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

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Answer 1 · 2018-07-07T11:12:14

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

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

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

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

multiply through by 5 to get whole numbers

$5 d - 4 c = 4000$ $5d - 4c = 4000$

$5 d - 4000 = 4 c$ $5d -4000 = 4c$

$4 c = 5 d - 4000$ $4c = 5d -4000$

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

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

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

substitute for c in (1).

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

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

$0.25 d > 1000$ $0.25d > 1000$

$d > 4000$ $d > 4000$

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

Combining both of these conditions gives:

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