SOLUTION: The ratio of the number of blue sticks to the number of green sticks in a box was 4:1. When David took out some blue and sticks and replaced them with an equal number of green s

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Question 1186883: The ratio of the number of blue sticks to the number of green sticks in a box was
4:1. When David took out some blue and sticks and replaced them with an equal
number of green sticks, the ratio of the number of blue sticks to the number of
green sticks became 3:1. If there were 185 green sticks in the box now,
(a) find the total number of blue and green sticks in the box,
(b) find the number of green sticks in the box at first.
Found 3 solutions by Octo-pie7, ikleyn, greenestamps:
Answer by Octo-pie7(11)   (Show Source): You can put this solution on YOUR website!
(a).
Now according to the given conditions:
x/7 = 4/1
x = 4y
Total number of stickers in the box is: 185
Therefore:
x + y = 185
x = 148
y = 36
Now let n number of blue stickers took from the box.
(148-n)/37+n = 3/1
solving this equation:
n = 37
Therefore:
Number of blue stickers = 111
Number of green stickers = 148
740 stickers in total
(b).
Number of green stickers at first = 37 * 4 = 148
Answer by ikleyn(52790)   (Show Source): You can put this solution on YOUR website!
.
The ratio of the number of blue sticks to the number of green sticks in a box was 4:1.
When David took out some blue and sticks and replaced them with an equal number of green sticks,
the ratio of the number of blue sticks to the number of green sticks became 3:1.
If there were 185 green sticks in the box now,
(a) find the total number of blue and green sticks in the box,
(b) find the number of green sticks in the box at first.
~~~~~~~~~~~~~~~~~~~


            This problem is good for mental solution,
            if to solve it in a backward manner. 


According to the condition, in the final state, there are 185 green sticks in the box, 

and the number of blue sticks is thrice of it, i.e. 185*3 = 555.

Thus the total number of sticks in the final state is  185 + 555 = 740.


    +------------------------------------------------------------------+
    |    Now notice that  total number of sticks in the final state    |
    |    is THE SAME as at the beginning, according to the problem.    |
    +------------------------------------------------------------------+



HENCE, there were 740 sticks in the box at the beginning, and the ratio of blue to green sticks was 4:1.


So, there were 4+1 = 5 equal groups; each group contained 740/5 = 148 sticks.


Thus the problem is just solved: at the beginning, there were 148 green sticks and 148*4 = 592 blue sticks.


ANSWER.  There were 148 green sticks and 148*4 = 592 blue sticks in the box at first.

         There were 740 sticks in the box (the same number at the beginning and at the end).

Solved  mentally,  without using any equation / equations.


-----------------


What is written in the post by  @Octo-pie7,  is  IRRELEVANT  to the problem.

So you better ignore it,  for your safety and for peace in your mind.



Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


The solution from the other (new) tutor contains several errors, including using the given information incorrectly. Don't spend too much time trying to figure out what he was trying to do with his solution.

Initially, the ratio of blue to green sticks was 4:1, so

let 4x = # of blue sticks initially
let x = # of blue sticks initially

David took out some blue sticks and replaced them with green sticks:

let y = # of sticks that were switched

Then

4x-y = # of blue sticks remaining
x+y = # of green sticks remaining

The number of blue sticks remaining was 3 times the number of green sticks:

4x-y=3(x+y)
4x-y=3x+3y
x=4y

Since x=4y, the initial numbers of sticks were 16y blue and 4y green.

After y sticks were switched from blue to green, the numbers of sticks were 15y blue and 5y green.

At the end, the number of green sticks was 185:

5y=185
y=37

Now we can answer the questions....

ANSWERS:

(a) total number of sticks: 16y+4y initially, or 15y+5y at the end, so 20y = 20(37)=740

(b) number of green sticks initially: 4y=4(37)=148
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