A pastry

A pastry chef made some tarts and cupcakes. 2/9 of the tarts were chocolate, and the rest were fruit. 3/7 out of the fruit tarts were lychee tarts, and the rest were peach tarts. There were 96 peach tarts. How many tarts did the pastry chef make altogether?

Final Answer:

n =  216

Step-by-step explanation:

$$\begin{gathered} a=\frac{2}{9}\doteq 0.2222 \\ b=1-a=1-\frac{2}{9}=\frac{1\cdot 9}{9}-\frac{2}{9}=\frac{9}{9}-\frac{2}{9}=\frac{9-2}{9}=\frac{7}{9}\doteq 0.7778 \\ c=\frac{3}{7}\cdot b=\frac{3}{7}\cdot \frac{7}{9}=\frac{3\cdot 7}{7\cdot 9}=\frac{21}{63}=\frac{1}{3}\doteq 0.3333 \\ d=b-c=\frac{7}{9}-\frac{1}{3}=\frac{7}{9}-\frac{3}{9}=\frac{7-3}{9}=\frac{4}{9}\doteq 0.4444 \\ d\cdot n=96 \\ 0.44444444444445\cdot n=96 \\ 0.444444n=96 \\ n=\frac{96}{0.44444444}=216 \\ n=216 \\ \text{ Verifying Solution: } \\ choco=\frac{2}{9}\cdot n=\frac{2}{9}\cdot 216=\frac{2\cdot 216}{9}=\frac{432}{9}=48 \\ fruit=n-choco=216-48=168 \\ {l}_{y}chee=\frac{3}{7}\cdot fruit=\frac{3}{7}\cdot 168=\frac{3\cdot 168}{7}=\frac{504}{7}=72 \\ peach=fruit-l_{y}chee=168-72=96 \end{gathered}$$

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