Rate: Using simple average instead of weighted average

💡 Quick Tips

  • Rate Problems: Always check units (mph, jobs/hour, etc.)
  • Work Problems: Add rates when working together
  • Relative Rates: Add for approaching/separating, subtract for same direction
  • Mixtures: Convert percentages to decimals first
  • Time Calculations: Be careful with unit conversions