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Ancient mathematical task from Lion Tolstoy
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<blockquote data-quote="Zorro" data-source="post: 975608" data-attributes="member: 80312"><p>The solution of the problem. In this case, in addition to the main unknown - the number of workers, which we will denote by x - it is convenient to introduce an auxiliary one, namely, the size of the plot mowed by one worker in 1 day; let's denote it by y.</p><p>Although the problem does not require its definition, it will make it easier for us to find the main unknown. Express in terms of x and y the area of a large meadow.</p><p>This meadow was mowed for half a day by x workers; they mowed - x × 1/2 × y = xy/4. In the second half of the day, only half of the group mowed it, that is, x / 2 workers; they mowed x/2 × 1/2 × y = xy/4.</p><p>Since the whole meadow was mown by evening, its area is equal to xy/2 + xy/4 = 3xy/4. Let us now express the area of the smaller meadow in terms of x and y. It was mowed by x/2 workers for half a day and the area was mowed by x/2 × 1/2 × y = xy/4.</p><p>Let us add an uncultivated area just equal to y (the area cultivated by one worker in 1 working day), and we get the area of the smaller meadow: xy/4 + y = (xy + 4y)/4.</p><p>It remains to translate into the language of algebra the phrase: "the first meadow is twice as large as the second" - and the equation is composed: 3xy/4 : (xy + 4y)/4 = 2, or 3xy/ (xy + 4y) = 2. Reduce the fraction on the left side of the equation by y; the auxiliary unknown is thus eliminated, and the equation takes the form 3x/(x + 4) = 2, or 3x = 2x + 8. In the end we get x = 8. There were 8 workers in the group.</p><p></p><p>You can also say this:</p><p></p><p><strong>Let "s" be the unit of area mowed by one worker in one day.</strong></p><p><strong>Let "x" be the number of workers equal to half the group</strong>.</p><p>Then, <strong>2x - is a whole group.</strong></p><p>Then, <strong>xs is the area mowed by half of the group for the whole day.</strong></p><p><strong>xs / 2 - the area that is mowed by half of the group in half a day.</strong></p><p>First half of the day: a whole gang mowed <strong>2 * (xs / 2) = xs of the area from a large meadow.</strong></p><p>Second half of the day: half of the artel mowed xs/2 area out of a large meadow.</p><p>The other half mowed the same area from a small meadow: <strong>xs/2.</strong></p><p>Second day: one worker mowed from a small meadow "s" square in one day.</p><p>Total area of a large meadow: <strong>S1 = xs + xs/2</strong></p><p>Total area of a small meadow: <strong>S2 = xs/2 + s</strong></p><p>Because a large meadow is twice as large as a small one, <strong>S1 = 2S2,</strong></p><p><strong><span style="color: rgb(184, 49, 47)">xs + xs/2 = 2 * (xs/2 + s)</span></strong></p><p><span style="color: rgb(184, 49, 47)"><strong>xs + xs/2 = 2*xs/2 + 2s</strong></span></p><p><span style="color: rgb(184, 49, 47)"><strong>xs - 2s = 2*xs/2 - xs/2</strong></span></p><p><span style="color: rgb(184, 49, 47)"><strong>xs - 2s = xs/2</strong></span></p><p><span style="color: rgb(184, 49, 47)"><strong>2(xs - 2s) = xs</strong></span></p><p><span style="color: rgb(184, 49, 47)"><strong>2xs - 4s = xs</strong></span></p><p><span style="color: rgb(184, 49, 47)"><strong>2xs - xs = 4s </strong></span></p><p><span style="color: rgb(184, 49, 47)"><strong>xs = 4s</strong></span></p><p><span style="color: rgb(184, 49, 47)"><strong>x = 4</strong></span> (workers) - this is half of the group.</p><p><strong><span style="color: rgb(184, 49, 47)">x*2 = 8</span></strong></p><p>Consequently, there were 8 workers in the group.</p></blockquote><p></p>
[QUOTE="Zorro, post: 975608, member: 80312"] The solution of the problem. In this case, in addition to the main unknown - the number of workers, which we will denote by x - it is convenient to introduce an auxiliary one, namely, the size of the plot mowed by one worker in 1 day; let's denote it by y. Although the problem does not require its definition, it will make it easier for us to find the main unknown. Express in terms of x and y the area of a large meadow. This meadow was mowed for half a day by x workers; they mowed - x × 1/2 × y = xy/4. In the second half of the day, only half of the group mowed it, that is, x / 2 workers; they mowed x/2 × 1/2 × y = xy/4. Since the whole meadow was mown by evening, its area is equal to xy/2 + xy/4 = 3xy/4. Let us now express the area of the smaller meadow in terms of x and y. It was mowed by x/2 workers for half a day and the area was mowed by x/2 × 1/2 × y = xy/4. Let us add an uncultivated area just equal to y (the area cultivated by one worker in 1 working day), and we get the area of the smaller meadow: xy/4 + y = (xy + 4y)/4. It remains to translate into the language of algebra the phrase: "the first meadow is twice as large as the second" - and the equation is composed: 3xy/4 : (xy + 4y)/4 = 2, or 3xy/ (xy + 4y) = 2. Reduce the fraction on the left side of the equation by y; the auxiliary unknown is thus eliminated, and the equation takes the form 3x/(x + 4) = 2, or 3x = 2x + 8. In the end we get x = 8. There were 8 workers in the group. You can also say this: [B]Let "s" be the unit of area mowed by one worker in one day. Let "x" be the number of workers equal to half the group[/B]. Then, [B]2x - is a whole group.[/B] Then, [B]xs is the area mowed by half of the group for the whole day. xs / 2 - the area that is mowed by half of the group in half a day.[/B] First half of the day: a whole gang mowed [B]2 * (xs / 2) = xs of the area from a large meadow.[/B] Second half of the day: half of the artel mowed xs/2 area out of a large meadow. The other half mowed the same area from a small meadow: [B]xs/2.[/B] Second day: one worker mowed from a small meadow "s" square in one day. Total area of a large meadow: [B]S1 = xs + xs/2[/B] Total area of a small meadow: [B]S2 = xs/2 + s[/B] Because a large meadow is twice as large as a small one, [B]S1 = 2S2, [COLOR=rgb(184, 49, 47)]xs + xs/2 = 2 * (xs/2 + s)[/COLOR][/B] [COLOR=rgb(184, 49, 47)][B]xs + xs/2 = 2*xs/2 + 2s xs - 2s = 2*xs/2 - xs/2 xs - 2s = xs/2 2(xs - 2s) = xs 2xs - 4s = xs 2xs - xs = 4s xs = 4s x = 4[/B][/COLOR] (workers) - this is half of the group. [B][COLOR=rgb(184, 49, 47)]x*2 = 8[/COLOR][/B] Consequently, there were 8 workers in the group. [/QUOTE]
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