Question

At a farm, there are hens, cows and bullocks and keepers to look after them. There are $69$ heads less than legs; the number of cows is doubled than that of the bullocks; the number of cows and hens is the same and there is one keeper per ten birds and cattle. The total number of hens plus cows and bullocks and their keepers does not exceed $50$. How many cows are there?

• A. $10$
• B. $14$
• C. $12$
• D. $16$

Answer 1

Answer: (C)

$12$

Let $H,C,B$ and $K$ represent the number of hens, cows, bullocks and keepers respectively.
The, as given, we have:
$H+C+B+K<50$ ... (1)
$C=2B$ ... (2)
$C=H$ ... (3)
$K=\frac{H+C+B}{10}$ ... (4)
From (2), (3) and (4), we have
$10K=H+C+B$
or, $10K=2C+B=2\times 2B+B$
or, $10K=5B$
or, $B=2K.$
Thus, $B=2K,C=2B=4K,H=C=4K$.
Total number of heads $=H+C+B+K$
Total number of legs $=2H+4C+4B+2K$
Therefore, $(2H+4C+4B+2K)-(H+C+B+K)=69$
or, $H+3C+3B+K=69$
or, $4K+12K+6K+K=69$
or, $23K=69$
or, $K=3$.
So, number of cows $=4K=4\times 3=12.$
Hence, (C) is the correct answer.