Let's denote the total weight of the tomatoes purchased as \( W \) pounds, and the percentage of the tomatoes that were ruined and discarded as \( P \). The cost of the tomatoes is \( 0.80W \) dollars.

The weight of the tomatoes that were not ruined is \( (1 - \frac{P}{100})W \) pounds.

The manager wants to make a profit equal to 10 percent of the cost of the tomatoes, which is \( 0.10 \times 0.80W = 0.08W \) dollars.

The selling price per pound is $0.977777777777778, so the revenue from selling the remaining tomatoes is \( 0.977777777777778 \times (1 - \frac{P}{100})W \) dollars.

To find the percentage of the tomatoes that were ruined, we set the revenue equal to the cost plus the desired profit:

\[ 0.977777777777778 \times (1 - \frac{P}{100})W = 0.80W + 0.08W \]
\[ 0.977777777777778 \times (1 - \frac{P}{100})W = 0.88W \]

Now we can divide both sides by \( W \) to get rid of the weight variable:

\[ 0.977777777777778 \times (1 - \frac{P}{100}) = 0.88 \]

Next, we solve for \( P \):

\[ 1 - \frac{P}{100} = \frac{0.88}{0.977777777777778} \]
\[ 1 - \frac{P}{100} = 0.8999999999999999 \]
\[ \frac{P}{100} = 1 - 0.8999999999999999 \]
\[ \frac{P}{100} = 0.1000000000000001 \]
\[ P = 0.1000000000000001 \times 100 \]
\[ P = 10.00000000000001 \]

So, approximately 10% of the tomatoes were ruined and discarded.
구매한 토마토의 총 무게를 \( W \) 파운드로 표시하고, 토마토 중 망가져 버려진 토마토의 비율을 \( P \)로 표시합니다. 토마토의 비용은 0.80달러입니다.

망가지지 않은 토마토의 무게는 \( (1 - \frac{P}{100})W \ 파운드입니다.

관리자는 토마토 비용의 10%에 해당하는 이윤을 얻고자 하는데, 이는 0.10 \times 0.80W = 0.08W 달러입니다.

1파운드당 판매 가격은 $0.977777777777778이므로 남은 토마토를 판매하여 얻은 수익은 \( 0.977777777777778 \times (1 - \frac{P}{100})W \ 달러입니다.

망가진 토마토의 비율을 구하기 위해 수익을 원가 및 원하는 이익과 같게 설정합니다:

\[ 0.977777777777778 \times (1 - \frac{P}{100})W = 0.80W + 0.08W \]
\[ 0.977777777777778 \times (1 - \frac{P}{100})W = 0.88W \]

이제 양쪽을 \( W \)로 나누어 무게 변수를 제거할 수 있습니다:

\[ 0.977777777777778 \times (1 - \frac{P}{100}) = 0.88 \]

다음으로 \( P \)를 풉니다:

\[ 1 - \frac{P}{100} = \frac{0.88}{0.977777777777778} \]
\[ 1 - \frac{P}{100} = 0.89999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999