TYPE-6 (LINEAR RACE) 86. In a 1500 m race, Anil beats Bakul by 150 m and in the same race Bakul beats Charles…

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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

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86. Step 1: When Anil finishes 1500 m, Bakul runs 1500 − 150 = 1350 m. Thus, v_A/v_B = 1500/1350 = 10/9. Step 2: When Bakul finishes 1500 m, Charles runs 1500 − 75 = 1425 m. Thus, v_B/v_C = 1500/1425 = 20/19. Step 3: v_A/v_C = 10/9 × 20/19 = 200/171. So, v_C/v_A = 171/200. Step 4: When Anil runs 1500 m, Charles runs 1500 × 171/200 = 1282.5 m. Anil beats Charles by 1500 − 1282.5 = 217.5 m. a) 217.50 m 87. Step 1: In 500 m race, A beats B by 50 m, so when A runs 500 m, B runs 450 m. v_A/v_B = 500/450 = 10/9. Step 2: In 600 m race, B beats C by 60 m, so when B runs 600 m, C runs 540 m. v_B/v_C = 600/540 = 10/9. Step 3: v_A/v_C = 10/9 × 10/9 = 100/81. v_C/v_A = 81/100. Step 4: In 400 m race, when A runs 400 m, C runs 400 × 81/100 = 324 m. A beats C by 400 − 324 = 76 m. c) 76 88. Step 1: Speed of A: v_A = 250/25 = 10 m/s. Speed of B: v_B = 250/30 ≈ 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 = 1000/10 = 100 s. Step 3: (1000 − x)/(v_B) = 100. 1000 − x = 100 × 250/30 = 100 × 25/3 ≈ 833.333. x = 1000 − 833.333 = 166.667 m. c) 166.67 m 89. Step 1: A beats B by 20 m in 100 m, so when A runs 100 m, B runs 80 m. v_A/v_B = 100/80 = 5/4. Step 2: B beats C by 5 m in 100 m, so when B runs 100 m, C runs 95 m. v_B/v_C = 100/95 = 20/19. Step 3: v_A/v_C = 5/4 × 20/19 = 25/19. v_C/v_A = 19/25. Step 4: When A runs 100 m, C runs 100 × 19/25 = 76 m. A beats C by 100 − 76 = 24 m. d) 24 m 90. Step 1: A beats B by 10 m in 100 m, so when A runs 100 m, B runs 90 m. v_A/v_B = 100/90 = 10/9. Step 2: B beats C by 10 m in 100 m, so when B runs 100 m, C runs 90 m. v_B/v_C = 100/90 = 10/9. Step 3: v_A/v_C = 10/9 × 10/9 = 100/81. v_C/v_A = 81/100. Step 4: When A runs 100 m, C runs 100 × 81/100 = 81 m. A beats C by 100 − 81 = 19 m. a) 19 m 91. Step 1: A beats B by 100 m in 1500 m, so when A runs 1500 m, B runs 1400 m. v_A/v_B = 1500/1400 = 15/14. Step 2: B beats C by 150 m in 1500 m, so when B runs 1500 m, C runs 1350 m. v_B/v_C = 1500/1350 = 10/9. Step 3: v_A/v_C = 15/14 × 10/9 = 25/21. v_C/v_A = 21/25. Step 4: When A runs 1500 m, C runs 1500 × 21/25 = 1260 m. A beats C by 1500 − 1260 = 240 m. d) 240 92. 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 = 4.5/13.5 = 1/3 h. Step 2: S runs 5 km in t. v_S = (5)/(1/3) = 15 km/h. a) 15 km/h 93. Step 1: Pankaj's time for 140 m: t_P = 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 = 120/60 = 2 m/s. c) 2 m/s 94. Step 1: A's time for 400 m: t_A = 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 = 385/35 = 11 m/s. a) 11 m/sec 95. Step 1: Q's head start in 2 s at 2 m/s: 2 × 2 = 4 m. Step 2: Q runs remaining 400 − 4 = 396 m at 2 m/s. Time t = 396/2 = 198 s. Step 3: P runs 400 m in 198 s. a) 198 seconds

Step-by-step solution

When Anil finishes $1500$ m, Bakul runs $1500 - 150 = 1350$ m.

When Bakul finishes $1500$ m, Charles runs $1500 - 75 = 1425$ m.

$\frac{v_A}{v_C} = \frac{10}{9} \times \frac{20}{19} = \frac{200}{171}.$

When Anil runs $1500$ m, Charles runs $1500 \times \frac{171}{200} = 1282.5$ m.

In $500$ m race, A beats B by $50$ m, so when A runs $500$ m, B runs $450$ m.

In $600$ m race, B beats C by $60$ m, so when B runs $600$ m, C runs $540$ m.

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

In $400$ m race, when A runs $400$ m, C runs $400 \times \frac{81}{100} = 324$ m.

Speed of A

In $1000$ m race for dead heat, let start given to B be $x$ m.

$\frac{1000 - x}{v_B} = 100.$

A beats B by $20$ m in $100$ m, so when A runs $100$ m, B runs $80$ m.

B beats C by $5$ m in $100$ m, so when B runs $100$ m, C runs $95$ m.

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

When A runs $100$ m, C runs $100 \times \frac{19}{25} = 76$ m.

A beats B by $10$ m in $100$ m, so when A runs $100$ m, B runs $90$ m.

B beats C by $10$ m in $100$ m, so when B runs $100$ m, C runs $90$ m.

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

When A runs $100$ m, C runs $100 \times \frac{81}{100} = 81$ m.

A beats B by $100$ m in $1500$ m, so when A runs $1500$ m, B runs $1400$ m.

B beats C by $150$ m in $1500$ m, so when B runs $1500$ m, C runs $1350$ m.

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

When A runs $1500$ m, C runs $1500 \times \frac{21}{25} = 1260$ m.

Race is $5$ km, T gets $500$ m = $0.5$ km head start.

S runs $5$ km in $t$ .

Pankaj's time for $140$ m

Arun gets $20$ m start, runs $120$ m.

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

A's time for $400$ m

B gets $15$ m start, runs $385$ m.

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

Q's head start in $2$ s at $2$ m/s

Q runs remaining $400 - 4 = 396$ m at $2$ m/s.

P runs $400$ m in $198$ s.

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Step-by-step solution

When Anil finishes 150015001500 m, Bakul runs 1500−150=13501500 - 150 = 13501500−150=1350 m.

When Bakul finishes 150015001500 m, Charles runs 1500−75=14251500 - 75 = 14251500−75=1425 m.

vAvC=109×2019=200171.\frac{v_A}{v_C} = \frac{10}{9} \times \frac{20}{19} = \frac{200}{171}.vC​vA​​=910​×1920​=171200​.

When Anil runs 150015001500 m, Charles runs 1500×171200=1282.51500 \times \frac{171}{200} = 1282.51500×200171​=1282.5 m.

In 500500500 m race, A beats B by 505050 m, so when A runs 500500500 m, B runs 450450450 m.

In 600600600 m race, B beats C by 606060 m, so when B runs 600600600 m, C runs 540540540 m.

vAvC=109×109=10081.\frac{v_A}{v_C} = \frac{10}{9} \times \frac{10}{9} = \frac{100}{81}.vC​vA​​=910​×910​=81100​.

In 400400400 m race, when A runs 400400400 m, C runs 400×81100=324400 \times \frac{81}{100} = 324400×10081​=324 m.

Speed of A

In 100010001000 m race for dead heat, let start given to B be xxx m.

1000−xvB=100.\frac{1000 - x}{v_B} = 100.vB​1000−x​=100.

A beats B by 202020 m in 100100100 m, so when A runs 100100100 m, B runs 808080 m.

B beats C by 555 m in 100100100 m, so when B runs 100100100 m, C runs 959595 m.

vAvC=54×2019=2519.\frac{v_A}{v_C} = \frac{5}{4} \times \frac{20}{19} = \frac{25}{19}.vC​vA​​=45​×1920​=1925​.

When A runs 100100100 m, C runs 100×1925=76100 \times \frac{19}{25} = 76100×2519​=76 m.

A beats B by 101010 m in 100100100 m, so when A runs 100100100 m, B runs 909090 m.

B beats C by 101010 m in 100100100 m, so when B runs 100100100 m, C runs 909090 m.

vAvC=109×109=10081.\frac{v_A}{v_C} = \frac{10}{9} \times \frac{10}{9} = \frac{100}{81}.vC​vA​​=910​×910​=81100​.

When A runs 100100100 m, C runs 100×81100=81100 \times \frac{81}{100} = 81100×10081​=81 m.

A beats B by 100100100 m in 150015001500 m, so when A runs 150015001500 m, B runs 140014001400 m.

B beats C by 150150150 m in 150015001500 m, so when B runs 150015001500 m, C runs 135013501350 m.

vAvC=1514×109=2521.\frac{v_A}{v_C} = \frac{15}{14} \times \frac{10}{9} = \frac{25}{21}.vC​vA​​=1415​×910​=2125​.

When A runs 150015001500 m, C runs 1500×2125=12601500 \times \frac{21}{25} = 12601500×2521​=1260 m.

Race is 555 km, T gets 500500500 m = 0.50.50.5 km head start.

S runs 555 km in ttt.

Pankaj's time for 140140140 m

Arun gets 202020 m start, runs 120120120 m.

Arun's speed vA=12060=2v_A = \frac{120}{60} = 2vA​=60120​=2 m/s.

A's time for 400400400 m

B gets 151515 m start, runs 385385385 m.

B's speed vB=38535=11v_B = \frac{385}{35} = 11vB​=35385​=11 m/s.

Q's head start in 222 s at 222 m/s

Q runs remaining 400−4=396400 - 4 = 396400−4=396 m at 222 m/s.

P runs 400400400 m in 198198198 s.

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More Mathematics Questions

When Anil finishes $1500$ m, Bakul runs $1500 - 150 = 1350$ m.

When Bakul finishes $1500$ m, Charles runs $1500 - 75 = 1425$ m.

$\frac{v_A}{v_C} = \frac{10}{9} \times \frac{20}{19} = \frac{200}{171}.$

When Anil runs $1500$ m, Charles runs $1500 \times \frac{171}{200} = 1282.5$ m.

In $500$ m race, A beats B by $50$ m, so when A runs $500$ m, B runs $450$ m.

In $600$ m race, B beats C by $60$ m, so when B runs $600$ m, C runs $540$ m.

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

In $400$ m race, when A runs $400$ m, C runs $400 \times \frac{81}{100} = 324$ m.

In $1000$ m race for dead heat, let start given to B be $x$ m.

$\frac{1000 - x}{v_B} = 100.$

A beats B by $20$ m in $100$ m, so when A runs $100$ m, B runs $80$ m.

B beats C by $5$ m in $100$ m, so when B runs $100$ m, C runs $95$ m.

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

When A runs $100$ m, C runs $100 \times \frac{19}{25} = 76$ m.

A beats B by $10$ m in $100$ m, so when A runs $100$ m, B runs $90$ m.

B beats C by $10$ m in $100$ m, so when B runs $100$ m, C runs $90$ m.

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

When A runs $100$ m, C runs $100 \times \frac{81}{100} = 81$ m.

A beats B by $100$ m in $1500$ m, so when A runs $1500$ m, B runs $1400$ m.

B beats C by $150$ m in $1500$ m, so when B runs $1500$ m, C runs $1350$ m.

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

When A runs $1500$ m, C runs $1500 \times \frac{21}{25} = 1260$ m.

Race is $5$ km, T gets $500$ m = $0.5$ km head start.

S runs $5$ km in $t$ .

Pankaj's time for $140$ m

Arun gets $20$ m start, runs $120$ m.

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

A's time for $400$ m

B gets $15$ m start, runs $385$ m.

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

Q's head start in $2$ s at $2$ m/s

Q runs remaining $400 - 4 = 396$ m at $2$ m/s.

P runs $400$ m in $198$ s.