Two grandmothers

Two grandmothers went to the market to sell eggs, and they had 100. When they sold all the eggs, they made the same money. The first grandmother said to the second, "If I sold my eggs for your price, I would earn 15 crowns. " The other grandmother replied, "If I sold my eggs for your price, I would get 6 and 2/3 crowns. " How many eggs did each of the bucks have? Try to find the result without solving a complex quadratic equation.

Correct answer:

a =  60
b =  40

Step-by-step explanation:

$$\begin{gathered} a+b=100 \\ a\cdot c_1=b\cdot c_2 \\ a\cdot c_2=15 \\ b\cdot c_1=6+2/3=\frac{20}{3}=6.666666 \\ a\cdot c_1=(100-a)\cdot c_2 \\ a\cdot c_2=15 \\ (100-a)\cdot c_1=6+2/3 \\ {c}_2=15/a \\ a\cdot c_1=(100-a)\cdot (15/a) \\ (100-a)\cdot c_1=6+2/3 \\ {c}_1=(100-a)\cdot (15/a)/a \\ (100-a)\cdot (100-a)\cdot (15/a)/a=6+2/3 \\ (100-a)\cdot (100-a)\cdot 15=a^2\cdot (6+2/3) \\ 3\cdot (100-a)\cdot (100-a)\cdot 15=a^2\cdot (6\cdot 3+2) \\ 3\cdot (100-a)\cdot (100-a)\cdot 15=a^2\cdot (6\cdot 3+2) \\ 25a^2-9000a+450000=0 \\ 25=5^2 \\ 9000=2^3\cdot 3^2\cdot 5^3 \\ 450000=2^4\cdot 3^2\cdot 5^5 \\ \text{GCD}(25,9000,450000)=5^2=25 \\ {a}^2-360a+18000=0 \\ p=1;q=-360;r=18000 \\ D=q^2-4pr=360^2-4\cdot 1\cdot 18000=57600 \\ D>0 \\ {a}_{1,2}=\frac{-q\pm \sqrt{D}}{2p}=\frac{360\pm \sqrt{57600}}{2} \\ {a}_{1,2}=\frac{360\pm 240}{2} \\ {a}_{1,2}=180\pm 120 \\ {a}_1=300 \\ {a}_2=60 \\ 0<a<100 \\ a=a_2=60 \\ {c}_1=(100-a)\cdot (15/a)/a=(100-60)\cdot (15/60)/60=\frac{1}{6}\doteq 0.1667 \\ {c}_2=15/a=15/60=\frac{1}{4}=0.25 \end{gathered}$$

Our quadratic equation calculator calculates it.

$$\begin{gathered} b=100-a=100-60=40 \\ \text{ Verifying Solution: } \\ {L}_1=a\cdot c_1=60\cdot 0.1667=10 \\ {P}_1=b\cdot c_2=40\cdot 0.25=10 \\ {L}_1=P_1 \\ {L}_2=a\cdot c_2=60\cdot 0.25=15 \\ {P}_2=b\cdot c_1=40\cdot 0.1667=\frac{20}{3}=6\frac{2}{3}\doteq 6.6667 \end{gathered}$$

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