# On a farm, the number of chickens was more than the number of ducks. After 20% of the chickens were sold, there were 800 less chickens than ducks on the farm. How many ducks were there on the farm? ___ducks

Jul 7, 2018

$d = \left(4000 + 4 x\right)$ where $x$ is an integer greater than 0

#### Explanation:

$c > d$ .......................................... (1)

$d - \left(c - \frac{20}{100} c\right) = 800$ ..............(2)

so from (2)
$d - 0.8 c = 800$

multiply through by 5 to get whole numbers

$5 d - 4 c = 4000$

$5 d - 4000 = 4 c$

$4 c = 5 d - 4000$

$c = \frac{5 d - 4000}{4}$

$c = \frac{5 d}{4} - \frac{4000}{4}$

$c = \frac{5 d}{4} - 1000$ .........................(3)

substitute for c in (1).

$1.25 d - 1000 > d$

$1.25 d - d - 1000 + 1000 > d - d + 1000$

$0.25 d > 1000$

$d > 4000$

but c must be a whole number so
in eqn (3) the term $\frac{5 d}{4}$ must be a whole number,
so $d$ must be divisible by 4.

Combining both of these conditions gives:

$d = 4000 + 4 x$ where $x$ is an integer greater than 0