Question

At a farm, the average number of chickens, ducks, rabbits and sheep is 315 per type of animal. The average number of chickens, ducks and rabbits is 320 per type of animal. The number of sheep is 60 more than the number of rabbits. There are half as many ducks as there are rabbits. How many chickens are there at the farm?

Answer 1

The average number of chickens, ducks, rabbits and sheep = 315 per type of animal.

The average number of chickens, ducks and rabbits = 320 per type of animal.

The number of sheep is 60 more than the number of rabbits.

Let us assume number of rabbits = x.

Number of sheep is 60 more than number of rabbits, that is = x+60.

There are half as many ducks as there are rabbits.

Therefore, number of ducks = x/2.

Let us assume number of chickens = y.

Adding  number of chickens, number of rabbits, number of sheep and number of ducks and dividing by 4 (because those are four types of animal).

$\frac{y+x+x+60+x/2}{4} = 315$

Multiplying both sides by 4, we get

y+x+x+60+x/2 = 1260

Subtracting 60 from both sides

y+x+x+60-60+x/2 = 1260-60

y+x+x+x/2 = 1200.

Combining like terms .

y + 5x/2 = 1200

Multiplying both sides by 2, we get

2y +5x = 2400  ------------equation (1)

Adding  number of chickens, number of rabbits and number of ducks and dividing by 3 (because those are three types of animal).

$\frac{y+x+60+x/2}{3} = 320$

Multiplying both sides by 3, we get

y+x+60+x/2 = 960

Subtracting 60 from both sides

y+ x+x/2 = 900

Combining like terms .

y+ 3x/2 = 900

Multiplying both sides by 2, we get

2y+3x = 1800 --------------------equation (2)

Multiplying first equation by 3 and second equation by -5, we get

(2y +5x = 2400 ) * 3   => 6y +15x = 7200

(2y+3x = 1800) *-5      =>

Adding

6y +15x = 7200

-10x + -15x = -9000.

_________________

-4y = -1800.

Dividing both sides by -4, we get

y=450.

Therefore, number of chickens are  450 there at the farm.