Complex word problem
1. The Work Rate Problem
Two workers, Alice and Bob, can complete a task together. Alice can finish the task in 6 hours, while Bob can finish it in 8 hours. How long will it take them to finish the task if they work together?
Solution:
- Alice's rate: 61 (task per hour)
- Bob's rate: 81 (task per hour)
- Combined rate: 61+81
- To combine rates, find the least common denominator: 61+81=244+243=247
- Time taken to complete the task together = 2471=724≈3.43 hours.
So, it will take them approximately 3.43 hours to finish the task together.
2. The Mixture Problem
A chemist needs to mix two solutions: one is 30% acid and the other is 50% acid. How many liters of each solution should be mixed together to obtain 10 liters of a 40% acid solution?
Solution:
Let x be the number of liters of the 30% acid solution, and 10−x be the number of liters of the 50% acid solution.
The total acid in the mixture is given by: 0.30x+0.50(10−x)=0.40×10 Simplify the equation: 0.30x+5−0.50x=4 Combine like terms: −0.20x+5=4 Solve for x: −0.20x=−1 x=5
Thus, x=5, so the chemist should mix 5 liters of the 30% acid solution and 5 liters of the 50% acid solution to get 10 liters of a 40% acid solution.
3. The Distance, Speed, and Time Problem
A train travels from City A to City B at a constant speed of 60 km/h. A second train leaves City A to City B 30 minutes after the first train, but travels at 80 km/h. How long will it take for the second train to catch up to the first train?
Solution:
Let t represent the time (in hours) it takes for the second train to catch up with the first train.
- In t hours, the second train will have traveled 80t km.
- In the same time, the first train will have traveled 60(t+0.5) km (since the second train started 30 minutes later).
To find when the second train catches up, set the distances equal: 80t=60(t+0.5) Simplify and solve for t: 80t=60t+30 20t=30 t=2030=1.5
Thus, it will take the second train 1.5 hours to catch up with the first train.
