Races and Games of Skill

UTKALPRESS

RACES & GAMES OF SKILL — ARITHMETIC HANDBOOK

PAGE 1 — BASIC CONCEPT OF RACES

Problems on Races are based mainly on the relationship between distance, speed and time. Two or more persons run over the same or different distances, and we compare their speeds or finishing times.

Speed = Distance ÷ Time
Time = Distance ÷ Speed
Distance = Speed × Time

1. Meaning of "A beats B by x metres"

If A beats B by x metres in a race of D metres, then when A finishes the race, B has covered only D − x metres.

Speed of A : Speed of B = D : (D − x)

Example: A beats B by 20 m in a 100 m race. Find the ratio of their speeds.

A : B = 100 : (100 − 20)
= 100 : 80
= 5 : 4

2. Finding the Distance Covered

If the speed ratio of A and B is known, and A completes a race of D metres, the distance covered by B in the same time can be found directly from the speed ratio.

Distance covered in same time ∝ Speed

If speed ratio = a : b,
distance ratio in the same time = a : b

Example: A and B run at speeds in the ratio 5 : 4. If A completes 500 m, how far does B run in the same time?

Distance ratio = 5 : 4
B's distance = 500 × 4/5
= 400 m

Therefore, A beats B by 100 m.

3. Time Ratio in a Race

For the same race distance, time is inversely proportional to speed.

For equal distance:
Time Ratio = Inverse of Speed Ratio

If Speed A : B = a : b,
then Time A : B = b : a

Example: The speeds of A and B are in the ratio 3 : 5. Find the ratio of their times for the same distance.

Time ratio = 5 : 3

Key idea: In race problems, always identify whether the question gives a distance margin, time margin, speed ratio or head start.

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RACES & GAMES OF SKILL

PAGE 2 — HEAD START, TIME MARGIN & DIFFERENT RACES

1. Head Start

A head start means one runner is allowed to start ahead of another runner by a certain distance.

If B gets a head start of x metres in a race of D metres,
B runs only (D − x) metres.

Example: A runs at 10 m/s and B runs at 8 m/s. In a 100 m race, B gets a head start of 20 m. Who reaches the finish line first?

A's time = 100/10 = 10 s
B's distance = 100 − 20 = 80 m
B's time = 80/8 = 10 s

Hence, both reach the finish line at the same time.

2. Head Start Given by A

If A can give B a head start of x metres in a race of D metres, then:

Speed A : Speed B = D : (D − x)

Example: A can give B a start of 25 m in a 100 m race. Find their speed ratio.

A : B = 100 : 75
= 4 : 3

3. Winning by Time

Sometimes the winner finishes a race a certain number of seconds before the other runner.

Time of slower runner − Time of faster runner = Winning time margin

Example: A completes a 200 m race in 20 seconds and B completes it in 25 seconds.

Time difference = 25 − 20
= 5 seconds

Therefore, A wins by 5 seconds.

4. When A Beats B by x Metres and y Seconds

Suppose both runners start together. A completes D metres while B completes only D − x metres in the time A takes.

Speed A = D/t
Speed B = (D − x)/t

Therefore:
Speed A : Speed B = D : (D − x)

If B takes y seconds more than A, then:

Time B = Time A + y

5. Different Race Lengths

If the same two runners participate in races of different lengths, calculate the time using:

Time = Distance ÷ Speed

Given Information	Useful Relation
A beats B by x metres in D metres	Speed ratio = D : (D − x)
Speed ratio a:b	Same-time distance ratio = a:b
Equal distance	Time ratio = inverse speed ratio
Head start x in D race	Runner covers D − x

Exam Tip: "Beats by x metres" refers to the distance between the two runners when the winner crosses the finish line.

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RACES & GAMES OF SKILL

PAGE 3 — CIRCULAR RACES & GAMES OF SKILL

1. Circular Race

In a circular race, runners move around a circular track. The important concept is relative speed.

Relative Speed in Opposite Directions = S₁ + S₂

Relative Speed in Same Direction = |S₁ − S₂|

Example: Two runners run around a circular track of length 400 m at 8 m/s and 6 m/s in opposite directions. When will they meet for the first time?

Relative speed = 8 + 6 = 14 m/s
Time = 400/14
= 28.57 seconds approximately

2. Meeting Again on a Circular Track

If two runners start together from the same point and run in the same direction, the faster runner catches the slower runner after covering one complete extra round relative to the slower runner.

Catch-up Time = Track Length ÷ Difference of Speeds

Example: Two runners run on a 600 m circular track in the same direction at 10 m/s and 7 m/s.

Relative speed = 10 − 7 = 3 m/s
Catch-up time = 600/3
= 200 seconds

3. Games of Skill — Basic Concept

In aptitude questions, a game of skill generally describes a competition between two players where their scores, points, goals, runs or performances are compared.

The mathematical approach is usually based on ratios, differences, percentages and proportional relationships.

Score Difference = Winner's Score − Loser's Score

Total Score = Score of Player A + Score of Player B

Example: In a game, A scores 60 points and B scores 45 points. Find the ratio of their scores.

A : B = 60 : 45
= 4 : 3

4. Ratio of Scores

If A : B = a : b and total score = T,

A's score = T × a/(a+b)
B's score = T × b/(a+b)

Example: The scores of A and B are in the ratio 5 : 3. Their total score is 64. Find their individual scores.

Total parts = 5 + 3 = 8
A = 64 × 5/8 = 40
B = 64 × 3/8 = 24

5. Points Won in Multiple Games

If each game gives a fixed number of points, total points can be calculated from the number of games won or lost.

Total Points = Number of Games × Points per Game

Shortcut: For circular races use relative speed. For score-based games use ratios, differences and total points according to the data given.

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FORMULA SHEET • SOLVED EXAMPLES • PRACTICE

PAGE 4 — COMPLETE FORMULA REVISION

``` ```

Concept	Formula / Relation
Basic Speed	Speed = Distance ÷ Time
Race Margin	A beats B by x in D race → A:B = D:(D−x)
Same-Time Distance	Distance ratio = Speed ratio
Same-Distance Time	Time ratio = Inverse speed ratio
Head Start	Runner with x m start covers D−x
Opposite Circular Race	Meeting time = Track length ÷ (S₁+S₂)
Same-Direction Circular Race	Catch-up time = Track length ÷ \|S₁−S₂\|
Score Ratio	If A:B=a:b, A=T×a/(a+b)

SOLVED REVISION EXAMPLES

Example 1: A beats B by 30 m in a 150 m race. Find their speed ratio.

A : B = 150 : 120
= 5 : 4

Example 2: A and B run at 12 m/s and 8 m/s respectively. A gives B a head start of 40 m in a 120 m race. Find whether they finish together.

A's time = 120/12 = 10 s
B's distance = 120 − 40 = 80 m
B's time = 80/8 = 10 s

Therefore, they finish together.

Example 3: Two players score in the ratio 7 : 5. If their total score is 144, find their scores.

Total parts = 12
First score = 144 × 7/12 = 84
Second score = 144 × 5/12 = 60

PRACTICE QUESTIONS

1. A beats B by 20 m in a 100 m race. Find the ratio of their speeds.

2. The speeds of A and B are in the ratio 5 : 4. If A runs 500 m, how far does B run in the same time?

3. A and B run at 10 m/s and 8 m/s. In a 100 m race, B gets a head start of 20 m. Do they finish together?

4. A beats B by 30 m in a 150 m race. Find their speed ratio.

5. Two runners move in opposite directions on a 500 m circular track at 6 m/s and 4 m/s. Find their first meeting time.

6. Two runners move in the same direction on a 600 m circular track at 10 m/s and 7 m/s. Find their catch-up time.

7. The scores of A and B are in the ratio 3 : 2. If their total score is 100, find their individual scores.

8. A completes a 200 m race in 20 seconds and B completes it in 25 seconds. By how many seconds does A win?

9. A and B have speeds in the ratio 4 : 3. If A completes 400 m, find the distance covered by B in the same time.

10. Two players score in the ratio 7 : 5. If the total score is 144, find the difference between their scores.

ANSWER KEY

``` 1. 5 : 4
2. 400 m
3. Yes, both take 10 seconds
4. 5 : 4
5. 50 seconds
6. 200 seconds
7. A = 60, B = 40
8. 5 seconds
9. 300 m
10. 24 points ```

EXAM SHORTCUTS

1. Same distance → time is inversely proportional to speed.

``` 2. Same time → distance is directly proportional to speed.

3. "A beats B by x metres in D metres" → use D : (D−x).

4. Circular track + opposite direction → add speeds.

5. Circular track + same direction → subtract speeds.

6. For score problems, convert ratios into total parts first.

7. Always keep distance and speed in compatible units before calculating time. ```