Trail mix

You have 12 cups of granola and 8 1/2 cups of peanuts to make trail mix. What is the greatest number of full batches of trail mix you can make? Each batch of trail mix requires 2 3/4 cups of granola and 1 1/3 cups of peanuts.

Final Answer:

n =  4

Step-by-step explanation:

$$\begin{gathered} A_1=12 \\ {A}_2=8\frac{1}{2}=8+\frac{1}{2}=\frac{8\cdot 2+1}{2}=\frac{16+1}{2}=\frac{17}{2}=8\frac{1}{2}=8.5 \\ {a}_1=2\frac{3}{4}=2+\frac{3}{4}=\frac{2\cdot 4+3}{4}=\frac{8+3}{4}=\frac{11}{4}=2\frac{3}{4}=2.75 \\ {a}_2=1\frac{1}{3}=1+\frac{1}{3}=\frac{1\cdot 3+1}{3}=\frac{3+1}{3}=\frac{4}{3}=1\frac{1}{3}\doteq 1.3333 \\ {n}_1=A_1/a_1=12/\frac{11}{4}=12:\frac{11}{4}=12\cdot \frac{4}{11}=\frac{12\cdot 4}{11}=\frac{48}{11}\doteq 4.3636 \\ {n}_2=A_2/a_2=\frac{17}{2}/\frac{4}{3}=\frac{17}{2}:\frac{4}{3}=\frac{17}{2}\cdot \frac{3}{4}=\frac{17\cdot 3}{2\cdot 4}=\frac{51}{8}=6.375 \\ {n}_1<n_2 \\ m=\min (n_1,n_2)=\min (4.3636,6.375)=\frac{48}{11}=4\frac{4}{11}\doteq 4.3636 \\ n=\lfloor m\rfloor =\lfloor 4.3636\rfloor =4 \end{gathered}$$

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