Towels Price Quantity

We bought towels for CZK 432. If there were six more towels for the same money, everyone would be 66 CZK cheaper. How many towels were there?

Final Answer:

n =  4

Step-by-step explanation:

$$\begin{gathered} n\cdot c=432 \\ (n+6)\cdot (c-66)=432 \\ 432/(n+6)+66=c \\ n\cdot (432/(n+6)+66)=432 \\ n\cdot (432+66\cdot (n+6))=432\cdot (n+6) \\ n\cdot (432+66\cdot (n+6))=432\cdot (n+6) \\ 66n^2+396n-2592=0 \\ 66=2\cdot 3\cdot 11 \\ 396=2^2\cdot 3^2\cdot 11 \\ 2592=2^5\cdot 3^4 \\ \text{GCD}(66,396,2592)=2\cdot 3=6 \\ 11n^2+66n-432=0 \\ a=11;b=66;c=-432 \\ D=b^2-4ac=66^2-4\cdot 11\cdot (-432)=23364 \\ D>0 \\ {n}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{-66\pm \sqrt{23364}}{22}=\frac{-66\pm 6\sqrt{649}}{22} \\ {n}_{1,2}=-3\pm 6.947858 \\ {n}_1=3.947857747 \\ {n}_2=-9.947857747 \\ n>0 \\ n=[n_1]=[3.9479]=4 \\ c=\frac{432}{n+6}+66=\frac{432}{4+6}+66=\frac{546}{5}=109\frac{1}{5}=109.2\text{kc} \end{gathered}$$

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