Sales of products

For 80 pieces of two quality products a total sales is 175 Eur. If the first quality product was sold for n EUR per piece (n natural number) and the second quality product after 2 EUR per piece. How many pieces of the first quality were sold?

Correct result:

a₁ =  15
a₂ =  5
a₃ =  3
a₄ =  1

Solution:

$$\begin{gathered} a+b=80 \\ n\cdot a+2b=175 \\ n\cdot a+2\cdot (80-a)=175 \\ n\cdot a+160-2\cdot a=175 \\ a(n-2)=15 \\ {n}_1=3 \\ {a}_1=15\mathrm{/}(n_1-2)=15\mathrm{/}(3-2)=15 \end{gathered}$$

$$\begin{gathered} n_2=5 \\ {a}_2=15\mathrm{/}(n_2-2)=15\mathrm{/}(5-2)=5 \end{gathered}$$

$$\begin{gathered} n_3=7 \\ {a}_3=15\mathrm{/}(n_3-2)=15\mathrm{/}(7-2)=3 \end{gathered}$$

$$\begin{gathered} n_4=17 \\ {a}_4=15\mathrm{/}(n_4-2)=15\mathrm{/}(17-2)=1 \end{gathered}$$

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