This document provides an overview of topics related to rates and work, including distance, speed, and time; average speed; relative speed; to and fro motion; and work, time, and efficiency. It includes definitions and formulas for these concepts, as well as example problems and solutions. The document aims to teach the relationships between distance, speed, and time when motion is constant, how to calculate average speed, how to determine relative speeds of objects moving in the same or opposite directions, and how to solve word problems involving work completed over time by individuals or groups.

1. GRE
Concept Session: Rates and Work
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2. Topics
• Distance, speed and time
• Average Speed
• Relative Speed
• To and fro motion
• Work, time and efficiency
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3. Distance, Speed and time
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Constant Time Speed Distance
Relationship
between the other
two variables
D varies directly
with S
D1/D2 = S1/S2
D varies directly
with T
D1/D2 = T1/T2
T varies Inversely
with S
T1/T2 = S2/S1

4. Problem
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Claire ran the first 10 kilometers of a 20-kilometer race at a constant
rate of 12 kilometers per hour. If she completed the entire 20-
kilometer race in 1 hour, at what constant rate did she run the last 10
kilometers of the race, in kilometers per hour?
A. 60
B. 50
C. 75
D. 45
E. 120
Answer : Option A

5. Average Speed
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𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑆𝑝𝑒𝑒𝑑 =
𝑇𝑜𝑡𝑎𝑙 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑇𝑜𝑡𝑎𝑙 𝑇𝑖𝑚𝑒
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When the total distance is in two parts and the distances covered are equal, let
the speeds be a and b over the two distances.
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑆𝑝𝑒𝑒𝑑 =
2𝑎𝑏
(𝑎 + 𝑏)

6. Problems
Chris drove from his house to office at 40 miles per hour, and
returned by the same route at 60 miles per hour.
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Answer : Option B
Quantity A Quantity B
Chris’ average speed for the
round trip, in miles per hour
50

7. Problems
Reek travels the first 2/3rd of a trip at 20 miles per hour (mph)
and the remainder of the trip at 30 mph. What is his average
speed over the whole journey?
A. 22 mph
B. 23 mph
C. 22.5 mph
D. 21.5 mph
E. Insufficient information
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Answer : Option C

8. Relative Speed
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When in same direction,
𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑆𝑝𝑒𝑒𝑑 = 𝑆𝑝𝑒𝑒𝑑 𝑓𝑎𝑠𝑡𝑒𝑟 − 𝑆𝑝𝑒𝑒𝑑 𝑆𝑙𝑜𝑤𝑒𝑟
When in opposite direction,
𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑆𝑝𝑒𝑒𝑑 = 𝑆𝑝𝑒𝑒𝑑 𝑓𝑎𝑠𝑡𝑒𝑟 + 𝑆𝑝𝑒𝑒𝑑 𝑆𝑙𝑜𝑤𝑒𝑟

9. Problems
A terrorist hijacked a train heading due north. At exactly the same time, a
police car located 100 miles south of the train started driving north
toward the train on an adjacent roadway parallel to the train track. If the
train travelled at a constant rate of 60 miles per hour, and the police car
travelled at a constant rate of 90 miles per hour, how long after the
hijacking did the police car catch up with the train?
A. 3 hours and 15 minutes
B. 3 hours and 20 minutes
C. 2 hours and 45 minutes
D. 3 hours
E. 3 hours and 30 minutes
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Answer : Option B

10. Problems
Two horses begin running on an oval course at the same time.
One runs each lap in 15 minutes; the other takes 12 minutes to
run each lap. How many minutes after the start will the faster
horse have a one lap lead?
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Answer : 60

11. To and fro motion in a straight line
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Two cars, A and B, started simultaneously from opposite ends of a
960-yard track. Car A travelled at a constant rate of 60 feet per
second; Cart B travelled at a constant rate of 36 feet per second.
After how many seconds of travel did the two carts collide? (1 yard
= 3 feet)
A. 100
B. 30
C. 10
D. 20
E. 25

12. Work, time and efficiency
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Work = time x efficiency
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13. Always work with efficiency
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14. Problems
Working alone at its own constant rate, a machine seals k
cartons in 8 hours, and working alone at its own constant
rate, a second machine seals k cartons in 4 hours. If the
two machines, each working at its own constant rate,
together sealed 6k cartons in t hours.
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Quantity A Quantity B
t 15

15. Problems
Twelve workers pack boxes at a constant rate of 120 boxes in 8
minutes. How many minutes would it take 24 workers to pack 180
boxes, if all workers work at the same constant rate?
(A) 12
(B) 8
(C) 14
(D) 6
(E) 10
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Answer : Option D

16. Wage proportional to efficiency
A and B can do a project in 12 and 15 days
respectively. They were told to complete the project in
5 days and were paid $ 1440. They took the help of C
and completed the work in time. Find C’s share in the
total money earned by them.
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1) 250 2) 360 3) 390 4) 450

17. Wage proportional to amount of work done
A,B and C can complete a work in 10, 20 and 30 days
respectively when they work individually. All three started
simultaneously. After 3 days A left, After two more days B left
and C completed the remaining work alone. If the total work is
for $ 6000. Find the amount received by C.
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1) 2700 2) 2500 3) 3500 4) 550