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A sum of $2345 was distributed to Aaron, Bill and Cavin. 1/2 of Aaron’s share was 1/3 of Bill’s share. 1/4 of Bill’s share was 1/5 of Cavin’s share. What was the ratio of Aaron’s share to Bill’s share to Cavin’s share? How much did Bill get?

 Feb 5, 2022
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We let Aaron's share be \(a\), Bill's share be \(b\), and Cavin's share be \(c\). From the problem, we get the equations \(\frac{a}{2} = \frac{b}{3}\)\(\frac{b}{4} = \frac{c}{5}\). To answer the first problem, we multiply both sides of the first equation by \(\frac{3}{4}\) to get \(\frac{3a}{4(2)} = \frac{b}{4}= \frac{3a}{8}\). Now, we have \(\frac{3a}{8} = \frac{b}{4} = \frac{c}{5}\). We want to find someting to multiply all of them to cancel out the fractions. The number we need is LCM(4, 8, 5) = 40 so we multiply them all by 40 to get \(\frac{120a}{8} = \frac{40b}{4} = \frac{40c}{5} = 15a = 10b = 8c\) so the answer to the first question is 15 : 10 : 8. 

 

For the second problem, we return to the equation \(\frac{3a}{8} = \frac{b}{4} = \frac{c}{5}\). We start from the equation \(\frac{a}{2} = \frac{b}{3}\). Multiplying all sides by 2 gives us \(a = \frac{2b}{3}\). Then, from the equation \(\frac{b}{4} = \frac{c}{5}\), we multiply both sides by 5 to get \(\frac{5b}{4} = c\). Suubsituting them all into the first thng they told us: \(a+b+c = 2345\) gives us \(\frac{2b}{3} + b + \frac{5b}{4} = \frac{2}{3}b + 1b + \frac{5}{4}b = (\frac{2}{3} + 1 +\frac{5}{4})b = (\frac{8}{12} + \frac{12}{12} + \frac{15}{12})b = \frac{8+12+15}{12}b = \frac{35}{12}b = 2345\). Multiplying by \(\frac{12}{35}\) gives us \(b = \frac{12 * 2345}{35} = \frac{28140}{35} = 804\) so Bill got $804.

 Feb 5, 2022

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