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# Ratio problem

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The ratio of the number of magnets Aspilla and Barry had was 2 : 3. When Aspilla bought 4 more magnets and Barry bought 13 more magnets, the ratio became 1 : 2. Find the number of magnets Barry had in the end.

May 28, 2021

#1
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The ratio of the number of magnets Aspilla and Barry had was 2 : 3.
When Aspilla bought 4 more magnets and Barry bought 13 more magnets,
the ratio became 1 : 2.
Find the number of magnets Barry had in the end.

$$\begin{array}{|rcll|} \hline \dfrac{A}{B} &=& \dfrac{2}{3} \\\\ \mathbf{A} &=& \mathbf{ \dfrac{2}{3}B } \\ \hline \dfrac{A+4}{B+13} &=& \dfrac{1}{2} \quad &|\quad \mathbf{A= \dfrac{2}{3}B } \\\\ \dfrac{\dfrac{2}{3}B+4}{B+13} &=& \dfrac{1}{2} \\\\ 2*\left(\dfrac{2}{3}B+4 \right) &=& B+13 \\\\ \dfrac{4}{3}B+8 &=& B+13 \\\\ \dfrac{4}{3}B-B &=& 5 \\\\ \dfrac{1}{3}B &=& 5 \quad &|\quad *3 \\\\ \mathbf{B} &=& \mathbf{15} \\ \hline A &=& \dfrac{2}{3}B \quad & | \quad \mathbf{B=15} \\\\ A &=& \dfrac{2}{3} * 15 \\\\ \mathbf{A} &=& \mathbf{10} \\ \hline \end{array}$$

The number of magnets Barry had in the end is $$B + 13 = 15+13 = \mathbf{28}$$ May 28, 2021
#2
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The ratio of the number of magnets Aspilla and Barry had was 2 : 3. When Aspilla bought 4 more magnets and Barry bought 13 more magnets, the ratio became 1 : 2. Find the number of magnets Barry had in the end.

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A : B = 2 : 3

(2x + 4) : (3x + 13) = 1 : 2

4x + 8 = 3x + 13

x = 5