Gift baskets

Alessia and Alexia made some gift baskets to sell. 2/5 of them were Yu-Gi-oh themed, and the rest were Barbie-themed. They sold 3/4 of the yu-gi-oh-themed baskets and 7/12 of the Barbie-themed baskets. They remained with 56 gift baskets. How many gift baskets did they make?

Correct answer:

x =  160

Step-by-step explanation:

$$\begin{gathered} \frac{3}{4}\cdot \frac{2}{5}\cdot x+(1-\frac{2}{5})\cdot \frac{7}{12}\cdot x+56=x \\ 3/4\cdot 2/5\cdot x+(1-2/5)\cdot 7/12\cdot x+56=x \\ 21x=3360 \\ x=\frac{3360}{21}=160 \\ x=160 \\ \text{ Verifying Solution: } \\ A=\frac{2}{5}\cdot x=\frac{2}{5}\cdot 160=\frac{2\cdot 160}{5}=\frac{320}{5}=64 \\ B=x-A=160-64=96 \\ a=\frac{3}{4}\cdot A=\frac{3}{4}\cdot 64=\frac{3\cdot 64}{4}=\frac{192}{4}=48 \\ b=\frac{7}{12}\cdot B=\frac{7}{12}\cdot 96=\frac{7\cdot 96}{12}=\frac{672}{12}=56 \\ r=x-a-b=160-48-56=56 \end{gathered}$$

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