In a zoo with 100rabbits ,with three kinds of rabbit 1kg,2kg,5kg. There are minimum of 10 rabbits of each kind and maximum of 60rabbits of each kind.
If 40 rabbits weighing a total of 148 kg are transferred from one zoo to another,then the remaining weight of 60 rabbits is 212kg.
1.what is the minimum number of 1kg rabbits transferred 2.what is the maximum number of 5kg rabbits that remain 3.if a total of 26 five kg rabbit were transferred then what is the maximum possible numbers of 1kg rabbits that remain in the zoo?
Could someone help me solve it or point me to a similar question?
a) Let $x_1,x_2,x_5$ be the numbers of 1kg, 2kg and 5kg rabbits, respectively, that were transferred. Then: $$\begin{cases}x_1+x_2+x_5=40 \\ x_1+2x_2+5x_5=148 \end{cases} \Rightarrow 4x_1+3x_2=52.$$ Let $x_1=1$ ($x_1\ne 0$, because $3\nmid 52 $), then $x_2=16$ and $x_5=23$.
On the other hand, let $x_1^*,x_2^*,x_5^*$ be the numbers of 1kg, 2kg and 5kg rabbits, respectively, that remained in the zoo. Then: $$\begin{cases}x_1^*+x_2^*+x_5^*=60 \\ x_1^*+2x_2^*+5x_5^*=212 \end{cases} \Rightarrow 4x_1^*+3x_2^*=88.$$ Let $x_1^*=10$ ($x_1^*\ge 9$, because $x_1+x_1^*\ge 10$ and $x_1^*\ne 9$, because $3\nmid 52$ ), then $x_2^*=16$ and $x_5^*=34$.
Hence the minimum number of 1kg rabbits that were transferred was $1$.