Answer to Question #107165 in Algebra for Akshat Patel

Question

One person decided to distribute 100 kilograms free rice to poor people. He wished to distribute grain to 100 person in such a way that old person get 3 kilograms , each young received 2 kilograms and each child half kg. Then how many old, young and children?

Answer 1

Let x = number of old person

y= number of young person

z = number of children

Number of person = 100

So x + y + z = 100 ..................................................eqn(1)

Each old person get 3 kg , young get 2kg and child get 0.5 kg of rice

Total amount of rice distributed among people = 100 kg

i.e. 3x + 2y + 0.5z = 100 kg

multiply both sides by 2

or 6x + 4y + z = 200.....................................eqn (2)

Also x, y, z are number of person they will take only positive integer value.

Now number of unknown is three and we have only two equation, so we need to assume value of one variable and solve for other two to see if they are integer (as person can't be fractional). Solving x and y in terms of z as follow

From equation (1) and (2)

x+y = 100-z ......................eqn (3)

6x+4y =200 -z ...............eqn (4)

Doing eqn4 - 4* eqn (3)

(6x+4y)-4(x+y)=200-z - 4(100-z)

6x+4y-4x-4y = 200 -z -400+4z

2x = 3z -200

x = 1.5z - 100

substituting value in eqn3

y = 100 - z - x = 100 - z - 1.5z +100 = 200 -2.5z

So general solution

x = 1.5z - 100

y= 200 - 2.5z

z = z

also old, young or child count will be either zero or positive integer.

$x\ge 0$ and $y\ge 0$ and $z\ge 0$

$1.5z-100\ge 0$ and $200-2.5z\ge 0$ and $z\ge 0$

$z\ge 200/3\approx 66.7$ , and $z\le 80$ and $z\ge 0$

common region

$200/3 \le z\le 80$

additionally in general solution ( x= 1.5z - 100 , y = 200 -2.5z) , if z is an odd integer then x and y will be fractional. So z must be an even integer between$[200/3 , 80]$

So solution will be

Number of old = 1.5z- 100

Number of young = 200-2.5z

Number of child = z , where z is an even number between 68 and 80 inclusive.

Verifying

let z = 70

number of child = 70

Number of old = 1.5 * 70 - 100 = 105 - 100 = 5

Number of young = 200 - 2.5 * 70 = 200 - 175 = 25

Total person = 70 + 5 + 25 = 100

Let z = 80

Number of child = 80

Number of old = 1.5*80 -100 = 120 - 100 =20

Number of young = 200 - 2.5 * 80 = 200 - 200 = 0

Total = 80+ 20+ 0 = 100

So it has multiple solutions.