Heads & Legs Problem


Q. In a forest, there are some peacocks and deer. The total number of heads is 60 and the total number of legs is 224. How many deer are there ?


A. 52   B. 58   C. 56   D. 4


Solution


The total number of heads = 60 and total number of legs = 224.
How many deer are there?


A) 52  B) 58  C) 56  D) 4


Solution


A peacock = 2 legs


A deer = 4 legs


First, assume all 60 animals are peacocks: 60 × 2 = 120 legs


But actual legs = 224


Extra legs: 224 − 120 = 104






Each deer has 2 extra legs compared to a peacock.


So,


Deer = 104 ÷ 2 = 52


Ans: A) 52 Deer


Trick / āĻŸā§ā§°āĻŋāĻ•


Deer = (Total Legs − Total Heads × 2) ÷ 2


= (224 − 60 × 2) ÷ 2


= 104 ÷ 2


= 52


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Q. A jar contains some crabs (8 legs each) and some rabbits (4 legs each). If the total number of legs is 100 and the total number of heads is 20, how many crabs are there ? āĻĒā§ā§°āĻļā§āύ: āĻāϟāĻž āϜāĻžā§°āϤ āĻ•āĻŋāϛ⧁āĻŽāĻžāύ āϕ⧇āρāϕ⧋⧰āĻž (āĻĒā§ā§°āϤāĻŋ āϕ⧇āρāϕ⧋⧰āĻžā§° 8āϟāĻž āĻ­ā§°āĻŋ) āφ⧰⧁ āĻ–ā§°āĻ—ā§‹āĻļ (āĻĒā§ā§°āϤāĻŋ āĻ–ā§°āĻ—ā§‹āĻļā§° 4āϟāĻž āĻ­ā§°āĻŋ) āφāϛ⧇āĨ¤ āϝāĻĻāĻŋ āĻŽā§āĻ  āĻ­ā§°āĻŋ = 100 āφ⧰⧁ āĻŽā§āĻ  āĻŽā§‚ā§° = 20, āϤ⧇āĻ¨ā§āϤ⧇ āϕ⧇āρāϕ⧋⧰āĻž āĻ•āĻŋāĻŽāĻžāύāϟāĻž ?


Options / āĻŦāĻŋāĻ•āĻ˛ā§āĻĒāϏāĻŽā§‚āĻš: (A) 15  (B) 5  (C) 16  (D) 4


Trick / āĻŸā§ā§°āĻŋāĻ•


Every animal has 1 head, so total animals = 20. : āĻĒā§ā§°āϤāĻŋāĻŸā§‹ āϜāĻ¨ā§āϤ⧁⧰ ā§§āϟāĻž āĻŽā§‚ā§° āĻĨāĻžāϕ⧇, āϏ⧇āϝāĻŧ⧇ āĻŽā§āĻ  āϜāĻ¨ā§āϤ⧁ = 20āĨ¤)


Assume all are rabbits (4 legs each). : āϧ⧰āĻŋ āϞāĻ“āρ āϏāĻ•āϞ⧋ āĻ–ā§°āĻ—ā§‹āĻļ (āĻĒā§ā§°āϤāĻŋ āĻ–ā§°āĻ—ā§‹āĻļā§° 4āϟāĻž āĻ­ā§°āĻŋ)āĨ¤


Legs (āĻ­ā§°āĻŋ) = 20 × 4 = 80


Actual legs (āĻĒā§ā§°āĻ•ā§ƒāϤ āĻ­ā§°āĻŋ) = 100


Extra legs (āĻ…āϤāĻŋā§°āĻŋāĻ•ā§āϤ āĻ­ā§°āĻŋ) = 100 − 80 = 20


Each crab has 4 extra legs than a rabbit (āĻāϟāĻž āϕ⧇āρāϕ⧋⧰āĻžā§° āĻ–ā§°āĻ—ā§‹āĻļāϤāĻ•ā§ˆ 4āϟāĻž āĻ­ā§°āĻŋ āĻŦ⧇āĻ›āĻŋ) (8 − 4 = 4).


Crabs (āϕ⧇āρāϕ⧋⧰āĻž) = Extra legs (āĻ…āϤāĻŋā§°āĻŋāĻ•ā§āϤ āĻ­ā§°āĻŋ) ÷ 4 = 20 ÷ 4 = 5


Ans / āωāĻ¤ā§āϤ⧰: (B) 5


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Q. A cage contains some rats (4 legs each) and some crabs (8 legs each). If the total number of heads is 30 and the total number of legs is 140, how many rats are there ? āĻĒā§ā§°āĻļā§āύ: āĻāϟāĻž āĻĒāĻŋāĻžā§āϜ⧰āĻžāϤ āĻ•āĻŋāϛ⧁āĻŽāĻžāύ āĻāĻ¨ā§āĻĻ⧁⧰ (āĻĒā§ā§°āϤāĻŋ āĻāĻ¨ā§āĻĻ⧁⧰⧰ 4āϟāĻž āĻ­ā§°āĻŋ) āφ⧰⧁ āϕ⧇āρāϕ⧋⧰āĻž (āĻĒā§ā§°āϤāĻŋ āϕ⧇āρāϕ⧋⧰āĻžā§° 8āϟāĻž āĻ­ā§°āĻŋ) āφāϛ⧇āĨ¤ āϝāĻĻāĻŋ āĻŽā§āĻ  āĻŽā§‚ā§° = 30 āφ⧰⧁ āĻŽā§āĻ  āĻ­ā§°āĻŋ = 140, āϤ⧇āĻ¨ā§āϤ⧇ āĻāĻ¨ā§āĻĻ⧁⧰ āĻ•āĻŋāĻŽāĻžāύāϟāĻž ?


Options / āĻŦāĻŋāĻ•āĻ˛ā§āĻĒāϏāĻŽā§‚āĻš: (A) 25  (B) 4  (C) 16  (D) 8


Trick / āĻŸā§ā§°āĻŋāĻ•


Every animal has 1 head, so total animals = 30. āĻĒā§ā§°āϤāĻŋāĻŸā§‹ āϜāĻ¨ā§āϤ⧁⧰ ā§§āϟāĻž āĻŽā§‚ā§° āĻĨāĻžāϕ⧇, āϏ⧇āϝāĻŧ⧇ āĻŽā§āĻ  āϜāĻ¨ā§āϤ⧁ = 30āĨ¤


Assume all are rats (4 legs each). āϧ⧰āĻŋ āϞāĻ“āρ āϏāĻ•āϞ⧋ āĻāĻ¨ā§āĻĻ⧁⧰ (āĻĒā§ā§°āϤāĻŋ āĻāĻ¨ā§āĻĻ⧁⧰⧰ 4āϟāĻž āĻ­ā§°āĻŋ)āĨ¤


Legs (āĻ­ā§°āĻŋ) = 30 × 4 = 120


Actual legs (āĻĒā§ā§°āĻ•ā§ƒāϤ āĻ­ā§°āĻŋ) = 140


Extra legs (āĻ…āϤāĻŋā§°āĻŋāĻ•ā§āϤ āĻ­ā§°āĻŋ) = 140 − 120 = 20


Each crab has 4 extra legs than a rat (āĻāϟāĻž āϕ⧇āρāϕ⧋⧰āĻžā§° āĻāĻ¨ā§āĻĻ⧁⧰āϤāĻ•ā§ˆ 4āϟāĻž āĻ­ā§°āĻŋ āĻŦ⧇āĻ›āĻŋ) (8 − 4 = 4).


Crabs (āϕ⧇āρāϕ⧋⧰āĻž) = 20 ÷ 4 = 5


Rats (āĻāĻ¨ā§āĻĻ⧁⧰) = 30 − 5 = 25


Ans / āωāĻ¤ā§āϤ⧰: (A) 25


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