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Explanation:
Let x and y be two positive whole numbers. They may or may not be equal.
The first bag has the ratio of red to blue as 3:5, which scales up to 3x:5x after multiplying both parts by x. We have 3x red and 5x blue.
The second bag has the ratio 3:2 which scales up to 3y:2y. This bag has 3y red and 2y blue.
We can make a handy table to keep track of all the expressions
[tex]\begin{array}{|c|c|c|}\cline{1-3}\text{ } & \text{Red} & \text{Blue}\\ \cline{1-3}\text{bag 1} & 3x & 5x\\ \cline{1-3}\text{bag 2} & 3y & 2y\\ \cline{1-3}\end{array}[/tex]
Based on that, we can then say:
Effectively, I added down along each column of that table.
Divide the two expressions for R and B to lead to the fraction 11/9, which is directly from the ratio 11:9
So,
(total red)/(total blue) = 11/9
R/B = 11/9
(3x+3y)/(5x+2y) = 11/9
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Let's cross multiply and then solve for y
(3x+3y)/(5x+2y) = 11/9
9(3x+3y) = 11(5x+2y)
27x+27y = 55x+22y
27y-22y = 55x-27x
5y = 28x
y = 28x/5
y = (28/5)x
y = 5.6x
We'll keep this in mind for later.
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Now let:
Recall that we had 3x red and 5x blue for the first bag. That means we have 3x+5x = 8x marbles in the first bag. Therefore, F = 8x.
Also, we had 3y red and 2y blue for the second bag.
So we have S = 3y+2y = 5y
Then we can apply further substitution to say
S = 5y
S = 5(5.6x) .... plug in y = 5.6x
S = 28x
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In short,
for some positive whole number x. We don't have enough info to pin down what x is exactly; however, it doesn't really matter. This is because the x variable will cancel when dividing F over S
F/S = (8x)/(28x) = 8/28 = 2/7
The fraction 2/7 leads directly to the final answer of the ratio 2:7
In other words, the ratio 8x:28x simplifies fully to 2:7 after dividing both parts by 4x.
