Races and Games of Skill — Extra Practice

Competitive Mathematics · Time, Work and Distance

7 Part13 worked examples172 practice questions

NoteThis is the Extra Practice set: lots of questions. To learn the methods first, read the main chapter of the same topic (the one without “Extra Practice” in its name), then practise here.

In a school 100 m race, if you are the fastest and your friend slower, he is allowed to start a little ahead for fairness — that is a start. If you run 100 m while he runs only 80 m, you gave him a start of 20 m. Every race question rests on one thing: how far each contestant covers in the same time. Their speeds are in the same ratio as the distances covered — as long as both ran for the same time.

∑ Formula

\frac{v _{A}}{v _{B}} = \frac{d _{A}}{d _{B}} (बराबर समय में), A जीते \Rightarrow d_{A} = L, d_{B} = L - हार का अंतर

The basic race rule

Starts and the length of the race

Full length − start

The one with a start runs a shorter race; in point games, the loser’s score.

Terms and direct questions

Start, margin, dead heat

Start, finishing point, winner, dead heat and the points game.

A gives B a start of $x$ m: A runs the full $L$ m, B only $(L - x)$ m.
A beats B by $x$ m: when A reaches the finish B is $x$ m behind — he has run $(L - x)$ m.
Dead heat: both finish at the same moment; the margin is zero.
A game of $N$ points: whoever scores $N$ first wins. If A scores $N$ and B scores $b$ , A can give B $(N - b)$ points.

Example 1 · Basic

In a 400 m race A gives B a start of 15 m. How far must B run?

Solution, step by step

1B’s run $= 400 - 15$ .
2 $= 385$ m

Answer 385 m

Example 2 · Basic

In a race P gives Q a start of 40 m, leaving Q a run of 260 m. What is the full length of the race?

Solution, step by step

1Q’s run $=$ full $-$ start.
2Full length $= 260 + 40 = 300$ m

Answer 300 m

Example 3 · Basic

In a game of 120 points A scores 120 while B scores only 85. How many points can A give B?

Solution, step by step

1A can give $120 - 85$ points.
2 $= 35$ points

Answer 35 points

BasicTry it yourself first

In a 400 m race A gives B a start of 15 m. How far does B have to run?

BasicTry it yourself first

In a 200 m race A gives B a start of 30 m. How far does B have to run?

StandardTry it yourself first

In a 800 m race A gives B a start of 45 m. How far does B have to run?

StandardTry it yourself first

In a 1000 m race A gives B a start of 120 m. How far does B have to run?

AdvancedTry it yourself first

In a 500 m race A gives B a start of 35 m. How far does B have to run?

BasicTry it yourself first

In a race P gives Q a start of 40 m, making the race 260 m for Q. What is the full length of the race?

BasicTry it yourself first

In a race P gives Q a start of 15 m, making the race 185 m for Q. What is the full length of the race?

StandardTry it yourself first

In a race P gives Q a start of 60 m, making the race 440 m for Q. What is the full length of the race?

AdvancedTry it yourself first

In a race P gives Q a start of 75 m, making the race 325 m for Q. What is the full length of the race?

BasicTry it yourself first

In a 400 m race M gives N some start and N's race becomes 360 m. How much start did N get?

StandardTry it yourself first

In a 500 m race M gives N some start and N's race becomes 455 m. How much start did N get?

StandardTry it yourself first

In a 1000 m race M gives N some start and N's race becomes 880 m. How much start did N get?

AdvancedTry it yourself first

In a 600 m race M gives N some start and N's race becomes 540 m. How much start did N get?

BasicTry it yourself first

In a game of 100 points A scores 100 while B scores only 72. How many points can A give B?

BasicTry it yourself first

In a game of 120 points A scores 120 while B scores only 85. How many points can A give B?

StandardTry it yourself first

In a game of 80 points A scores 80 while B scores only 53. How many points can A give B?

StandardTry it yourself first

In a game of 150 points A scores 150 while B scores only 98. How many points can A give B?

AdvancedTry it yourself first

In a game of 60 points A scores 60 while B scores only 41. How many points can A give B?

The margin: distance and time

Tie it to the loser’s speed

From “beat by x m or T s” find the time and speed, and win even after giving a start.

Margin, dead heat and speed ratio

B needs

T

seconds to run

x

The distance by which the loser is behind, run at his own speed, takes exactly the margin in time.

∑ Formula

t_{A} = \frac{T ( L - x )}{x}

A beats B by

x

m or

T

s in an

L

m race

From “x m or T s” the loser’s speed: $v_{B} = \frac{x}{T}$ .
B’s full time $= \frac{L}{v _{B}} = \frac{L T}{x}$ ; A’s time $=$ B’s time $- T$ .
If both times are given find how far the slower covers in the faster’s time; the rest is the margin.
Start for a dead heat: $L - \frac{t _{A}}{t _{B}} L$ m (or a time start of $t_{B} - t_{A}$ seconds).
Winning even after the start: A’s time $\frac{L}{v _{A}}$ , B’s time $\frac{L - s}{v _{B}}$ ; B’s time $=$ A’s time $+ T$ .

Example 4 · Standard

In a 1000 m race A beats B by 25 m or 5 seconds. What is A’s time?

Solution, step by step

1B’s speed $= \frac{25}{5} = 5$ m/s; B’s time $= \frac{1000}{5} = 200$ s.
2A’s time $= 200 - 5 = 195$ s

Answer 195 seconds

Example 5 · Standard

P runs 1000 m in 4 min 30 s and Q in 5 min. By how many metres does P beat Q?

Solution, step by step

1P’s time $= 270$ s, Q’s $= 300$ s.
2In that time Q covers $\frac{270}{300} \times 1000 = 900$ m; margin $= 100$ m

Answer 100 m

Example 6 · Standard

In a 400 m race A runs at 8 m/s. He gives B a 20 m start and wins by 26 seconds. Find B’s speed.

Solution, step by step

1A’s time $= \frac{400}{8} = 50$ s; B’s time $= 50 + 26 = 76$ s.
2B’s speed $= \frac{400 - 20}{76} = 5$ m/s

Answer 5 m/s

StandardTry it yourself first

In a 500 m race A beats B by 20 m or 4 seconds. What is A's time over the course?

StandardTry it yourself first

In a 400 m race A beats B by 16 m or 5 seconds. What is A's time over the course?

StandardTry it yourself first

In a 1000 m race A beats B by 40 m or 8 seconds. What is A's time over the course?

AdvancedTry it yourself first

In a 800 m race A beats B by 32 m or 4 seconds. What is A's time over the course?

AdvancedTry it yourself first

In a 600 m race A beats B by 30 m or 5 seconds. What is A's time over the course?

BasicTry it yourself first

P runs 1000 m in 4 min and Q runs the same distance in 5 min 20 s. By how many metres does P beat Q?

StandardTry it yourself first

P runs 1000 m in 7 min 30 s and Q runs the same distance in 8 min 20 s. By how many metres does P beat Q?

StandardTry it yourself first

P runs 500 m in 2 min and Q runs the same distance in 2 min 30 s. By how many metres does P beat Q?

AdvancedTry it yourself first

P runs 800 m in 5 min and Q runs the same distance in 6 min 40 s. By how many metres does P beat Q?

AdvancedTry it yourself first

P runs 1200 m in 9 min and Q runs the same distance in 10 min. By how many metres does P beat Q?

BasicTry it yourself first

A runs 1000 m in 6 min 40 s and B in 8 min 20 s. How many metres start can A give B in a 1000 m race so that it ends in a dead heat?

StandardTry it yourself first

A runs 800 m in 4 min and B in 5 min. How many metres start can A give B in a 800 m race so that it ends in a dead heat?

StandardTry it yourself first

A runs 1000 m in 7 min 30 s and B in 10 min. How many metres start can A give B in a 1000 m race so that it ends in a dead heat?

AdvancedTry it yourself first

A runs 500 m in 1 min 40 s and B in 2 min 5 s. How many metres start can A give B in a 500 m race so that it ends in a dead heat?

AdvancedTry it yourself first

A runs 1200 m in 6 min and B in 7 min 30 s. How many metres start can A give B in a 1200 m race so that it ends in a dead heat?

BasicTry it yourself first

A runs 1000 m in 6 min 40 s and B in 8 min 20 s. How many seconds' time-start must A give B for a dead heat?

StandardTry it yourself first

A runs 800 m in 4 min and B in 5 min. How many seconds' time-start must A give B for a dead heat?

AdvancedTry it yourself first

A runs 1200 m in 6 min and B in 7 min 30 s. How many seconds' time-start must A give B for a dead heat?

StandardTry it yourself first

In a 400 m race A runs at 8 m/s. A gives B a start of 20 m and still beats him by 26 seconds. Find B's speed.

StandardTry it yourself first

In a 300 m race A runs at 6 m/s. A gives B a start of 15 m and still beats him by 7 seconds. Find B's speed.

AdvancedTry it yourself first

In a 500 m race A runs at 10 m/s. A gives B a start of 50 m and still beats him by 25 seconds. Find B's speed.

AdvancedTry it yourself first

In a 200 m race A runs at 4 m/s. A gives B a start of 8 m and still beats him by 14 seconds. Find B's speed.

BasicTry it yourself first

While A runs 35 m, B runs 40 m. In a 1000 m race by how many metres does B beat A?

StandardTry it yourself first

While A runs 18 m, B runs 20 m. In a 1000 m race by how many metres does B beat A?

StandardTry it yourself first

While A runs 28 m, B runs 32 m. In a 1000 m race by how many metres does B beat A?

AdvancedTry it yourself first

While A runs 48 m, B runs 50 m. In a 1000 m race by how many metres does B beat A?

AdvancedTry it yourself first

While A runs 54 m, B runs 60 m. In a 500 m race by how many metres does B beat A?

AdvancedTry it yourself first

While A runs 45 m, B runs 48 m. In a 1000 m race by how many metres does B beat A?

Three contestants and point games

A chain of ratios

When A beats two others, their standing between themselves; races of different lengths and point games.

The A, B, C chain

While A runs

L

Write all three distances at one moment and form their mutual ratios.

A gives B a start of $x$ and C $y$ : when A runs $L$ , B runs $L - x$ and C runs $L - y$ ⟹ B : C $= (L - x) : (L - y)$ .
B’s start to C: when B runs $L$ C runs $L \frac{L - y}{L - x}$ ; start $= \frac{L ( y - x )}{L - x}$ .
A beats B by $x$ and B beats C by $y$ (same $L$ ): A beats C by $x + y - \frac{x y}{L}$ .
Races of different lengths: write the speed ratio from each race and multiply them.
Point games follow the same logic: when A scores $N$ , B has $N - a$ and C $N - b$ ; B : C in that ratio.

Example 7 · Standard

In a 1000 m race A can give B a start of 100 m and C a start of 136 m. How much start can B give C?

Solution, step by step

1When A runs 1000 m, B runs 900 m and C runs 864 m ⟹ B : C $= 900 : 864$ .
2When B runs 1000 m, C runs $1000 \times \frac{864}{900} = 960$ m; start $= 40$ m

Answer 40 m

Example 8 · Standard

In a 500 m race A beats B by 25 m and B beats C by 20 m. By how many metres does A beat C?

Solution, step by step

1A runs 500 ⟹ B runs 475; B runs 500 ⟹ C runs 480 ⟹ A runs 500 ⟹ C runs $475 \times \frac{480}{500} = 456$ m.
2Margin $= 500 - 456 = 44$ m (rule: $25 + 20 - \frac{25 \times 20}{500} = 44$ ✓)

Answer 44 m

Example 9 · Standard

In a game of 150 A can give B 30 points and C 54 points. How many points can B give C in a game of 80?

Solution, step by step

1When A has 150, B has 120 and C has 96 ⟹ B : C $= 5 : 4$ .
2When B has 80, C has 64 ⟹ B can give $80 - 64 = 16$ points

Answer 16 points

StandardTry it yourself first

A, B and C run a 1000 m race. A can give B a start of 100 m and C a start of 136 m. How much start can B give C?

StandardTry it yourself first

A, B and C run a 1000 m race. A can give B a start of 50 m and C a start of 145 m. How much start can B give C?

AdvancedTry it yourself first

A, B and C run a 1000 m race. A can give B a start of 200 m and C a start of 360 m. How much start can B give C?

AdvancedTry it yourself first

A, B and C run a 800 m race. A can give B a start of 80 m and C a start of 152 m. How much start can B give C?

AdvancedTry it yourself first

A, B and C run a 600 m race. A can give B a start of 30 m and C a start of 87 m. How much start can B give C?

StandardTry it yourself first

A, B and C run a 400 m race. A can give B a start of 20 m and C a start of 58 m. How much start can B give C?

StandardTry it yourself first

In a 300 m race A beats B by 30 m and B beats C by 24 m. By how many metres does A beat C in the same race?

StandardTry it yourself first

In a 500 m race A beats B by 25 m and B beats C by 20 m. By how many metres does A beat C in the same race?

AdvancedTry it yourself first

In a 400 m race A beats B by 40 m and B beats C by 16 m. By how many metres does A beat C in the same race?

AdvancedTry it yourself first

In a 1000 m race A beats B by 50 m and B beats C by 40 m. By how many metres does A beat C in the same race?

AdvancedTry it yourself first

In a 600 m race A beats B by 30 m and B beats C by 12 m. By how many metres does A beat C in the same race?

BasicTry it yourself first

In a 250 m race A beats B by 25 m and B beats C by 10 m. By how many metres does A beat C in the same race?

AdvancedTry it yourself first

In a 800 m race A beats B by 80 m and in a 500 m race B beats C by 25 m. By how many metres does A beat C in a 600 m race?

AdvancedTry it yourself first

In a 600 m race A beats B by 30 m and in a 400 m race B beats C by 20 m. By how many metres does A beat C in a 300 m race?

AdvancedTry it yourself first

In a 1000 m race A beats B by 100 m and in a 500 m race B beats C by 50 m. By how many metres does A beat C in a 400 m race?

AdvancedTry it yourself first

In a 400 m race A beats B by 20 m and in a 250 m race B beats C by 25 m. By how many metres does A beat C in a 500 m race?

StandardTry it yourself first

In a game of 100 points A can give B 20 points and C 36 points. How many points can B give C in a game of 60?

StandardTry it yourself first

In a game of 150 points A can give B 30 points and C 54 points. How many points can B give C in a game of 80?

AdvancedTry it yourself first

In a game of 160 points A can give B 16 points and C 40 points. How many points can B give C in a game of 72?

AdvancedTry it yourself first

In a game of 90 points A can give B 10 points and C 25 points. How many points can B give C in a game of 64?

StandardTry it yourself first

In a game of 100 points A can give B 10 points and C 28 points. How many points can B give C in a game of 50?

Circular tracks, overtaking and the winning post

Lead and the fairness condition

Counting overtakes on a circular track, catching up on the clock, and the distance that ends in a tie.

Mixed questions

Laps, the clock and the post

One overtake per full lap of lead; arriving together means equal times.

On a circular track: divide the distance the winner gained over the loser by the length of one lap; the whole-number quotient is the number of overtakes.
Starting at different times: find the first runner’s lead, then lead $\div$ relative speed; add to the clock.
Where the post must be for a tie: if A is $\frac{p}{q}$ times as fast and the start is $s$ , then $\frac{L}{p} = \frac{L - s}{q}$ ⟹ $L = \frac{s p}{p - q}$ .
Margin with a start: subtract the start from the starter’s run and find the other’s distance.

Example 10 · Standard

A 7500 m race is held on a 600 m circular track. P and Q run at speeds in the ratio 5 : 4. How many times does the winner overtake the loser?

Solution, step by step

1When P runs 7500 m, Q runs 6000 m; lead $= 1500$ m.
2 $\frac{1500}{600} = 2.5$ ⟹ whole laps $2$ ⟹ 2 times

Answer 2 times

Example 11 · Standard

Kiran starts running at 7:00 at 6 km/h and Mohit from the same place in the same direction at 8:30 at 10 km/h. When does Mohit catch Kiran?

Solution, step by step

1By 8:30 Kiran’s lead $= 6 \times 1.5 = 9$ km; relative speed $= 4$ .
2 $\frac{9}{4} = 2.25$ hours $= 2$ h 15 min ⟹ 10:45 am

Answer 10:45 am

Example 12 · Standard

A runs

\frac{7}{4}

times as fast as B. A gives B a start of 24 m. How far must the winning post be for a tie?

Solution, step by step

1 $\frac{L}{7} = \frac{L - 24}{4}$ ⟹ $4 L = 7 L - 168$ .
2 $3 L = 168$ ⟹ $L = 56$ m

Answer 56 m

BasicTry it yourself first

In a 800 m race the speeds of L and M are in the ratio 2 : 3. L has a start of 300 m. By how many metres does L win?

StandardTry it yourself first

In a 1000 m race the speeds of L and M are in the ratio 4 : 5. L has a start of 240 m. By how many metres does L win?

StandardTry it yourself first

In a 600 m race the speeds of L and M are in the ratio 4 : 5. L has a start of 150 m. By how many metres does L win?

StandardTry it yourself first

In a 240 m race the speeds of L and M are in the ratio 1 : 2. L has a start of 130 m. By how many metres does L win?

AdvancedTry it yourself first

In a 600 m race the speeds of L and M are in the ratio 2 : 3. L has a start of 220 m. By how many metres does L win?

AdvancedTry it yourself first

In a 1200 m race the speeds of L and M are in the ratio 3 : 4. L has a start of 360 m. By how many metres does L win?

StandardTry it yourself first

A 12000 m race is held on a circular track of 600 m. The speeds of P and Q are in the ratio 3 : 2. How many times does the winner overtake the loser?

StandardTry it yourself first

A 9000 m race is held on a circular track of 750 m. The speeds of P and Q are in the ratio 5 : 3. How many times does the winner overtake the loser?

AdvancedTry it yourself first

A 14000 m race is held on a circular track of 700 m. The speeds of P and Q are in the ratio 7 : 5. How many times does the winner overtake the loser?

AdvancedTry it yourself first

A 18000 m race is held on a circular track of 900 m. The speeds of P and Q are in the ratio 3 : 2. How many times does the winner overtake the loser?

AdvancedTry it yourself first

A 12000 m race is held on a circular track of 800 m. The speeds of P and Q are in the ratio 6 : 5. How many times does the winner overtake the loser?

StandardTry it yourself first

Kiran and Mohit start running from the same place in the same direction: Kiran at 6:00 am at 10 km/h and Mohit at 7:00 am at 15 km/h. At what time does Mohit overtake Kiran?

StandardTry it yourself first

Kiran and Mohit start running from the same place in the same direction: Kiran at 8:00 am at 5 km/h and Mohit at 9:00 am at 9 km/h. At what time does Mohit overtake Kiran?

AdvancedTry it yourself first

Kiran and Mohit start running from the same place in the same direction: Kiran at 6:30 am at 9 km/h and Mohit at 8:00 am at 15 km/h. At what time does Mohit overtake Kiran?

AdvancedTry it yourself first

Kiran and Mohit start running from the same place in the same direction: Kiran at 6:00 am at 6 km/h and Mohit at 7:30 am at 10 km/h. At what time does Mohit overtake Kiran?

BasicTry it yourself first

A runs

\frac{3}{2}

times as fast as B. A gives B a start of 40 m. How far must the winning post be so that both reach it together?

StandardTry it yourself first

A runs

\frac{5}{4}

times as fast as B. A gives B a start of 20 m. How far must the winning post be so that both reach it together?

StandardTry it yourself first

A runs

\frac{9}{5}

times as fast as B. A gives B a start of 36 m. How far must the winning post be so that both reach it together?

AdvancedTry it yourself first

A runs

\frac{4}{3}

times as fast as B. A gives B a start of 15 m. How far must the winning post be so that both reach it together?

AdvancedTry it yourself first

A runs

\frac{8}{5}

times as fast as B. A gives B a start of 18 m. How far must the winning post be so that both reach it together?

Traps and fast tricks

Slips · tricks

Where marks go in race questions.

The commonest mistakes

Six traps

The slips that cost marks.

Trap 1Taking the margin as B’s run. If A beats B by

x

m, B ran

L - x

Trap 2Adding the start. The runner with a start runs less, not more.

Trap 3Using “x m or T s” for the winner. It gives the loser’s speed

\frac{x}{T}

, not the winner’s.

Trap 4Taking the wrong starting ratio with three runners. Always write the distances at the moment A finishes.

Trap 5Counting a fraction of a lap as an overtake. Only whole laps of lead count as overtakes.

Trap 6Using the scored points instead of the start in a game. “A can give” means

N - b

Fast tricks

A snapshot of one moment

Habits for speed.

Pick the moment the winner finishes and write everyone’s distances — the speed ratios appear right there.
When speeds are in a ratio $v_{A} : v_{B} = L : (L - x)$ is the heart of the race.
Rules: with three runners $x + y - \frac{x y}{L}$ and B’s start to C $\frac{L ( y - x )}{L - x}$ .
Check: the winner’s distance is always $L$ and the loser’s is less.

Example 13 · Standard

A beat B by 50 m in a 1000 m race and B beat C by 40 m in the same race. By how many metres did A beat C?

Solution, step by step

1Rule: $x + y - \frac{x y}{L} = 50 + 40 - \frac{50 \times 40}{1000}$ .
2 $= 90 - 2 = 88$ m

Answer 88 m

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

BasicTry it yourself first

In a 300 m race A gives B a start of 25 m. How far does B have to run?

BasicTry it yourself first

In a 600 m race A gives B a start of 40 m. How far does B have to run?

BasicTry it yourself first

In a race P gives Q a start of 35 m, making the race 265 m for Q. What is the full length of the race?

BasicTry it yourself first

In a race P gives Q a start of 20 m, making the race 280 m for Q. What is the full length of the race?

BasicTry it yourself first

In a 500 m race M gives N some start and N's race becomes 420 m. How much start did N get?

BasicTry it yourself first

In a 800 m race M gives N some start and N's race becomes 740 m. How much start did N get?

BasicTry it yourself first

In a game of 100 points A scores 100 while B scores only 58. How many points can A give B?

BasicTry it yourself first

In a game of 90 points A scores 90 while B scores only 63. How many points can A give B?

BasicTry it yourself first

P runs 1000 m in 6 min and Q runs the same distance in 6 min 40 s. By how many metres does P beat Q?

BasicTry it yourself first

P runs 600 m in 2 min 30 s and Q runs the same distance in 3 min 20 s. By how many metres does P beat Q?

BasicTry it yourself first

A runs 600 m in 2 min 30 s and B in 3 min 20 s. How many metres start can A give B in a 600 m race so that it ends in a dead heat?

BasicTry it yourself first

A runs 1000 m in 5 min and B in 6 min 40 s. How many metres start can A give B in a 1000 m race so that it ends in a dead heat?

BasicTry it yourself first

A runs 600 m in 2 min 30 s and B in 3 min 20 s. How many seconds' time-start must A give B for a dead heat?

BasicTry it yourself first

While A runs 24 m, B runs 30 m. In a 500 m race by how many metres does B beat A?

BasicTry it yourself first

While A runs 36 m, B runs 40 m. In a 1000 m race by how many metres does B beat A?

BasicTry it yourself first

In a 400 m race A beats B by 20 m or 4 seconds. What is A's time over the course?

BasicTry it yourself first

In a 200 m race A beats B by 10 m or 2 seconds. What is A's time over the course?

BasicTry it yourself first

In a 500 m race the speeds of L and M are in the ratio 1 : 2. L has a start of 270 m. By how many metres does L win?

BasicTry it yourself first

In a 600 m race the speeds of L and M are in the ratio 2 : 3. L has a start of 210 m. By how many metres does L win?

BasicTry it yourself first

In a 100 m race A runs at 5 m/s. A gives B a start of 5 m and still beats him by 3.75 seconds. Find B's speed.

BasicTry it yourself first

A runs

\frac{5}{3}

times as fast as B. A gives B a start of 24 m. How far must the winning post be so that both reach it together?

BasicTry it yourself first

A, B and C run a 400 m race. A can give B a start of 20 m and C a start of 39 m. How much start can B give C?

BasicTry it yourself first

In a 200 m race A beats B by 20 m and B beats C by 10 m. By how many metres does A beat C in the same race?

BasicTry it yourself first

In a game of 100 points A can give B 10 points and C 19 points. How many points can B give C in a game of 50?

BasicTry it yourself first

A 7500 m race is held on a circular track of 600 m. The speeds of P and Q are in the ratio 5 : 4. How many times does the winner overtake the loser?

BasicTry it yourself first

Kiran and Mohit start running from the same place in the same direction: Kiran at 6:00 am at 6 km/h and Mohit at 7:00 am at 9 km/h. At what time does Mohit overtake Kiran?

Level 2 — Standard

Two ideas combined

Exam-level questions.

StandardTry it yourself first

In a 1500 m race A gives B a start of 85 m. How far does B have to run?

StandardTry it yourself first

In a race P gives Q a start of 55 m, making the race 345 m for Q. What is the full length of the race?

StandardTry it yourself first

In a 1200 m race M gives N some start and N's race becomes 1085 m. How much start did N get?

StandardTry it yourself first

In a game of 140 points A scores 140 while B scores only 97. How many points can A give B?

StandardTry it yourself first

P runs 1000 m in 5 min 30 s and Q runs the same distance in 6 min 15 s. By how many metres does P beat Q?

StandardTry it yourself first

P runs 1500 m in 8 min and Q runs the same distance in 10 min. By how many metres does P beat Q?

StandardTry it yourself first

A runs 1500 m in 8 min and B in 10 min. How many metres start can A give B in a 1500 m race so that it ends in a dead heat?

StandardTry it yourself first

A runs 1000 m in 5 min 50 s and B in 6 min 40 s. How many metres start can A give B in a 1000 m race so that it ends in a dead heat?

StandardTry it yourself first

A runs 1000 m in 5 min 50 s and B in 6 min 40 s. How many seconds' time-start must A give B for a dead heat?

StandardTry it yourself first

While A runs 49 m, B runs 56 m. In a 1000 m race by how many metres does B beat A?

StandardTry it yourself first

While A runs 63 m, B runs 72 m. In a 800 m race by how many metres does B beat A?

StandardTry it yourself first

In a 800 m race A beats B by 40 m or 10 seconds. What is A's time over the course?

StandardTry it yourself first

In a 1000 m race A beats B by 50 m or 10 seconds. What is A's time over the course?

StandardTry it yourself first

In a 900 m race the speeds of L and M are in the ratio 6 : 7. L has a start of 150 m. By how many metres does L win?

StandardTry it yourself first

In a 1000 m race the speeds of L and M are in the ratio 4 : 5. L has a start of 300 m. By how many metres does L win?

StandardTry it yourself first

In a 600 m race A runs at 6 m/s. A gives B a start of 30 m and still beats him by 14 seconds. Find B's speed.

StandardTry it yourself first

In a 800 m race A runs at 8 m/s. A gives B a start of 40 m and still beats him by 52 seconds. Find B's speed.

StandardTry it yourself first

A runs

\frac{7}{5}

times as fast as B. A gives B a start of 30 m. How far must the winning post be so that both reach it together?

StandardTry it yourself first

A, B and C run a 1000 m race. A can give B a start of 60 m and C a start of 107 m. How much start can B give C?

StandardTry it yourself first

A, B and C run a 800 m race. A can give B a start of 40 m and C a start of 59 m. How much start can B give C?

StandardTry it yourself first

In a 800 m race A beats B by 40 m and B beats C by 32 m. By how many metres does A beat C in the same race?

StandardTry it yourself first

In a 600 m race A beats B by 24 m and B beats C by 18 m. By how many metres does A beat C in the same race?

StandardTry it yourself first

In a game of 120 points A can give B 20 points and C 32 points. How many points can B give C in a game of 75?

StandardTry it yourself first

In a game of 200 points A can give B 40 points and C 56 points. How many points can B give C in a game of 100?

StandardTry it yourself first

In a 500 m race A beats B by 25 m and in a 400 m race B beats C by 20 m. By how many metres does A beat C in a 300 m race?

StandardTry it yourself first

In a 1000 m race A beats B by 50 m and in a 800 m race B beats C by 80 m. By how many metres does A beat C in a 600 m race?

StandardTry it yourself first

A 10000 m race is held on a circular track of 500 m. The speeds of P and Q are in the ratio 4 : 3. How many times does the winner overtake the loser?

StandardTry it yourself first

A 16500 m race is held on a circular track of 750 m. The speeds of P and Q are in the ratio 7 : 5. How many times does the winner overtake the loser?

StandardTry it yourself first

Kiran and Mohit start running from the same place in the same direction: Kiran at 7:00 am at 8 km/h and Mohit at 8:00 am at 14 km/h. At what time does Mohit overtake Kiran?

StandardTry it yourself first

Kiran and Mohit start running from the same place in the same direction: Kiran at 6:30 am at 6 km/h and Mohit at 7:30 am at 10 km/h. At what time does Mohit overtake Kiran?

Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

AdvancedTry it yourself first

In a race P gives Q a start of 85 m, making the race 515 m for Q. What is the full length of the race?

AdvancedTry it yourself first

In a 1500 m race M gives N some start and N's race becomes 1305 m. How much start did N get?

AdvancedTry it yourself first

In a game of 175 points A scores 175 while B scores only 128. How many points can A give B?

AdvancedTry it yourself first

P runs 1000 m in 4 min 45 s and Q runs the same distance in 5 min. By how many metres does P beat Q?

AdvancedTry it yourself first

P runs 2000 m in 12 min and Q runs the same distance in 13 min 20 s. By how many metres does P beat Q?

AdvancedTry it yourself first

A runs 2000 m in 12 min and B in 13 min 20 s. How many metres start can A give B in a 2000 m race so that it ends in a dead heat?

AdvancedTry it yourself first

A runs 1500 m in 7 min and B in 8 min 20 s. How many metres start can A give B in a 1500 m race so that it ends in a dead heat?

AdvancedTry it yourself first

A runs 1500 m in 7 min and B in 8 min 20 s. How many seconds' time-start must A give B for a dead heat?

AdvancedTry it yourself first

While A runs 56 m, B runs 63 m. In a 900 m race by how many metres does B beat A?

AdvancedTry it yourself first

While A runs 75 m, B runs 80 m. In a 1200 m race by how many metres does B beat A?

AdvancedTry it yourself first

In a 1200 m race A beats B by 60 m or 12 seconds. What is A's time over the course?

AdvancedTry it yourself first

In a 1500 m race A beats B by 75 m or 15 seconds. What is A's time over the course?

AdvancedTry it yourself first

In a 1500 m race the speeds of L and M are in the ratio 8 : 9. L has a start of 200 m. By how many metres does L win?

AdvancedTry it yourself first

In a 1000 m race the speeds of L and M are in the ratio 5 : 6. L has a start of 180 m. By how many metres does L win?

AdvancedTry it yourself first

In a 800 m race A runs at 10 m/s. A gives B a start of 40 m and still beats him by 20 seconds. Find B's speed.

AdvancedTry it yourself first

In a 600 m race A runs at 12 m/s. A gives B a start of 24 m and still beats him by 22 seconds. Find B's speed.

AdvancedTry it yourself first

A runs

\frac{11}{8}

times as fast as B. A gives B a start of 36 m. How far must the winning post be so that both reach it together?

AdvancedTry it yourself first

A, B and C run a 1200 m race. A can give B a start of 80 m and C a start of 108 m. How much start can B give C?

AdvancedTry it yourself first

A, B and C run a 1000 m race. A can give B a start of 75 m and C a start of 112 m. How much start can B give C?

AdvancedTry it yourself first

In a 1500 m race A beats B by 60 m and B beats C by 45 m. By how many metres does A beat C in the same race?

AdvancedTry it yourself first

In a 900 m race A beats B by 36 m and B beats C by 27 m. By how many metres does A beat C in the same race?

AdvancedTry it yourself first

In a game of 180 points A can give B 30 points and C 48 points. How many points can B give C in a game of 100?

AdvancedTry it yourself first

In a game of 240 points A can give B 40 points and C 60 points. How many points can B give C in a game of 140?

AdvancedTry it yourself first

In a 800 m race A beats B by 40 m and in a 500 m race B beats C by 25 m. By how many metres does A beat C in a 300 m race?

AdvancedTry it yourself first

In a 900 m race A beats B by 45 m and in a 600 m race B beats C by 30 m. By how many metres does A beat C in a 1000 m race?

AdvancedTry it yourself first

A 22500 m race is held on a circular track of 900 m. The speeds of P and Q are in the ratio 5 : 4. How many times does the winner overtake the loser?

AdvancedTry it yourself first

A 13000 m race is held on a circular track of 650 m. The speeds of P and Q are in the ratio 8 : 5. How many times does the winner overtake the loser?

AdvancedTry it yourself first

Kiran and Mohit start running from the same place in the same direction: Kiran at 6:45 am at 7 km/h and Mohit at 8:15 am at 13 km/h. At what time does Mohit overtake Kiran?

AdvancedTry it yourself first

Kiran and Mohit start running from the same place in the same direction: Kiran at 6:15 am at 9 km/h and Mohit at 7:45 am at 15 km/h. At what time does Mohit overtake Kiran?

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.