Question

I have a box of chocolates with white, milk and dark chocolate in. The ratio of White to Not White is 3:4. The Ratio of Not dark to dark is 17:4. What is the ratio of White: Milk : Dark?

Answer 1

Let the number of white, milk and dark chocolates in the box be x, y and z respectively. Since the ratio of White to Not White is 3:4, therefore x/(y+z) =3/4 or, 4x =3(y+z) or, 4x-3y-3z= 0…(1). Further, the Not dark to dark is 17:4 so that (x+y)/z = 17/4 or, 4x+4y-17z = 0…(2). The coefficient matrix of this linear system is A (say) =

4	-3	-3
4	4	-17

To solve this linear system, we will reduce A t5o its RREF as under:

Multiply the 1st row by ¼

Add -4 times the 1st row to the 2nd row

Multiply the 2nd row by 1/7

Add 3/4 times the 2nd row to the 1st row

Then the RREF of A is

1	0	-9/4
0	1	-2

Thus, the above linear system is equivalent to x -9z/4 = 0 or, x = 9z/4 and y -2z = 0 or, y = 2z. Then x:y:z = 9z/4: 2z:z = 9/4: 2:1 = 9:8:2. Thus, the ratio of White: Milk : Dark is 9:8:2