Question

At a farm, there are hens, cows and bullocks, and the keepers to look after them. There are $69$ heads less than legs; the number of cows and hens is the same , number of cows are twice as number of bullocks 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. $16$
• B. $14$
• C. $12$
• D. $10$

Answer 1

Answer: (C)

$12$

Let $H,C,B$ and $K$ represent the number of hens, cows, bullocks and keepers respectively.
Then, 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:
$10K=H+C+B$
$\Rightarrow 10K=2C+B$
$=2\times 2B+B$
$\Rightarrow 10K=5B$
$\Rightarrow B=2K$
Thus, $B=2K,C=2B=4K,H=C=4K$
Total number of heads $=H+C+B+K$
Total number of legs $=2H+4C+4B+2K$
$\therefore (2H+4C+4B+2K)-(H+C+B+K)=69$
$\Rightarrow H+3C+3B+K=69$
$\Rightarrow 4K+12K+6K+K=69$
$\Rightarrow 23K=69$
$\Rightarrow K=3$
Hence, number of cows $=C=4K=(4\times 3)=12$.