Races and Games of Skill: Formulas, Shortcuts and Solved Examples — MCQs with Answers (Set 2)

Let the race distance = 500 m. Divya beats Karan by 50 m, so when Divya covers 500 m, Karan covers 500-50 = 450 m. This gives the distance ratio Divya:Karan = 500:450. Karan beats Neha by 50 m over the same 500 m course, so Karan:Neha = 500:450. To combine, find how far Neha runs when Karan covers only 450 m instead of the full 500 m. Neha’s distance scales down in proportion: 450 x (450)/500 = 405 m. So when Divya finishes 500 m, Neha has covered 405 m. Divya beats Neha by 500-405 = 95 m. Check: 95 m is more than 50 m and more than 50 m, as expected when two gaps combine.

Let the race distance = 500 m. Arjun beats Priya by 100 m, so when Arjun covers 500 m, Priya covers 500-100 = 400 m. This gives the distance ratio Arjun:Priya = 500:400. Priya beats Neha by 30 m over the same 500 m course, so Priya:Neha = 500:470. To combine, find how far Neha runs when Priya covers only 400 m instead of the full 500 m. Neha’s distance scales down in proportion: 400 x (470)/500 = 376 m. So when Arjun finishes 500 m, Neha has covered 376 m. Arjun beats Neha by 500-376 = 124 m. Check: 124 m is more than 100 m and more than 30 m, as expected when two gaps combine.

Let the race distance = 100 m. Priya beats Neha by 20 m, so when Priya covers 100 m, Neha covers 100-20 = 80 m. This gives the distance ratio Priya:Neha = 100:80. Neha beats Meera by 10 m over the same 100 m course, so Neha:Meera = 100:90. To combine, find how far Meera runs when Neha covers only 80 m instead of the full 100 m. Meera’s distance scales down in proportion: 80 x (90)/100 = 72 m. So when Priya finishes 100 m, Meera has covered 72 m. Priya beats Meera by 100-72 = 28 m. Check: 28 m is more than 20 m and more than 10 m, as expected when two gaps combine.

Let the race distance = 400 m. Arjun beats Priya by 80 m, so when Arjun covers 400 m, Priya covers 400-80 = 320 m. This gives the distance ratio Arjun:Priya = 400:320. Priya beats Neha by 10 m over the same 400 m course, so Priya:Neha = 400:390. To combine, find how far Neha runs when Priya covers only 320 m instead of the full 400 m. Neha’s distance scales down in proportion: 320 x (390)/400 = 312 m. So when Arjun finishes 400 m, Neha has covered 312 m. Arjun beats Neha by 400-312 = 88 m. Check: 88 m is more than 80 m and more than 10 m, as expected when two gaps combine.

Let the race distance = 500 m. Meera beats Neha by 100 m, so when Meera covers 500 m, Neha covers 500-100 = 400 m. This gives the distance ratio Meera:Neha = 500:400. Neha beats Priya by 10 m over the same 500 m course, so Neha:Priya = 500:490. To combine, find how far Priya runs when Neha covers only 400 m instead of the full 500 m. Priya’s distance scales down in proportion: 400 x (490)/500 = 392 m. So when Meera finishes 500 m, Priya has covered 392 m. Meera beats Priya by 500-392 = 108 m. Check: 108 m is more than 100 m and more than 10 m, as expected when two gaps combine.

Let the race distance = 100 m. Sanjay beats Karan by 15 m, so when Sanjay covers 100 m, Karan covers 100-15 = 85 m. This gives the distance ratio Sanjay:Karan = 100:85. Karan beats Rohan by 20 m over the same 100 m course, so Karan:Rohan = 100:80. To combine, find how far Rohan runs when Karan covers only 85 m instead of the full 100 m. Rohan’s distance scales down in proportion: 85 x (80)/100 = 68 m. So when Sanjay finishes 100 m, Rohan has covered 68 m. Sanjay beats Rohan by 100-68 = 32 m. Check: 32 m is more than 15 m and more than 20 m, as expected when two gaps combine.

Let the race distance = 1000 m. Priya takes 114 seconds and Sanjay takes 120 seconds to run it. Sanjay’s speed = 1000/120 m per second. When Priya finishes, at time 114 seconds, Sanjay has been running for only 114 seconds too. Priya beats Sanjay by 120-114 = 6 seconds of running time still left for Sanjay. In those 6 seconds, Sanjay would have covered 1000 x 6/120 = 50 m more. So Priya beats Sanjay by 50 m. Check: at Sanjay’s speed, 50 m takes 50/(1000/120) = 6 seconds, matching the time gap.

Let the race distance = 1000 m. Rohan takes 166 seconds and Divya takes 200 seconds to run it. Divya’s speed = 1000/200 m per second. When Rohan finishes, at time 166 seconds, Divya has been running for only 166 seconds too. Rohan beats Divya by 200-166 = 34 seconds of running time still left for Divya. In those 34 seconds, Divya would have covered 1000 x 34/200 = 170 m more. So Rohan beats Divya by 170 m. Check: at Divya’s speed, 170 m takes 170/(1000/200) = 34 seconds, matching the time gap.

Let the race distance = 500 m. Sanjay takes 194 seconds and Karan takes 200 seconds to run it. Karan’s speed = 500/200 m per second. When Sanjay finishes, at time 194 seconds, Karan has been running for only 194 seconds too. Sanjay beats Karan by 200-194 = 6 seconds of running time still left for Karan. In those 6 seconds, Karan would have covered 500 x 6/200 = 15 m more. So Sanjay beats Karan by 15 m. Check: at Karan’s speed, 15 m takes 15/(500/200) = 6 seconds, matching the time gap.

Let the race distance = 400 m. Karan takes 133 seconds and Priya takes 140 seconds to run it. Priya’s speed = 400/140 m per second. When Karan finishes, at time 133 seconds, Priya has been running for only 133 seconds too. Karan beats Priya by 140-133 = 7 seconds of running time still left for Priya. In those 7 seconds, Priya would have covered 400 x 7/140 = 20 m more. So Karan beats Priya by 20 m. Check: at Priya’s speed, 20 m takes 20/(400/140) = 7 seconds, matching the time gap.

Let the race distance = 200 m. Divya takes 108 seconds and Priya takes 135 seconds to run it. Priya’s speed = 200/135 m per second. When Divya finishes, at time 108 seconds, Priya has been running for only 108 seconds too. Divya beats Priya by 135-108 = 27 seconds of running time still left for Priya. In those 27 seconds, Priya would have covered 200 x 27/135 = 40 m more. So Divya beats Priya by 40 m. Check: at Priya’s speed, 40 m takes 40/(200/135) = 27 seconds, matching the time gap.

Let the race distance = 200 m. Divya takes 93 seconds and Neha takes 124 seconds to run it. Neha’s speed = 200/124 m per second. When Divya finishes, at time 93 seconds, Neha has been running for only 93 seconds too. Divya beats Neha by 124-93 = 31 seconds of running time still left for Neha. In those 31 seconds, Neha would have covered 200 x 31/124 = 50 m more. So Divya beats Neha by 50 m. Check: at Neha’s speed, 50 m takes 50/(200/124) = 31 seconds, matching the time gap.

Let the race distance = 500 m. Meera takes 221 seconds and Karan takes 260 seconds to run it. Karan’s speed = 500/260 m per second. When Meera finishes, at time 221 seconds, Karan has been running for only 221 seconds too. Meera beats Karan by 260-221 = 39 seconds of running time still left for Karan. In those 39 seconds, Karan would have covered 500 x 39/260 = 75 m more. So Meera beats Karan by 75 m. Check: at Karan’s speed, 75 m takes 75/(500/260) = 39 seconds, matching the time gap.

Let the race distance = 800 m. Neha takes 70 seconds and Priya takes 100 seconds to run it. Priya’s speed = 800/100 m per second. When Neha finishes, at time 70 seconds, Priya has been running for only 70 seconds too. Neha beats Priya by 100-70 = 30 seconds of running time still left for Priya. In those 30 seconds, Priya would have covered 800 x 30/100 = 240 m more. So Neha beats Priya by 240 m. Check: at Priya’s speed, 240 m takes 240/(800/100) = 30 seconds, matching the time gap.

Let the race distance = 800 m. Meera beats Divya by 75 m, and also by 18 seconds. This means when Meera finishes, Divya has run only 800-75 = 725 m, taking 18 extra seconds to cover the last 75 m. So Divya’s speed = 75 m per 18 seconds. Divya’s time for the full 800 m = 18 x 800/75 = 192 seconds. Meera finishes 18 seconds before Divya, so Meera’s time = 192-18 = 174 seconds. Check: in 174 seconds, Divya (at 75/18 m per second) covers 174×75/18 = 725 m, 75 m short of 800, as expected.

Let the race distance = 400 m. Meera beats Sanjay by 32 m, and also by 10 seconds. This means when Meera finishes, Sanjay has run only 400-32 = 368 m, taking 10 extra seconds to cover the last 32 m. So Sanjay’s speed = 32 m per 10 seconds. Sanjay’s time for the full 400 m = 10 x 400/32 = 125 seconds. Meera finishes 10 seconds before Sanjay, so Meera’s time = 125-10 = 115 seconds. Check: in 115 seconds, Sanjay (at 32/10 m per second) covers 115×32/10 = 368 m, 32 m short of 400, as expected.

Let the race distance = 200 m. Karan beats Rohan by 10 m, and also by 16 seconds. This means when Karan finishes, Rohan has run only 200-10 = 190 m, taking 16 extra seconds to cover the last 10 m. So Rohan’s speed = 10 m per 16 seconds. Rohan’s time for the full 200 m = 16 x 200/10 = 320 seconds. Karan finishes 16 seconds before Rohan, so Karan’s time = 320-16 = 304 seconds. Check: in 304 seconds, Rohan (at 10/16 m per second) covers 304×10/16 = 190 m, 10 m short of 200, as expected.

Let the race distance = 500 m. Meera beats Arjun by 25 m, and also by 20 seconds. This means when Meera finishes, Arjun has run only 500-25 = 475 m, taking 20 extra seconds to cover the last 25 m. So Arjun’s speed = 25 m per 20 seconds. Arjun’s time for the full 500 m = 20 x 500/25 = 400 seconds. Meera finishes 20 seconds before Arjun, so Meera’s time = 400-20 = 380 seconds. Check: in 380 seconds, Arjun (at 25/20 m per second) covers 380×25/20 = 475 m, 25 m short of 500, as expected.

Let the race distance = 200 m. Neha beats Divya by 20 m, and also by 20 seconds. This means when Neha finishes, Divya has run only 200-20 = 180 m, taking 20 extra seconds to cover the last 20 m. So Divya’s speed = 20 m per 20 seconds. Divya’s time for the full 200 m = 20 x 200/20 = 200 seconds. Neha finishes 20 seconds before Divya, so Neha’s time = 200-20 = 180 seconds. Check: in 180 seconds, Divya (at 20/20 m per second) covers 180×20/20 = 180 m, 20 m short of 200, as expected.

Let the race distance = 200 m. Arjun beats Rohan by 25 m, and also by 8 seconds. This means when Arjun finishes, Rohan has run only 200-25 = 175 m, taking 8 extra seconds to cover the last 25 m. So Rohan’s speed = 25 m per 8 seconds. Rohan’s time for the full 200 m = 8 x 200/25 = 64 seconds. Arjun finishes 8 seconds before Rohan, so Arjun’s time = 64-8 = 56 seconds. Check: in 56 seconds, Rohan (at 25/8 m per second) covers 56×25/8 = 175 m, 25 m short of 200, as expected.

Let the race distance = 400 m. Rohan beats Karan by 17 m, and also by 17 seconds. This means when Rohan finishes, Karan has run only 400-17 = 383 m, taking 17 extra seconds to cover the last 17 m. So Karan’s speed = 17 m per 17 seconds. Karan’s time for the full 400 m = 17 x 400/17 = 400 seconds. Rohan finishes 17 seconds before Karan, so Rohan’s time = 400-17 = 383 seconds. Check: in 383 seconds, Karan (at 17/17 m per second) covers 383×17/17 = 383 m, 17 m short of 400, as expected.

Let the race distance = 1500 m. Arjun beats Sanjay by 40 m, and also by 20 seconds. This means when Arjun finishes, Sanjay has run only 1500-40 = 1460 m, taking 20 extra seconds to cover the last 40 m. So Sanjay’s speed = 40 m per 20 seconds. Sanjay’s time for the full 1500 m = 20 x 1500/40 = 750 seconds. Arjun finishes 20 seconds before Sanjay, so Arjun’s time = 750-20 = 730 seconds. Check: in 730 seconds, Sanjay (at 40/20 m per second) covers 730×40/20 = 1460 m, 40 m short of 1500, as expected.

Let the race distance = 400 m. Divya beats Rohan by 20 m, and also by 3 seconds. This means when Divya finishes, Rohan has run only 400-20 = 380 m, taking 3 extra seconds to cover the last 20 m. So Rohan’s speed = 20 m per 3 seconds. Rohan’s time for the full 400 m = 3 x 400/20 = 60 seconds. Divya finishes 3 seconds before Rohan, so Divya’s time = 60-3 = 57 seconds. Check: in 57 seconds, Rohan (at 20/3 m per second) covers 57×20/3 = 380 m, 20 m short of 400, as expected.

Let the race distance = 500 m. Divya beats Meera by 10 m, and also by 5 seconds. This means when Divya finishes, Meera has run only 500-10 = 490 m, taking 5 extra seconds to cover the last 10 m. So Meera’s speed = 10 m per 5 seconds. Meera’s time for the full 500 m = 5 x 500/10 = 250 seconds. Divya finishes 5 seconds before Meera, so Divya’s time = 250-5 = 245 seconds. Check: in 245 seconds, Meera (at 10/5 m per second) covers 245×10/5 = 490 m, 10 m short of 500, as expected.

Let the race distance = 400 m. Priya beats Arjun by 10 m, and also by 11 seconds. This means when Priya finishes, Arjun has run only 400-10 = 390 m, taking 11 extra seconds to cover the last 10 m. So Arjun’s speed = 10 m per 11 seconds. Arjun’s time for the full 400 m = 11 x 400/10 = 440 seconds. Priya finishes 11 seconds before Arjun, so Priya’s time = 440-11 = 429 seconds. Check: in 429 seconds, Arjun (at 10/11 m per second) covers 429×10/11 = 390 m, 10 m short of 400, as expected.