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Aminah, Getha and Muthu share a sum of money. The ratio of the amount of money Aminah has to the total amount of money Getha and Muthu have is . The ratio of the amount of money Getha has to the total amount of money of Aminah and Muthu have is . Muthu has more than Aminah. Find the sum of money.
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Q. Aminah, Getha and Muthu share a sum of money. The ratio of the amount of money Aminah has to the total amount of money Getha and Muthu have is . The ratio of the amount of money Getha has to the total amount of money of Aminah and Muthu have is . Muthu has more than Aminah. Find the sum of money.
- Set Ratios: Let's denote the amount of money Aminah has as , Getha as , and Muthu as . According to the problem, we have the following ratios:Aminah's amount to the total amount of Getha and Muthu: Getha's amount to the total amount of Aminah and Muthu: Additionally, we know that Muthu has more than Aminah: $M = A + \$\(51\)
- First Equation: From the first ratio, we can write the equation:\(\newline\)\(A = \frac{1}{4} \times (G + M)\)
- Second Equation: From the second ratio, we can write the equation: \(G = \frac{5}{16} \times (A + M)\)
- Substitute M: Substitute the expression for M from the third piece of information into the first equation:\(\newline\)\(A = \frac{1}{4} \times (G + A + \$(51))\)
- Solve for A: Rearrange the equation to solve for A:\(\newline\)\(A - \frac{1}{4} \cdot A = \frac{1}{4} \cdot G + \frac{1}{4} \cdot (\$)51\)\(\newline\)\(\frac{3}{4} \cdot A = \frac{1}{4} \cdot G + (\$)12.75\)\(\newline\)\(A = \frac{1}{3} \cdot G + (\$)17\)
- Substitute \(A\): Now substitute the expression for \(A\) from the above step into the second equation: \(G = \frac{5}{16} \times \left(\left(\frac{1}{3} \times G + \$(17)\right) + M\right)\)
- Distribute \(5\)/\(16\): Distribute the \(5/16\):\[G = \frac{5}{16} \times \left(\frac{1}{3} \times G\right) + \frac{5}{16} \times (\$17) + \frac{5}{16} \times M\]\[G = \frac{5}{48} \times G + (\$5.3125) + \frac{5}{16} \times M\]
- Rearrange for G: Rearrange the equation to solve for G:\(\newline\)\(G - \frac{5}{48} G = (\$)5.3125 + \frac{5}{16} M\)\(\newline\)\(\frac{43}{48} G = (\$)5.3125 + \frac{5}{16} M\)\(\newline\)\(G = \left(\frac{48}{43}\right) \left((\$)5.3125 + \frac{5}{16} M\right)\)
- Substitute M: Substitute the expression for M \(M = A + (\$)51\) into the equation:\(\newline\)\[G = \left(\frac{48}{43}\right) \times \left((\$)5.3125 + \frac{5}{16} \times (A + (\$)51)\right)\]
- Solve Simultaneously: Now we have two equations with two variables \(A\) and \(G\). We need to solve these equations simultaneously. However, at this point, we realize that we have not used the information that \(M = A + (\$)51\) effectively in our equations. We need to go back and correct this oversight.
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