Rates & Word Problems (Advanced)

Harder items at competition-adjacent depth.

Phase 2 · ACT Math

Rates & Word Problems: core idea

Piecewise schedules.

Two trains $420$ miles apart travel toward each other at $55$ mph and $50$ mph. How many hours until they meet?

Hint: Use their combined speed. Answer: 4.

1. A pipe fills a tank in $8$ hours, but a drain empties it in $12$ hours. Starting empty with both open, how long to fill?
2. A merchant mixes $8$ lb of $6$ dollar coffee with $12$ lb of $10$ dollar coffee. What is the cost per pound of the blend?
3. A runner's average speed for first $6$ miles is $8$ mph. To average $9$ mph for $10$ miles total, what speed is needed for final $4$ miles?
4. A factory's production rises from $480$ units to $600$ units. What percent increase is this?
5. A $30,000$ dollar loan earns simple interest at $4\%$ per year. How much interest accrues in $3$ years?
6. One worker can paint a room in $5$ hours and another in $10$ hours. After working together for $2$ hours, what fraction remains?
7. A car travels $120$ miles at $40$ mph and returns $120$ miles at $60$ mph. What is its average speed for the round trip?
8. A store needs $1,000$ dollars in revenue. It sells small items for $8$ and large items for $14$ . If it sells $90$ items total, how many large items must it sell?
9. A solution is $25\%$ alcohol. How many liters of pure alcohol should be added to $10$ L to make a $40\%$ solution?