Rates

Question 1

8 Types of Rate Problems

Answer 1

1. Elementary rate
2. Average rate
3. Converging rate
4. Diverging rate
5. Round trip rate
6. Catch-up rate
7. Relative motion rate
8. If/then question

Question 2

Average Rate

Answer 2

Average Rate = Total Distance / Total Time

Average Rate = (t₁/T)v₁ + (t₂/T)v₂

Question 3

Average Rate: Hard Question

Answer 3

William drove from Town A to Town B and then drove along the same road from Town B back to Town A. Was William’s average speed for the entire trip at least 60 miles per hour?

1. The distance from Town A to Town B is 117 miles.
2. William’s average speed while driving from Town A to Town B was 30 miles per hour.

Answer = B. If William averaged 30mph (half of 60) for one-half of the trip, it is impossible for the average speed for the entire trip to be 60 mph.

Question 4

Converging Rate

Answer 4

Distance₁ + Distance₂ = Total Distance_{1 and 2}

Question 5

Converging Rate: Two objects leave at different times

Answer 5

• T = the travel time of the object that leaves later
• T + difference between departure time = the travel time of the object that leaves earlier

Question 6

Converging Rate: Relative Rate

Answer 6

If x rate is ¼ greater than y rate → x = y(1 + ¼) = 5/4y

Question 7

Diverging Rate

Answer 7

Distance₁ + Distance₂ = Total Distance_{1 and 2}

Question 8

Round-trip Rate

Answer 8

The distance an object travels from the starting point to the destination equals the distance the object travels back from the destination to the starting point.

Question 9

Catch-up Rate

Answer 9

1. Type 1: When two objects start from the same point and catch up to each other, both objects will have traveled the same distance when they meet.
2. Type 2: The faster object’s distance is equal to the slower object’s distance plus any difference in starting points and distance by which the faster object must pass the slower object.
   → Time = ∆ Distance / ∆ Rate

Question 10

Relative Motion Rate

Answer 10

• An object travels relatively faster when it’s moving along with an outside force than when it is traveling under its own power.
• An object travels relatively slower when it’s moving against with an outside force than when it is traveling under its own power.

Question 11

Fast vs. Slow

Answer 11

Faster Time + Time Difference = Slower Time

Question 12

If/then Rate

Answer 12

If [object] had traveled [some rate], it would have [saved/added] t hours to its time.

Question 13

Relationships in Rate Problems

Answer 13

RT = D
→ Rate is inversely proportional to time.
→ Rate is directly proportional to distance.
→ Time is directly proportional to distance.

Examples:

• Did it take Brian more than 15 minutes to walk 600 feet?
  → D = 600, T > 15 min, so is R < 40 ft/min?
• Did it take Brian less than 15 minutes to walk 600 feet?
  → D = 600, T < 15 min, so is R > 40 ft/min?

Question 14

Constant Amount

Answer 14

Grow by a constant amount means adding a constant amount

Question 15

Constant Factor

Answer 15

Grow by a constant factor means multiplying by a constant factor

Question 16

Growth

Answer 16

Don’t make assumptions about the growth driver!

Question 17

Counting in Line

Answer 17

If you are the m^th person counted from the beginning of the line and the n^th person counted from the end of the line, then the number of people waiting in the line is m + n - 1

Be careful when there are more than 1 person in the line. There can be multiple line formation scenarios (e.g., person A could be either in front of or behind person B) if the question doesn’t specify.