Games and Races Related Problems

A race is a competition in which competitors must traverse a certain distance in the quickest time possible. Driving, biking, diving, sailing, rowing, and other activities may be included in a race. A racecourse is a path on which a race takes place. The starting place of a race is called the starting point, while the winning point, or goal, is where the race ends. Races and games are subtopics of time, speed, and distance. So all the formulas of time, speed, and distance apply to problems regarding races and games.

A Glossary of Terms Used in Races

Here are some key concepts of races and games:

1. ‘A gives B an x-meter start’: This sentence implies that A begins the race at the starting line, while B begins x meters ahead of A. If x is 50, then A will have to travel 100 meters, but B will only cover 50 meters to complete a 100-meter race (100 – x).

2. ‘A wins by x m over B’: This sentence implies that while A reached the winning position, B is x m behind A. In this scenario, A has completed 100 meters, while B has only 50 meters to cover a 100-meter race (100 – x ).

3. A can give B a t-minute headstart: This sentence implies that A will begin t minutes after B has already started from the beginning. Both A and B will arrive simultaneously at the finish line.

4. B x meters and t minutes are given by A: This statement suggests that A and B leave the starting line simultaneously, but A arrives at the finish line x meters ahead of B, and B takes t minutes longer to complete the race than A. As a result, B spends extra t minutes covering the remaining x meters. B’s speed is calculated as x/t.

5. Dead Heat: When all participants arrive simultaneously at the finish line, it is called dead heat.

6. At P, A and B start together: Both runners started from the same spot. The distance covered by the runners in the time it takes the winner to cross the finish line is the ratio of their speed.

7. A wins by t seconds over B: Once A arrives at the finish line, B will take t seconds longer to traverse the remaining distance. The winner’s time is equal to the loser’s time minus t.

Reference Questions for Games and Races

1. In a 200-meter race, P provides Q a 20-meter headstart and R a 56-meter lead. As a result, Q can offer R a headstart of how many meters?

We may accomplish this by using a ratio. P’s ratio to Q and R is presented to us. P:Q is thus 200:180, while P:R is 200:144. 

Q/P × P/R = 180/200 × 200/144

Q/R = 180/144

As a result, R will run 144 meters when Q runs 180 meters. 

R will run 144/180 × 100 = 80 meters when Q runs.

2. In a 50-meter race, A can give B a 5-meter start and C a 14-meter start. How much of a headstart can B offer C in the same race?

A covers 50 meters, B spans (50 – 5) = 45 meters, and C spans (50-14)=36 meters at the same time. If the race is just between B and C, the unitary technique predicts that B will run 45 meters and C will run 36 meters. As a result, when B runs 50 meters, C runs (36/4550 = 40). That signifies that B gives C a 10-meter head start.

3. Bhairav may give Akshay a 128-meter start in a mile race. If Bhairav can give Chinmay a four-meter head start in a 100-meter dash, who among Akshay and Chinmay will win a one-and-a-half-mile race, and what will be the winner’s final lead over the loser? A mile is 1,600 meters.

Bhairav may give Akshay a 128-meter start in a mile stretch. Bhairav can begin after Akshay has covered 128 meters and still finish the mile with him. In other terms, Bhairav can cover one mile (1,600 meters) in the same amount of time as Akshay (1,600 – 128) = 1,472 meters. As a result, the ratio of Bhairav and Akshay’s speeds equals the ratio of both lengths travelled in the same time= 1600/1472 = 25: 23.

Bhairav can provide Chinmay with a 4-mile start. When Bhairav runs 100 meters, Chinmay is only running 96 meters. As a result, the ratio between Bhairav’s and Chinmay’s speeds is 96:100 = 25:24.

As a result, B:A = 25:23  and B:C = 25:24 are obtained. As a result, A:B:C = 23:25:24. Although Chinmay covers 24 meters, Akshay only covers 23 meters in the same amount of time. Chinmay is, in other terms, speedier than Akshay. So, if both of them race for 1.5 miles (2,400 meters), Chinmay will finish first, and Akshay will only finish 2,300 meters. In other terms, Chinmay would win by 100 meters (1/16 mile) over Akshay.

4. In a 10-kilometer race, the gold, silver, and bronze medals go to A, B, and C, all running simultaneously. How many meters does A beat C by if A beats B by 1 km and B beats C by 1 km?

A beats B by 1 kilometer, which means A goes 10 kilometers simultaneously as B. The ratio of A and B speeds is 10:9. Similarly, B and C have a 10:9 speed ratio.

The ratio of their speeds, A:B:C = 100:90:81

A:B:C = 10:9 B:C = 10:9

When A travels 100 meters, C travels 81 meters; when A travels 10,000 meters, C travels 8,100 meters; and when A travels 10,000 meters, C travels 8,100 meters or A beats C by 1,900 meters.

Conclusion

Races are part of competitive games where one can observe practical applications of mathematics. One should have a clear concept of speed and time to solve questions regarding races and games.