Grandfather 3253

Grandfather keeps 40 animals — geese, pigs, goats, and chickens—in total. There are 3 geese for every goat. If there were 8 fewer chickens, there would be as many geese and pigs combined. If Grandpa exchanged a quarter of the geese for chickens at a ratio of 3 chickens to 1 goose, he would have a total of 46 animals. How many animals does Grandpa keep?

h =  12
k =  4
p =  2
s =  22

Step-by-step explanation:

h+p+k+s=40
3k = h
s-8=h+p
h-h/4+3(h/4)+p+k+s=46

h+p+k+s=40
3·k = h
s-8=h+p
h-h/4+3·(h/4)+p+k+s=46

h+k+p+s = 40
h-3k = 0
h+p-s = -8
6h+4k+4p+4s = 184

Pivot: Row 1 ↔ Row 4
6h+4k+4p+4s = 184
h-3k = 0
h+p-s = -8
h+k+p+s = 40

Row 2 - 1/6 · Row 1 → Row 2
6h+4k+4p+4s = 184
-3.6667k-0.6667p-0.6667s = -30.6667
h+p-s = -8
h+k+p+s = 40

Row 3 - 1/6 · Row 1 → Row 3
6h+4k+4p+4s = 184
-3.6667k-0.6667p-0.6667s = -30.6667
-0.6667k+0.3333p-1.6667s = -38.6667
h+k+p+s = 40

Row 4 - 1/6 · Row 1 → Row 4
6h+4k+4p+4s = 184
-3.6667k-0.6667p-0.6667s = -30.6667
-0.6667k+0.3333p-1.6667s = -38.6667
0.3333k+0.3333p+0.3333s = 9.3333

Row 3 - -0.66666667/-3.66666667 · Row 2 → Row 3
6h+4k+4p+4s = 184
-3.6667k-0.6667p-0.6667s = -30.6667
0.4545p-1.5455s = -33.0909
0.3333k+0.3333p+0.3333s = 9.3333

Row 4 - 0.33333333/-3.66666667 · Row 2 → Row 4
6h+4k+4p+4s = 184
-3.6667k-0.6667p-0.6667s = -30.6667
0.4545p-1.5455s = -33.0909
0.2727p+0.2727s = 6.5455

Row 4 - 0.27272727/0.45454545 · Row 3 → Row 4
6h+4k+4p+4s = 184
-3.6667k-0.6667p-0.6667s = -30.6667
0.4545p-1.5455s = -33.0909
1.2s = 26.4

s = 26.4/1.2 = 22
p = -33.09090909+1.5454545454545s/0.45454545 = -33.09090909+1.54545455 · 22/0.45454545 = 2
k = -30.66666667+0.66666666666667p+0.66666666666667s/-3.66666667 = -30.66666667+0.66666667 · 2+0.66666667 · 22/-3.66666667 = 4
h = 184-4k-4p-4s/6 = 184-4 · 4-4 · 2-4 · 22/6 = 12

h = 12
k = 4
p = 2
s = 22

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