I had a total of 129 marbles in 3 boxes at first. Then I threw away 2/3 of the marbles from Box 1, added 15 marbles to Box 2, and added marbles to Box 3 until the numbers in box 3 tripled. As a result the number of marbles in box 1 to the number in Box 2 to the number of marbles in box 3 became 2:3:9 Wat is the total number of marbles in the 3 boxes?
let the number of marbles be as follows:
Box1 x
Box3 y
Box2 129-x-y
after the described action
box1 = x/3 (you took 2/3 away, so 1/3 remains)
box2 = 129-x-y + 15 = 144-x-y
box3 = 3y
then (x/3) : 144-x-y : 3y = 2:3:9
so then (x/3)/(144-x-y) = 2/3
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3x + 2y = 288 (equ#1)
also (x/3)/3y = 2/9
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x = 2y (equ#2) sub that into equ#1 to get
y = 36
then x = 72
check:
original
box1 72
box2 21
box3 36 --- total 129
after action:
box1 = 24
box2 = 36
box3 = 108
24:36:108 = 2:3:9
After: 24:36:108 (total 168)
Before: 72:21:36 (total 129)
Method used: trial and error
Start with "after" combinations with the right ratios, determine the required operations and "before" compositions, and see which one adds up to 129 "after".
It only took three attempts.
simply just wat is says above 168
To find the total number of marbles in the three boxes, we will solve this problem step by step.
Let's begin by assigning variables to each box:
- Let x represent the number of marbles in Box 1 initially.
- Let y represent the number of marbles in Box 2 initially.
- Let z represent the number of marbles in Box 3 initially.
According to the given information, initially Box 1 had 2/3 of the total marbles in the three boxes. Therefore, we can write the equation:
x = 2/3 * (x + y + z)
Next, we know that marbles were added to Box 2, resulting in it having 15 more marbles:
y + 15 = y
Furthermore, marbles were added to Box 3 until the number of marbles in it tripled:
3z = z
Finally, the ratio of the marbles in Box 1, Box 2, and Box 3 is given as 2:3:9, which can be expressed as:
x:y:z = 2:3:9
Now we can solve these equations simultaneously to find the values of x, y, and z.
First, simplify the equation for Box 1:
x = 2/3 * (x + y + z)
Multiplying through by 3 to eliminate the fraction:
3x = 2(x + y + z)
3x = 2x + 2y + 2z
3x - 2x = 2y + 2z
x = 2y + 2z
Rewriting the ratio equation for Box 1, Box 2, and Box 3:
x:y:z = 2:3:9
From the equation x = 2y + 2z, substitute it into the ratio equation:
(2y + 2z):y:z = 2:3:9
Since the ratio of x:y:z is equal to the ratio 2:3:9, we can conclude that 2y + 2z = 2.
Now, substitute x = 2y + 2z back into the equation x = 2y + 2z:
2y + 2z = 2
Simplifying this equation:
2(y + z) = 2
y + z = 1
Since y + z = 1, we can assign values to y and z to satisfy this equation. Let's set y = 1 and z = 0.
Now, substitute the values of y = 1 and z = 0 back into the equation x = 2y + 2z:
x = 2(1) + 2(0)
x = 2
Therefore, we have found that x = 2, y = 1, and z = 0.
To calculate the total number of marbles in the three boxes, you add up the values of x, y, and z:
Total = x + y + z
Total = 2 + 1 + 0
Total = 3
Therefore, the total number of marbles in the three boxes is 3.