Welcome! The aim of this topic is to help you solve the various kinds of problems on Races and Games. The first lesson in this topic is an introduction to Races and Games where we have shared some important points and terminologies involved . The exercises contain solved question and answers on Races and Games. When solving exercises, we recommend you to try solving the problems on your own, then view the solution.
By the end of this post, you will learn to solve the following types of problems:
- A and B are two friends who plan to run a race of X mts. B being slow, A gives B a lead of y mts. Now if you know the speed of A can you find the speed of B?
- In the above problem, if you know the speeds of A and B and lead given by A to B, can you find the length of the race.
- In a game of m points. If A gives B x points and B gives C y points. How many points can A give C
Introduction to Races
Starting Point: The point from which the race begins is the starting point.
Winning Point (or Goal): The point where the race ends is the winning point or goal.
Dead Heat: If all the contestants in a race reach the ending point at exactly the same time, then the race is said to end in dead heat.
In a race between 2 contestants A and B,
- If A gives B a start of ‘x’ metres, then A is at the starting point and B is ahead of A by ‘x’ metres.
- If A beats B by ‘x’ metres, say, in a 100m race, then while A runs 100 metres B runs 100-x metres.
- If A gives B a start of ‘t’ seconds, then A starts ‘t’ seconds after B starts from the same starting point.
- If A beats B by ‘t’ seconds, then A finishes the race ‘t’ seconds before B
Note : From the above points we see that, for a race of 100 metres the following statements are equivalent – “A beats B by 10 metres” or “A gives B a start of 10 metres’ or “A can give B a start of 10 metres”
Introduction to Games
In a game of 100, if A scores 100 points first, then A is the winner.
In a game of 100, If A gives B 10 points, it implies that when A scores 100 points, B scores 90 points.
Exercise
Races and Games - Exercise - Blog
15 questions
- In a race of 1 km, A can beat B by 100m or by 10 secs. What is the time taken by B to finish the race?
- 90 seconds
- 10 seconds
- 100 seconds
- 110 seconds
Show the answer
Answer: 100 seconds
If A beats B by 100m or 10 secs, it implies that B can cover 100m in 10secs. Hence speed of B is $\frac{100}{10}$ = 10 m/s.
To cover 1km or 1000m, B takes $\frac{1000}{10}$ = 100 secs.
- In a race of 100m, A can beat B by 10m or by 2 secs, what is the speed of A?
- 20 kms/hr
- 15 kms/hr
- 30 kms/hr
- 18 kms/hr
Show the answer
Answer: 20 kms/hr
Speed of B is $\frac{10}{2}$ = 5 m/s
Time taken by B to race 100m = $\frac{100}{5}$ = 20 secs.
Since, A beats B by 2 secs, A takes 18 secs to coves 100m.
Hence speed of A = $\frac{100}{18}$ m/s = $\frac{100}{18}$ x $\frac{18}{5}$ km/hr = 20 km/hr
- In a 100m race A gives B a lead of 10m and B gives C a lead of 10m. Then A can give C a lead of?
- 10m
- 19m
- 20m
- 25m
Show the answer
Answer: 19m
As per the problem,
If A runs 100m, then in the same time B runs 90m and
If B runs 100m, then in the same time C runs 90m.
To calculate the lead given by A to C, we have to find the distance run by C when B runs 90m.
Since when B runs 100m, C runs 90m,
when B runs 90m, C runs 90 x $\frac{90}{100}$ = 81m
i.e, when A runs 100m, B runs 90m and C runs 81m.
Hence, A gives C a lead of 19m
- In a 100m race, A gives B a lead of 12m and C a lead of 23m. What is the lead given by B to C?
- 10m
- 12.5m
- 20m
- 23m
Show the answer
Answer: 12.5m
To find the lead given by B to C, we have to find the distance covered by C when B runs 100m.
From the problem, we see that, when A runs 100m, B runs 88m and C runs 77m.
Hence, when B runs 88m, then C runs 77m.
So, when B runs 100m, C runs 77 x $\frac{100}{88}$ = 87.5m
Hence B gives C a lead of 12.5m
- In a certain game of 500 points, A gives B 50 points and B gives C 100 points. How many points does A give C.
- 50 points
- 100 points
- 120 points
- 140 points
Show the answer
Answer: 140 points
To find the number of points given by A to C, we have to find the number of points scored by C when A scores 500 points.
Given,
When A scores 500 points, B scores 450.
When B scores 500 points, C score 400.
Hence, when B scores 450 points, C scores 400 x $\frac{450}{500}$ = 360.
Hence, when A scores 500, B scores 450 and when B scores 450, C scores 360 which implies that when A scores 500 C score 360 or that A gives C 140 points.
- Speed of A and B is in ratio 2:3. In a 1 km race A gives B a lead of 130m. By what distance was A behind B when B reached end point?
- 580m
- 870m
- 780m
- 850m
Show the answer
Answer: 580m
Since A gives B a lead of 130m, B must travel 870m to reach the end point.
Now, given speeds of A and B is in ratio 2:3. i.e,
When B travels a distance of 3 units, A travels a distance of 2 units.
Then, when B travels a distance of 870m, A travels a distance of 2 x $\frac{870}{3}$ = 580m.
- A is 1 1/3 times faster than B. In 500m race A gives B a lead of 80 m. A beats B by what distance?
- 80m
- 45m
- 125m
- 70m
Show the answer
Answer: 45m
Since A is 1$\frac{1}{3}$ times faster than B,
speed of A and B is in ratio 4:3.
i.e when A travels 4m, B travels 3m
So when A travels 500 , B travels 3 x $\frac{500}{4}$ = 375m
Since, A gives B a lead of 80m, A beats B by 500 - (375+80) = 45m.
- If A takes 20 secs to cover 100 m while B takes 25 secs to cover the same distance. If A gives B a lead of 160 m. Find the time in which A beats B in 1 km race.
- 25 secs
- 20 secs
- 15 secs
- 10 secs
Show the answer
Answer: 10 secs
Speed of A = $\frac{100}{20}$ = 5 m/s
Speed of B = $\frac{100}{25}$ = 4 m/s
In a race of 1 km, A takes $\frac{1000}{5}$ = 200 secs to reach the end point and since A gives B a lead of 160m, to cover a distance of 840m,B takes $\frac{840}{4}$ = 210 secs.
Hence, A beats B by 210 - 200 = 10 seconds.
- In a race of 200 m, A completes the race in 40 secs and B in 50 secs. A beats B by what distance?
- 20m
- 30m
- 40m
- 50m
Show the answer
Answer: 40m
Hence A beats B by 10 seconds.
Distance covered by B in 10 seconds is $\frac{200}{50}$ x 10 = 40m.
Hence A beats B by 40m.
- In a race of 500 m, A beats B by 50 m and C by 100 m. In a race of 144m, by how much distance will B beat C?
- 50m
- 35m
- 24m
- 16m
Show the answer
Answer: 16m
Given that,
If A covers 500m, B covers 450m and C covers 400m.
which implies that, when B covers 450m, C covers 400m.
Then, when B covers 144m, C covers 400 x $\frac{144}{450}$ = 128m.
Hence, in a 144m race B beats C by 144 - 128 = 16m.
- A runs 1 km in 2 mins 5 secs and B runs the same distance in 3 min 20 secs. What is the lead given by A to B, if the race ends in a dead heat.
- 125m
- 250m
- 375m
- 500m
Show the answer
Answer: 375m
A's speed = $\frac{1000}{125}$ = 8 m/s {1km = 1000m and 2 mins 5 secs = 125 secs}
B's speed = $\frac{1000}{200}$ = 5 m/s {1km = 1000m and 3 min 20 secs = 200secs}
Let the lead given by A to B be $x$m. Then, if the race ends in a dead heat, B must travel (1000-x)m in the same time as A takes to cover 1000m. Hence,
$\frac{1000-x}{5}$ = 125
$x$ = 375m.
- A runs at a speed of 6 m/s and B at a speed of 8 m/s. If B gives A a lead of 100 m. What must be the length of the race so that the race ends in a dead heat.
- 400m
- 500m
- 100m
- 1000m
Show the answer
Answer: 400m
Let the length of the race be $x$m. Then in the race, since B gives A a lead of 100m, A has to cover ($x$-100)m and B has to cover $x$m.
For the race to end in dead heat,
$\frac{x-100}{6}$ = $\frac{x}{8}$
$6x$ = $8x$ - $800$
$x$ = $400$m
- In a game of 100 points, A gives B 10 points and C gives A 15 points. Find the number of points C can give to B in a game of 400 points.
- 10
- 25
- 50
- 94
Show the answer
Answer: 94
When C scores 100 points A scores 85 points.
So, when C scores 400 points A scores 340 points.
Now, to find the number of points given by C to B, find the number of points given by A to B when A scores 340 points.
Given, when A scores 100 points, B scores 90 points and
when A scores 340 points, B scores 90 x $\frac{340}{100}$ = 306 points.
Hence, in a game of 400 points when C scores 400 points, A scores 340 points and when A scores 340 points B scores 306 points, which implies that when C scores 400 points B scores 306 points.
Hence, in a game of 400 points C can give 400-306 = 94 points to B.
- In a certain race, A gives B a lead of 20 m and C a lead of 30 m. B gives C a lead of 15 m. Find the length of the race.
- 60m
- 80m
- 100m
- 120m
Show the answer
Answer: 60m
If the race is for $x$m, then
when A runs $x$m, B runs $x-20$m and C runs $x-30$m ___ 1
when B runs $x$m, C runs $x-15$m ___ 2
i.e,
From 1 => B : C = $x-20$ : $x-30$
From 2 => B : C = $x$ : $x-15$
∴ $\frac{x-20}{x-30}$ =$\frac{x}{x-15}$
$(x-20)(x-15)$ = $x(x-30)$
$x^2-35x+300$ = $x^2-30x$
5$x$ = 300
$x$ = 60m
- In a race of 1000 m. A runs at a speed of 5 m/s and gives B a start of 100 m. What is the speed of B if the race ends in a dead rubber.
- 5 m/s
- 4.5 m/s
- 4 m/s
- 3 m/s
Show the answer
Answer: 4.5 m/s
Let the speed of B be $x$ m/s.
If the race ends in a dead rubber, then the time taken by A covers 1000m will be same as the time taken by B to cross 900m(since 100m is the lead given by A to B).
i.e.
$\frac{1000}{5}$ = $\frac{900}{x}$
$x$ = 4.5 m/s
