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RATE PROBLEMS

RATE PROBLEMS. INTRODUCTION. Several types of problems fall into the category known as “rate problems”: Distance Work Percent problems Mixture problems. Distance Problems. Rate x Time = Distance Within the category of distance problems, we encounter 3 types of questions:

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RATE PROBLEMS

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  1. RATE PROBLEMS

  2. INTRODUCTION • Several types of problems fall into the category known as “rate problems”: • Distance • Work • Percent problems • Mixture problems

  3. Distance Problems • Rate x Time = Distance • Within the category of distance problems, we encounter 3 types of questions: • Opposite direction • Same direction • Round-trip

  4. Opposite Direction Key Words: opposite direction, towards each other, away from each other Example: Two trains start from the same point and travel in opposite directions. The northbound train averages 45 mph and starts 2 hours before the southbound train, which averages 50 mph. How long after the northbound train starts will the trains be 470 miles apart?

  5. RATE TIME DISTANCE N 45 X 45x S 50 X-2 50(x – 2) Opposite Direction (Cont.) What do we know? Organize the information: Northbound– 45 mph Southbound– 50 mph Northbound train starts 2 hours earlier. What do we want to find out? “How long…” indicates time.

  6. Opposite Direction Practice • Two boats leave at 7:00 AM from ports that are 252 miles apart and cruise toward each other at speeds of 30 mph and 26 mph. At what time will they pass each other? • Answer: 11:30

  7. Same Direction Key words: same direction, pass, overtake, catch up to Example: In a cross country race, Adam left the starting line at 7:00 AM and drove an average speed of 75 km/h. At 8:30, Manuel left the starting point and drove the same route, averaging 90 km/h. At what time did Manuel overtake Adam?

  8. RATE TIME DISTANCE A 75 x+1.5 75(x+1.5) M 90 X 90x Same Direction (Cont.) What do we know? Organize the information: Adam– 75 km/h Manuel– 90 km/h Manuel left 1.5 hours later. What do we want to find out? “What time…” indicates time.

  9. Same Direction Practice • In a cycling race, Mary left the starting line at 6:00 AM averaging 25 mph. At 6:30 AM, Peter left the starting point and traveled the same route at 30 mph. At what time did Peter pass Mary? • Answer: 9:00

  10. Round Trip Key words: round trip, to and from, there and back Example: A ski lift carried Maria up a slope at the rate of 6 km/h and she skied back down parallel to the lift at 34 km/h. The round trip took 30 min. How far did she ski?

  11. RATE TIME DISTANCE U 6 ½ - x 6(1/2 – x) D 34 x 34x Round Trip (Cont.) What do we know? Organize the information: Uphill– 6 km/h Downhill (ski) – 34 km/h Round trip took 30 min. What do we want to find out? “How far…” indicates distance. “How long…” indicates time

  12. Practice 1) At noon, a plane leaves Hawaii for California at 555 km/h. At the same time, a plane leaves California for Hawaii at 400 km/h. The distance between the two cities is about 3800 km. At what time do the planes meet? SOLUTION:

  13. Practice 2) Maria and Tom start at the park. They drive in opposite directions for 3 hours. The are then 510 km apart. Maria’s speed is 80 km/h. What is Tom’s speed? SOLUTION:

  14. Practice 3) Kara floats from A to B at 6 km/h. She returns by motorboat at 18 km/h. The float trip is 4 hours longer than the motorboat trip. How far is it from A to B? SOLUTION: The question asks us to find distance. Since distance = rate x time, D = 18(2) = 36 km

  15. Head Wind- Tail Wind • Some distance problems add the variable of wind speed or water current. • Tail Wind– Wind blowing in the same direction as the vehicle (increases speed) • Head Wind– Wind blowing in the opposite direction (decreases speed) • Upstream– Craft is traveling against the current (decreases speed) • Downstream– Craft is traveling with the current (increases speed)

  16. Example # 1 A boat traveled downstream 160 km in 8 hours. It took the boat 10 hours to travel the same distance upstream. Find the speed of the boat and the current. Let x = speed of the boat y = speed of the current

  17. Example # 2 When flying with a tailwind, a plane traveled 1200 miles in 3 hours. Flying with a headwind, the plane traveled the same distance in 4 hours. Find the plane’s speed and the speed of the wind. Let x = speed of the plane y = speed of the wind

  18. Work Rate • Work Rate problems are directly related to distance problems: • Rate x Time = Work Done • When setting up the table, • Rate = amount of work completed per unit of time • Time = amount of time working together

  19. Example #1 An office has two envelope stuffing machines.  Machine A can stuff a batch of envelopes in 5 hours, while Machine B can stuff a batch of envelopes in 3 hours.  How long would it take the two machines working together to stuff a batch of envelopes?

  20. Example # 1 Cont. What do we know? Machine A takes 5 hours. Machine B takes 3 hours.

  21. Example #2 Mary can clean an office complex in 5 hours.  Working together John and Mary can clean the office complex in 3.5 hours.  How long would it take John to clean the office complex by himself?

  22. Example # 2 Cont. What do we know? Mary can clean the office in 5 hours. Working together, it takes 3.5 hours.

  23. Mixture Problems • Mixture problems fall into 2 different categories: • Concentration/ percent • Money/ Cost • There are 2 different strategies for solving mixture problems: • Table • Diagram

  24. Example #1 CONCENTRATION A solution is 10% acid. How much pure acid must be added to 8 L of solution to get a solution that is 20% acid?

  25. Example # 1 (Cont.) Strategy: TABLE Use the last column in the table to write an equation. What is the resulting Concentration? You need to add 1 L of acid.

  26. Example # 1 (Cont.) Strategy: DIAGRAM Start Add Result 8L x 8+x + = .10 1 .20 To write the equation, Multiply the percent on The outside with the Amount on the inside!

  27. Example #2 Cost/ Money A mixture of raisins and peanuts sells for $4 a kilogram. The raisins sell for $3.60 a kilogram. The peanuts sell for $4.20 a kilogram. How many kilograms of each are used in 12 kilograms of the mixture?

  28. Example # 2 (Cont.) Strategy: TABLE Use the last column in the table to write an equation. You need 4 kg of raisins, and 8 kg of peanuts.

  29. Example # 2 (Cont.) Strategy: DIAGRAM Raisins Peanuts Mixture R 12-R 12 + = 3.60 4.20 4 To write the equation, Multiply the cost on The outside with the Amount on the inside!

  30. Example #3 CONCENTRATION You have 40g of a 50% solution of acid in water. How much water must you add to make a 10% acid solution?

  31. Example # 3 (Cont.) Strategy: DIAGRAM Start Add Result 40g x 40+x + = .50 0 .10 To write the equation, Multiply the percent on The outside with the Amount on the inside! We need 160g of water!

  32. Example 4-- Investment • Mervin Invested an amount of money at 5% interest and another amount at 8.25% interest. The amount invested at 5% was $1000 more than the amount at 8.25%. If his total income from simple interest for 1 year was $315, how much did he invest at each rate?

  33. Example 4 Cont. STRATEGY: TABLE $3000 invested at 5%. $2000 invested at 8.25%

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