Apples

60% of first-quality apples were picked in the upper orchard, and 90% of first-quality apples were in the lower orchard. How many% of the first quality were achieved in total if the total harvest was 240 tons of apples and in the lower orchard, the harvest was 40% richer?

Correct answer:

p =  77.5 %

Step-by-step explanation:

$$\begin{gathered} m=240\text{t} \\ (a+b)+(c+d)=m \\ a=0.60\cdot (a+b) \\ c=0.90\cdot (c+d) \\ c+d=1.4\cdot (a+b) \\ (a+b)+(c+d)=240 \\ a=0.60\cdot (a+b) \\ c=0.90\cdot (c+d) \\ c+d=1.4\cdot (a+b) \\ a+b+c+d=240 \\ 0.4a-0.6b=0 \\ 0.1c-0.9d=0 \\ 1.4a+1.4b-c-d=0 \\ Pivot:Row1\leftrightarrow Row4 \\ 1.4a+1.4b-c-d=0 \\ 0.4a-0.6b=0 \\ 0.1c-0.9d=0 \\ a+b+c+d=240 \\ Row2-\frac{0.4}{1.4}\cdot Row1\to Row2 \\ 1.4a+1.4b-c-d=0 \\ -b+0.2857c+0.2857d=0 \\ 0.1c-0.9d=0 \\ a+b+c+d=240 \\ Row4-\frac{1}{1.4}\cdot Row1\to Row4 \\ 1.4a+1.4b-c-d=0 \\ -b+0.2857c+0.2857d=0 \\ 0.1c-0.9d=0 \\ 1.7143c+1.7143d=240 \\ Pivot:Row3\leftrightarrow Row4 \\ 1.4a+1.4b-c-d=0 \\ -b+0.2857c+0.2857d=0 \\ 1.7143c+1.7143d=240 \\ 0.1c-0.9d=0 \\ Row4-\frac{0.1}{1.71428571}\cdot Row3\to Row4 \\ 1.4a+1.4b-c-d=0 \\ -b+0.2857c+0.2857d=0 \\ 1.7143c+1.7143d=240 \\ -d=-14 \\ d=\frac{-14}{-1}=14 \\ c=\frac{240-1.7142857142857d}{1.71428571}=\frac{240-1.71428571\cdot 14}{1.71428571}=126 \\ b=\frac{0-0.28571428571429c-0.28571428571429d}{-1}=\frac{0-0.28571429\cdot 126-0.28571429\cdot 14}{-1}=40 \\ a=\frac{0-1.4b+c+d}{1.4}=\frac{0-1.4\cdot 40+126+14}{1.4}=60 \\ a=60 \\ b=40 \\ c=126 \\ d=14 \\ p=100\cdot \frac{a+c}{m}=100\cdot \frac{60+126}{240}=\frac{155}{2}\%=77.5\% \end{gathered}$$

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