Rates and Work (Hard)

Late AMC 10/12 through early AIME — dense computation.

Rates and Work: core idea

Tricky schedules.

A train goes $180$ mi at $60$ mph and returns at $45$ mph. Find average speed.

Hint: Write each rate as work or distance per unit time. Answer: 360/7.

1. Pipes fill a tank in $6$ and $4$ hours. Find hours together.
2. A boat goes $30$ mi downstream in $2$ hr and upstream in $3$ hr. Find current speed.
3. A $12$ mph cyclist leaves; 2 hr later an $18$ mph cyclist follows. Find catch-up time after second starts.
4. Add pure sugar to $20$ oz of $40\%$ syrup to make $50\%$ . How many oz?
5. Workers need $10,15,30$ days alone. Find days together.
6. A $100$ price gets $20\%$ off then $10\%$ up. Find final price.
7. Run equal halves at $8$ and $12$ mph. Find average speed.
8. Ten liters of $30\%$ acid evaporate to $50\%$ acid. How many liters evaporate?
9. A machine makes $240$ parts in $8$ hr. How many in $3.5$ hr?