Question

a shepherd had goats and sheeps when asked how many goats does he have he told he had 112 legs and 32 heads find the number of hens he had ​

Answer 1

Answer-

He had 8 hens.

$\rule{100}2$

Explanation-

Let the number of hens be x and goats & sheeps be y.

We know that goat, sheep and hen has one head only. So,

x + y = 32

Also, hen has two legs, while goat and sheep has four legs. So,

2x + 4y = 112

According to question, the two equation are :-

x + y = 32 and 2x + 4y = 112

→ x + y = 32

→ x = 32 - y ...(1)

Similarly,

→ 2x + 4y = 112

→ x + 2y = 56

→ 32 - y + 2y = 56 [From (1)]

→ 32 + y = 56

→ y = 24

Therefore, number of goats and sheeps = 24

Substitute value of y in x

→ x = 32 - 24

→ x = 8

Therefore, number of hens = 8

Answer 2

Correct Question :--- a shepherd had goats and hens when asked how many goats does he have he told he had 112 legs and 32 heads find the number of hens he had ?

Solution :---

→ Since, goats have 4 legs and hens have 2 legs ...

Let total number of Hens be = x .

Than Number of goats left = (32-x) { As their were 32 heads total .}

Now,

→ Total Legs All goats have = 4*(32-x)

→ Total legs all hens have = 2x.

Now, it is given that, Total legs were 112.

so, According to Question now,

→ 2x + 4(32-x) = 112

→ 2x + 128 - 4x = 112

→ (-2x) = 112 - 128

→ (-2x) = (-16)

Dividing both sides by (-2) we get ,

→ x = 8

→ Total Number of Hens = x = 8 (Ans)..

Hence, their were 8 hens ...

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*** Shortcut *** :------

→ For this problem in Question we have given animals with 4 legs and 2 legs always..

So, to Find Total number of 2 legs animals , we can use :---

→ 2 legs Head = [ ( Total Heads * 4 - Total legs ) /2]

puttings values from Question we get,

→ Total Hens (2 legs) = [ ( 32*4 - 112)/2]

→ Total Hens = (128-112)/2

→ Total hens = 16/2 = 8

Hence , Their were 8 Hens ...