Question

There are blue red and green buttons in a box. The ratio of blue to red buttons is 3:4. There are 12 more green buttons than blue buttons.
6 blue buttons and 1/3 of the green buttons were removed from the box. The number of buttons removed were replaced by the same number of red buttons, such that the total number of buttons remains unchanged. The ratio of blue to red buttons eventually became 3:12.
a) How many blue buttons were there initially?
b) What is the ratio of green to red buttons eventually?

Answer 1

There are blue red and green buttons in a box.

The ratio of blue to red buttons is 3:4. Let blue button 3x,red button 4x. Then green button=(3x)+12.

Total number of buttons=(3x)+(4x)+(3x)+12=(10x)+12

Again 6 blue buttons and 1/3 of the green buttons were removed from the box.

Now new blue button=(3x)-6.

New green button=((3x)+12)-((3x)+12)/3)=(2x+8)

New red button =(4x)+(6)+((x)+4)=(5x)+10

The ratio of blue to red buttons eventually became 3:12

(a) (3x - 6)/(5x + 10) = 3/12
=> 12x - 24 = 5x + 10
=> 7x = 34
=> x = 34/7
= 5 (approx)

Therefore initial blue buttons =15 (approx)

(b) Ratio of green to red = (2x + 8)/5x + 10) = (2 * 34/7 + 8)/(5 * 34/7 + 10) = (68 + 56)/(170 + 80) = 124/240
= 31/60