Grandmother and grandfather

Grandmother baked cakes. Grandfather ate half, then a quarter of the rest ate Peter, and Paul ate half of the rest. There were six cakes left for parents. How many cakes were made by the grandmother?

Final Answer:

x =  32

Step-by-step explanation:

$$\begin{gathered} x-\frac{x}{2}-\frac{1}{4}\cdot \frac{x}{2}-\frac{1}{2}\cdot (x-\frac{x}{2}-\frac{1}{4}\cdot \frac{x}{2})=6 \\ x-\frac{x}{2}-\frac{1}{8}\cdot x-\frac{1}{2}\cdot \frac{3}{8}\cdot x=6 \\ \frac{x}{2}-\frac{1}{8}\cdot x-\frac{3}{16}\cdot x=6 \\ \frac{3}{8}\cdot x-\frac{3}{16}\cdot x=6 \\ \frac{3}{8}\cdot x-\frac{3}{16}\cdot x=6 \\ \frac{3}{16}\cdot x=6 \\ x=6\cdot \frac{16}{3}=\frac{6\cdot 16}{3}=\frac{96}{3}=32 \\ \text{ Verifying Solution: } \\ a=x/2=32/2=16 \\ b=(x-a)/4=(32-16)/4=4 \\ c=(x-a-b)/2=(32-16-4)/2=6 \\ d=6 \\ S=a+b+c+d=16+4+6+6=32 \end{gathered}$$

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