How many pounds of grain need to be ground to have exactly 100 pounds of flour left after paying for the work - 10% of the milling? 

There are no losses during milling

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Think about what part of the payment will be from the final revenue, not the initial revenue.

## Solution

The remaining 100 pounds constitute $90\%$ (that is, $9/10$) of the weight of the grain. Therefore, the weight of the grain was $100 \cdot \frac{10}{9} = 111 \frac{1}{9}$ pounds.

## Answer

$111 \frac{1}{9}$ pounds.

![](https://cdn.mathpix.com/cropped/2024_05_06_9c89379ec21d997cc715g-01.jpg?height=117&width=2038&top_left_y=1136&top_left_x=-1)

For what values of $a$ and $b$ does the expression $p=2a^2-8ab+17b^2-16a-4b+2044$ take its minimum value? What is this value?

## Solution

$p=2(a-2b-4)^2 + 9(b-2)^2 + 1976 \geq 1976$. Equality $p=1976$ is possible if $b=2, a-2b-4=0$, that is, when $a=8, b=2$.

## Answer

For $a=8, b=2; 1976$.
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