every marble in a jar has either a dot, a stripe, or both. the ratio of striped marbles to non striped marbles is 3:1, and the ratio of dotted marbles to non dotted marbles is 2:3. if six marbles have both a dot and a stripe, how many marbles are there all together?
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s=number of striped marbles (includes both striped and dotted)
d=number of dotted marbles (includes both striped and dotted)
6=number of both
d-6=number of non-striped marbles
s-6=number of non-dotted marbles
s/(d-6)=3/1 => s=3(d-6) ...(1)
d/(s-6)=2/3 => 3d=2(s-6) ...(2)
solve for s and d
To check your answer, number of striped is slightly less than twice the dotted ones, or put back you solutions into (1) and (2) and see if the equations hold.
Let the number of marbles with a dot be \(x\) and the number of marbles with a stripe be \(y\). Then the number of non-dotted marbles is y+6 and the number of non-striped marbles is x+6. Employing the given ratios we have\[3(y+6)=x\]\[2(x+6)=3y\]Solve this system for \(x\) and \(y\), and add 6 to the total.