TYPE-6 (LINEAR RACE) 86. In a 1500 m race, Anil beats Bakul by 150 m and in the same race Bakul beats Charles by 75 in. By what distance does Anil beat Charles? a) 217.50 m b) 200.15 m c) 293.50 m d) 313.75 m 87. In a linear race of 500 m, A can beat B by 50 m and in a race of 600 m, B can beat C by 60 m. By how many metres will A beat C in a race of 400 m ? a) 70 b) 68 c) 76 d) 72 88. A can run 250 m in 25 sec and B in 30 sec. How many metres of start can A give to B in a km race so that the race may end in a dead - heat? a) 169.53 m b) 173.82 m c) 166.67 m d) 186.34 m 89. In a 100-m race, A beats B by 20 m and B beats C by 5 m. in the same race, A beats C by: a) 26 m b) 22 m c) 25 m d) 24 m 90. In a race of 100 m, A beats B by 10m. In a race of 100 m, B beats C by 10 m. By what distance will A beat C in a race of 100 m? a) 19 m b) 21 m c) 18 m d) 20 m 91. In a 1500-m race, if A beats B by 100 m and B beats C by 150 m, then by what distance (in m) does A beat C? a) 140 c) 100 b) 150 d) 240 92. In a 5 km race, S gave a 500 m head start to T. If they finished the race at the same time and the speed of T is 13.5 km/h, the speed of S is: a) 15 km/h c) 17.5 km/h b) 20 km/h d) 16.5 km/h 93. In a 140 m race, Pankaj runs at a speed of 2.5 m/s. He gives a start of 20 m to Arun and still defeats him by 4 seconds. The speed of Arun is: a) 1.75 m/s c) 2 m/s b) 1.25 m/s d) 1.5 m/s 94. In a 400 metre race, A runs at a speed of 16 m/sec. If A gives B a start of 15 metres and still beats him by 10 sec, then what is the speed of B? a) 11 m/sec c) 13 m/sec b) 10 m/sec d) 9 m/sec 95. P gives Q a head-start of 2 seconds in a 400 m race, but both finish the race at the same time. Find the time taken by P to finish the race if the speed of Q is 2 m/sec. a) 198 seconds b) 199 seconds c) 200 seconds d) 195 seconds

Step 1: When Anil finishes $1500$ m, Bakul runs $1500 - 150 = 1350$ m.
Thus, $\frac{v_A}{v_B} = \frac{1500}{1350} = \frac{10}{9}.$

Step 2: When Bakul finishes $1500$ m, Charles runs $1500 - 75 = 1425$ m.
Thus, $\frac{v_B}{v_C} = \frac{1500}{1425} = \frac{20}{19}.$

Step 3: $\frac{v_A}{v_C} = \frac{10}{9} \times \frac{20}{19} = \frac{200}{171}.$
So, $\frac{v_C}{v_A} = \frac{171}{200}.$

Step 4: When Anil runs $1500$ m, Charles runs $1500 \times \frac{171}{200} = 1282.5$ m.
Anil beats Charles by $1500 - 1282.5 = 217.5$ m.

a) 217.50 m

Step 1: In $500$ m race, A beats B by $50$ m, so when A runs $500$ m, B runs $450$ m.
$\frac{v_A}{v_B} = \frac{500}{450} = \frac{10}{9}.$

Step 2: In $600$ m race, B beats C by $60$ m, so when B runs $600$ m, C runs $540$ m.
$\frac{v_B}{v_C} = \frac{600}{540} = \frac{10}{9}.$

Step 3: $\frac{v_A}{v_C} = \frac{10}{9} \times \frac{10}{9} = \frac{100}{81}.$
$\frac{v_C}{v_A} = \frac{81}{100}.$

Step 4: In $400$ m race, when A runs $400$ m, C runs $400 \times \frac{81}{100} = 324$ m.
A beats C by $400 - 324 = 76$ m.

c) 76

Step 1: Speed of A: $v_A = \frac{250}{25} = 10$ m/s.
Speed of B: $v_B = \frac{250}{30} \approx 8.333$ m/s.

Step 2: In $1000$ m race for dead heat, let start given to B be $x$ m.
B runs $1000 - x$ m in time A runs $1000$ m.
$t = \frac{1000}{10} = 100$ s.

Step 3: $\frac{1000 - x}{v_B} = 100.$
$1000 - x = 100 \times \frac{250}{30} = 100 \times \frac{25}{3} \approx 833.333.$
$x = 1000 - 833.333 = 166.667$ m.

c) 166.67 m

Step 1: A beats B by $20$ m in $100$ m, so when A runs $100$ m, B runs $80$ m.
$\frac{v_A}{v_B} = \frac{100}{80} = \frac{5}{4}.$

Step 2: B beats C by $5$ m in $100$ m, so when B runs $100$ m, C runs $95$ m.
$\frac{v_B}{v_C} = \frac{100}{95} = \frac{20}{19}.$

Step 3: $\frac{v_A}{v_C} = \frac{5}{4} \times \frac{20}{19} = \frac{25}{19}.$
$\frac{v_C}{v_A} = \frac{19}{25}.$

Step 4: When A runs $100$ m, C runs $100 \times \frac{19}{25} = 76$ m.
A beats C by $100 - 76 = 24$ m.

d) 24 m

Step 1: A beats B by $10$ m in $100$ m, so when A runs $100$ m, B runs $90$ m.
$\frac{v_A}{v_B} = \frac{100}{90} = \frac{10}{9}.$

Step 2: B beats C by $10$ m in $100$ m, so when B runs $100$ m, C runs $90$ m.
$\frac{v_B}{v_C} = \frac{100}{90} = \frac{10}{9}.$

Step 3: $\frac{v_A}{v_C} = \frac{10}{9} \times \frac{10}{9} = \frac{100}{81}.$
$\frac{v_C}{v_A} = \frac{81}{100}.$

Step 4: When A runs $100$ m, C runs $100 \times \frac{81}{100} = 81$ m.
A beats C by $100 - 81 = 19$ m.

a) 19 m

Step 1: A beats B by $100$ m in $1500$ m, so when A runs $1500$ m, B runs $1400$ m.
$\frac{v_A}{v_B} = \frac{1500}{1400} = \frac{15}{14}.$

Step 2: B beats C by $150$ m in $1500$ m, so when B runs $1500$ m, C runs $1350$ m.
$\frac{v_B}{v_C} = \frac{1500}{1350} = \frac{10}{9}.$

Step 3: $\frac{v_A}{v_C} = \frac{15}{14} \times \frac{10}{9} = \frac{25}{21}.$
$\frac{v_C}{v_A} = \frac{21}{25}.$

Step 4: When A runs $1500$ m, C runs $1500 \times \frac{21}{25} = 1260$ m.
A beats C by $1500 - 1260 = 240$ m.

d) 240

Step 1: Race is $5$ km, T gets $500$ m = $0.5$ km head start.
T runs $4.5$ km at $13.5$ km/h.
Time $t = \frac{4.5}{13.5} = \frac{1}{3}$ h.

Step 2: S runs $5$ km in $t$.
$v_S = \frac{5}{1/3} = 15$ km/h.

a) 15 km/h

Step 1: Pankaj's time for $140$ m: $t_P = \frac{140}{2.5} = 56$ s.

Step 2: Arun gets $20$ m start, runs $120$ m.
Pankaj beats by $4$ s, so Arun's time for $120$ m is $56 + 4 = 60$ s.

Step 3: Arun's speed $v_A = \frac{120}{60} = 2$ m/s.

c) 2 m/s

Step 1: A's time for $400$ m: $t_A = \frac{400}{16} = 25$ s.

Step 2: B gets $15$ m start, runs $385$ m.
A beats by $10$ s, so B's time for $385$ m is $25 + 10 = 35$ s.

Step 3: B's speed $v_B = \frac{385}{35} = 11$ m/s.

a) 11 m/sec

Step 1: Q's head start in $2$ s at $2$ m/s: $2 \times 2 = 4$ m.

Step 2: Q runs remaining $400 - 4 = 396$ m at $2$ m/s.
Time $t = \frac{396}{2} = 198$ s.

Step 3: P runs $400$ m in $198$ s.

a) 198 seconds