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# Fractions

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Allan, Beth, Charis and Dollah had some paper clips each. The number of paper clips Allan had was 1/2 of the total number of paper clips Beth, Charis and Dollah had. Beth had 1/5 of the total number of paper clips Allan, Charis and Dollah had. Charis had 2/7 of the total number of paper clips Allan, Beth and Dollah had. If Dollah had 145 paper clips, find the total number of paper clips Allan and Beth had altogether.

Jun 5, 2021

#1
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Using the given information, we can write 4 equations.  Let Allan's paper clip count = a, Beth's = b, Charis' = c, and Dollah's = d

$$a=\frac{1}{2}(b+c+d)$$

$$b=\frac{1}{5}(a+c+d)$$

$$c=\frac{2}{7}(a+b+d)$$

$$d=145$$

Substituting d  as 145 into each of the 3 equations, we have

$$a=\frac{1}{2}(b+c+145)$$

$$b=\frac{1}{5}(a+c+145)$$

$$c=\frac{2}{7}(a+b+145)$$

From here, it is simply substitution, elimination, to solve for each of the variables.  In the end, we have

(a, b, c, d) = (174, 87, 116, 145), so Allan+Beth = 174+87 = 261

Jun 5, 2021