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?
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
To solve this problem, we can use the information given and solve it in two steps:
Step 1: Determine the initial number of blue buttons.
Step 2: Determine the ratio of green to red buttons eventually.
Let's start with step 1:
a) How many blue buttons were there initially?
Given:
- The ratio of blue to red buttons is 3:4.
- There are 12 more green buttons than blue buttons.
Let's assume the number of blue buttons initially is 3x.
Then, the number of red buttons initially is 4x.
And the number of green buttons initially is 3x + 12.
After removing 6 blue buttons and 1/3 of the green buttons, we need to replace the same number of buttons, resulting in the same total number of buttons.
The original number of buttons is:
3x (blue) + 4x (red) + (3x + 12) (green).
After removing 6 blue buttons, we have:
3x - 6 (blue) + 4x (red) + (3x + 12) (green).
This leads to a total of:
10x + 12 (green).
Now, let's account for the replacement:
3x - 6 (blue) + 4x (red) + (3x + 12) (green) - (1/3)(3x + 12) (green) = 10x + 12 (green) + 6 (red).
Simplifying the equation:
3x - 6 + 4x + 3x + 12 - (1/3)(3x + 12) = 10x + 12 + 6.
Expanding and simplifying further:
3x - 6 + 4x + 3x + 12 - x - 4 = 10x + 12 + 6,
6x + 2 = 10x + 18,
4x = 16,
x = 4.
Therefore, there were initially 3x = 3 * 4 = 12 blue buttons.
Moving on to step 2:
b) What is the ratio of green to red buttons eventually?
Given:
- The ratio of blue to red buttons eventually is 3:12.
From step 1, we found that initially, there were 12 blue buttons.
After removing 6 blue buttons and replacing them with an equal number of red buttons, the ratio became 3:12 for blue to red buttons.
This means that after the replacement, there are 12 - 6 = 6 blue buttons and 6 red buttons.
To find the ratio of green to red buttons eventually, we need to determine the number of green buttons.
Initially, there were 3x + 12 green buttons, where x = 4.
So, initially, there were 3 * 4 + 12 = 24 green buttons.
After removing 1/3 of the green buttons, there are 2/3 remaining, which is 2/3 * 24 = 16 green buttons.
Therefore, the ratio of green to red buttons eventually is 16:6, which simplifies to 8:3.
a) There were initially 12 blue buttons.
b) The ratio of green to red buttons eventually is 8:3.