In a 100 m race, A beats B by 20 m. If A runs at 5 m/s, how many seconds after A does B finish?

5 s. A runs 100 m in 1005 = 20 s, in which B covers 80 m, so B's speed is 4 m/s. B still has 20 m, taking 204 = 5 s.

In a 200 m race, A beats B by 10 m or 2 seconds. What is B's speed?

5 m/s. 'Beats B by 10 m or 2 s' means B covers those last 10 m in 2 s, so B's speed is 102 = 5 m/s.

In a 100 m race, A beats B by 10 m and B beats C by 10 m. By how many metres does A beat C in a 100 m race?

19 m. When A covers 100 m, B covers 90 m. When B covers 90 m, C covers 90 90100 = 81 m. So A beats C by 100 - 81 = 19 m.

A runs 1 23 times as fast as B. If A gives B a start of 80 m, how long must the race course be for them to finish together?

200 m. A : B speeds = 5 : 3. For a dead heat, LL - 80 = 53, so 3L = 5L - 400 and L = 200 m.

In a 1 km race, A beats B by 40 m and B beats C by 25 m. In the same race, by how many metres does A beat C?

64 m. When A runs 1000 m, B runs 960 m. C covers 9751000 of B's distance, so C runs 960 0.975 = 936 m. A beats C by 1000 - 936 = 64 m.

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Quantitative Aptitude · Time, Work & Distance

Races and Games of Skill — Questions & Answers

In a race, 'A beats B by $x$ m' means when A finishes, B still has $x$ m to run. Convert every statement into how far each runner covers in the same time, then take the ratio.

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Key formulas included

Key formulas

'A beats B by $x$ m': in A's finishing time, A runs the full course and B runs (course $- x$ )
'A beats B by $t$ s': B covers the last stretch in $t$ s at B's speed
'A gives B a start of $x$ m': B only has to run (course $- x$ )

5 solved Races and Games of Skill questions

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01
In a 100 m race, A beats B by 20 m. If A runs at 5 m/s, how many seconds after A does B finish?
3 steps · reveal one at a time
1. Step 1: Calculate A's Time
  First, let's find out how long A takes to finish the 100 m race. A runs at $5 \text{ m/s}$ , so A's time to cover $100 \text{ m}$ is $\frac{100}{5} = 20 \text{ s}$ .
2. Step 2: Find B's Speed
  When A finishes in $20 \text{ s}$ , B has covered $100 - 20 = 80 \text{ m}$ . Since B runs for the same $20 \text{ s}$ , B's speed is $\frac{80}{20} = 4 \text{ m/s}$ .
3. Step 3: The answer
  So the answer is (B) 5 s. B still needs to cover the remaining $20 \text{ m}$ . At B's speed of $4 \text{ m/s}$ , this will take $\frac{20}{4} = 5 \text{ s}$ . This is how many seconds after A that B finishes.
02
In a 200 m race, A beats B by 10 m or 2 seconds. What is B's speed?
4 steps · reveal one at a time
1. Step 1: Interpret the Lead
  The phrase "A beats B by 10 m or 2 seconds" means that when A crosses the finish line, B is still 10 meters behind A. This also implies that B would take an additional 2 seconds to cover those last 10 meters.
2. Step 2: Relate Distance and Time
  From the problem statement, we can directly infer that B covers a distance of 10 meters in a time of 2 seconds. This is the crucial piece of information needed to find B's speed.
3. Step 3: Calculate B's Speed
  Speed is calculated by dividing the distance traveled by the time taken. For B, this is $10 \text{ m} \div 2 \text{ s}$ . Therefore, B's speed is $5 \text{ m/s}$ .
4. Step 4: The answer
  So the answer is (C) 5 m/s. This is because B's speed is directly found by dividing the 10 m distance by the 2 s time it takes B to cover that distance.
03
In a 100 m race, A beats B by 10 m and B beats C by 10 m. By how many metres does A beat C in a 100 m race?
4 steps · reveal one at a time
1. Step 1: A's and B's distance
  First, let's look at A and B. A beats B by 10 m in a 100 m race. This means that when A finishes the 100 m race, B has covered $100 - 10 = 90$ m.
2. Step 2: B's and C's distance
  Next, consider B and C. B beats C by 10 m in a 100 m race. So, when B covers 100 m, C covers $100 - 10 = 90$ m. This implies C covers $90/100$ of the distance B covers.
3. Step 3: C's distance when A finishes
  We know that when A covers 100 m, B covers 90 m. Since C covers $90/100$ of B's distance, C will cover $90 \times (90/100)$ m when B covers 90 m. This calculates to $81$ m.
4. Step 4: The answer
  So the answer is (A) 19 m. When A finishes the 100 m race, C has covered 81 m, meaning A beats C by $100 - 81 = 19$ m.
04
A runs $1\tfrac{2}{3}$ times as fast as B. If A gives B a start of 80 m, how long must the race course be for them to finish together?
5 steps · reveal one at a time
1. Step 1: Understand speeds
  First, let's figure out the ratio of their speeds. A runs $1\tfrac{2}{3}$ times as fast as B. Converting the mixed fraction, $1\tfrac{2}{3} = \frac{5}{3}$ . So, if B's speed is 3 units, A's speed is 5 units. The speed ratio of A to B is $5 : 3$ .
2. Step 2: Set up distances
  A gives B a start of 80 m. This means B starts 80 m ahead of A. If the total race course length is $L$ meters, A must run $L$ meters. B, however, only needs to run $L - 80$ meters to reach the finish line.
3. Step 3: Formulate the equation
  For them to finish together (a dead heat), they must take the same amount of time. Since time is distance divided by speed, we can set up the equation: $\frac{\text{Distance A}}{\text{Speed A}} = \frac{\text{Distance B}}{\text{Speed B}}$ . Using our variables, this is $\frac{L}{5} = \frac{L - 80}{3}$ .
4. Step 4: Solve for L
  Now, we solve the equation for $L$ . Cross-multiply to get $3L = 5(L - 80)$ . Distribute the 5: $3L = 5L - 400$ . Subtract $5L$ from both sides: $-2L = -400$ . Finally, divide by -2 to find $L = 200$ m.
5. Step 5: The answer
  So the answer is (C) 200 m. This is the length of the race course required for A and B to finish simultaneously, given A's speed advantage and B's head start.
05
In a 1 km race, A beats B by 40 m and B beats C by 25 m. In the same race, by how many metres does A beat C?
4 steps · reveal one at a time
1. Step 1: A's lead over B
  When A completes the $1$ km ( $1000$ m) race, B is $40$ m behind. This means that when A runs $1000$ m, B runs $1000 - 40 = 960$ m.
2. Step 2: B's lead over C
  Similarly, when B completes $1000$ m, C is $25$ m behind. So, C runs $1000 - 25 = 975$ m in the time B runs $1000$ m. This means C covers $\frac{975}{1000}$ of B's distance.
3. Step 3: C's distance when A finishes
  We know B runs $960$ m when A finishes. C covers $0.975$ of B's distance. So, C runs $960 \times 0.975 = 936$ m when A runs $1000$ m.
4. Step 4: The answer
  So the answer is (D) 64 m. A beats C by the difference between their distances: $1000 - 936 = 64$ m.

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