Kilograms 81234

The price of 1 kg of pears is 7 crowns higher than the price of 1 kg of apples. The seller sold 2 kg more apples than pears. He received the same amount for pears and apples, namely 420 CZK. How many kilograms of apples and how many kg of pears did he sell?

Correct answer:

a =  12 kg
b =  10 kg

Step-by-step explanation:

$$\begin{gathered} a\cdot c=420 \\ b\cdot (c+7)=420 \\ a=2+b \\ a\cdot c=420 \\ (a-2)\cdot (c+7)=420 \\ (a-2)\cdot (420/a+7)=420 \\ (a-2)\cdot (420+7a)=420a \\ (a-2)\cdot (420+7a)=420a \\ 7a^2-14a-840=0 \\ 7...\text{ prime number} \\ 14=2\cdot 7 \\ 840=2^3\cdot 3\cdot 5\cdot 7 \\ \text{GCD}(7,14,840)=7 \\ {a}^2-2a-120=0 \\ p=1;q=-2;r=-120 \\ D=q^2-4pr=2^2-4\cdot 1\cdot (-120)=484 \\ D>0 \\ {a}_{1,2}=\frac{-q\pm \sqrt{D}}{2p}=\frac{2\pm \sqrt{484}}{2} \\ {a}_{1,2}=\frac{2\pm 22}{2} \\ {a}_{1,2}=1\pm 11 \\ {a}_1=12 \\ {a}_2=-10 \\ a=a_1=12=12\text{kg} \end{gathered}$$

Our quadratic equation calculator calculates it.

$$\begin{gathered} b=a-2=12-2=10=10\text{kg} \\ \text{ Verifying Solution: } \\ {c}_1=420/a=420/12=35\text{Kc/kg} \\ {c}_2=420/b=420/10=42\text{Kc/kg} \end{gathered}$$

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