{"id":4733,"date":"2025-11-04T19:22:29","date_gmt":"2025-11-04T22:22:29","guid":{"rendered":"https:\/\/colegadeclasse.com.br\/blog\/?p=4733"},"modified":"2025-11-04T19:24:55","modified_gmt":"2025-11-04T22:24:55","slug":"variacao-de-grandezas-razao-proporcao-e-regra-de-tres","status":"publish","type":"post","link":"https:\/\/colegadeclasse.com.br\/blog\/2025\/11\/04\/variacao-de-grandezas-razao-proporcao-e-regra-de-tres\/","title":{"rendered":"Varia\u00e7\u00e3o de Grandezas: Raz\u00e3o, Propor\u00e7\u00e3o e Regra de Tr\u00eas"},"content":{"rendered":"<div style=\"display:flex; gap:10px;justify-content:flex-end\" class=\"wps-pgfw-pdf-generate-icon__wrapper-frontend\">\n\t\t<a  href=\"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/posts\/4733?action=genpdf&amp;id=4733\" class=\"pgfw-single-pdf-download-button\" ><img src=\"https:\/\/colegadeclasse.com.br\/blog\/wp-content\/plugins\/pdf-generator-for-wp\/admin\/src\/images\/PDF_Tray.svg\" title=\"Gerar PDF  \" style=\"width:auto; height:45px;\"><\/a>\n\t\t<\/div>\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p class=\"\">A varia\u00e7\u00e3o de grandezas \u00e9 um dos t\u00f3picos mais recorrentes em provas de concursos p\u00fablicos, aparecendo em quest\u00f5es de racioc\u00ednio l\u00f3gico, matem\u00e1tica e at\u00e9 em situa\u00e7\u00f5es pr\u00e1ticas de c\u00e1lculos administrativos. Dominar raz\u00e3o, propor\u00e7\u00e3o e regra de tr\u00eas \u00e9 fundamental para resolver problemas do dia a dia e garantir pontos preciosos em sua prova.<\/p>\n\n\n\n<p class=\"\">Neste artigo, voc\u00ea encontrar\u00e1 uma explica\u00e7\u00e3o completa, desde os conceitos b\u00e1sicos at\u00e9 quest\u00f5es avan\u00e7adas no estilo dos principais concursos do Brasil.<\/p>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h2 class=\"wp-block-heading\">1. RAZ\u00c3O E PROPOR\u00c7\u00c3O<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">1.1 O que \u00e9 Raz\u00e3o?<\/h3>\n\n\n\n<p class=\"\"><strong>Raz\u00e3o<\/strong> \u00e9 a compara\u00e7\u00e3o entre duas grandezas atrav\u00e9s de uma divis\u00e3o. \u00c9 expressa na forma de fra\u00e7\u00e3o ou divis\u00e3o.<\/p>\n\n\n\n<p class=\"\"><strong>F\u00f3rmula:<\/strong> <\/p>\n\n\n\n<p class=\"\">Onde:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\">a = antecedente<\/li>\n\n\n\n<li class=\"\">b= consequente (sempre diferente de zero)<\/li>\n<\/ul>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Exemplos Pr\u00e1ticos:<\/h4>\n\n\n\n<p class=\"\"><strong>Exemplo 1:<\/strong> Em uma sala h\u00e1 20 homens e 30 mulheres. Qual a raz\u00e3o entre homens e mulheres?<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$\\text{Raz\u00e3o} = \\frac{a}{b} \\text{ ou } a:b$$<script src=\"https:\/\/colegadeclasse.com.br\/blog\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/colegadeclasse.com.br\/blog\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<p class=\"\">Interpreta\u00e7\u00e3o: Para cada 2 homens, h\u00e1 3 mulheres.<\/p>\n\n\n\n<p class=\"\"><strong>Exemplo 2:<\/strong> Um carro percorre 150 km em 2 horas. Qual a raz\u00e3o entre dist\u00e2ncia e tempo (velocidade m\u00e9dia)?<\/p>\n\n\n\n<p class=\"\"><span class=\"MathJax_Preview\" style=\"color: inherit;\"><\/span><span class=\"mjx-chtml MJXc-display\" style=\"text-align: center;\"><span id=\"MathJax-Element-4-Frame\" class=\"mjx-chtml MathJax_CHTML\" tabindex=\"0\" data-mathml=\"<math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;<mtext&gt;Velocidade<\/mtext&gt;<mo&gt;=<\/mo&gt;<mfrac&gt;<mrow&gt;<mn&gt;150<\/mn&gt;<mtext&gt;&#xA0;km<\/mtext&gt;<\/mrow&gt;<mrow&gt;<mn&gt;2<\/mn&gt;<mtext&gt;&#xA0;h<\/mtext&gt;<\/mrow&gt;<\/mfrac&gt;<mo&gt;=<\/mo&gt;<mn&gt;75<\/mn&gt;<mtext&gt;&#xA0;km\/h<\/mtext&gt;<\/math&gt;\" role=\"presentation\" style=\"font-size: 113%; text-align: center; position: relative;\"><span id=\"MJXc-Node-34\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-35\" class=\"mjx-mrow\"><span id=\"MJXc-Node-36\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">Velocidade<\/span><\/span><span id=\"MJXc-Node-37\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.101em; padding-bottom: 0.297em;\">=<\/span><\/span><span id=\"MJXc-Node-38\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\" style=\"width: 3.311em; padding: 0px 0.12em;\"><span class=\"mjx-numerator\" style=\"width: 3.311em; top: -1.396em;\"><span id=\"MJXc-Node-39\" class=\"mjx-mrow\"><span id=\"MJXc-Node-40\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">150<\/span><\/span><span id=\"MJXc-Node-41\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">\u00a0km<\/span><\/span><\/span><\/span><span class=\"mjx-denominator\" style=\"width: 3.311em; bottom: -0.711em;\"><span id=\"MJXc-Node-42\" class=\"mjx-mrow\"><span id=\"MJXc-Node-43\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">2<\/span><\/span><span id=\"MJXc-Node-44\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">\u00a0h<\/span><\/span><\/span><\/span><span class=\"mjx-line\" style=\"border-bottom: 1.3px solid; top: -0.281em; width: 3.311em;\"><\/span><\/span><span class=\"mjx-vsize\" style=\"height: 2.106em; vertical-align: -0.711em;\"><\/span><\/span><span id=\"MJXc-Node-45\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.101em; padding-bottom: 0.297em;\">=<\/span><\/span><span id=\"MJXc-Node-46\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">75<\/span><\/span><span id=\"MJXc-Node-47\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.445em; padding-bottom: 0.592em;\">\u00a0km\/h<\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML MJX_Assistive_MathML_Block\" role=\"presentation\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtext>Velocidade<\/mtext><mo>=<\/mo><mfrac><mrow><mn>150<\/mn><mtext>\u00a0km<\/mtext><\/mrow><mrow><mn>2<\/mn><mtext>\u00a0h<\/mtext><\/mrow><\/mfrac><mo>=<\/mo><mn>75<\/mn><mtext>\u00a0km\/h<\/mtext><\/math><\/span><\/span><\/span><script>\\text{Velocidade} = \\frac{150 \\text{ km}}{2 \\text{ h}} = 75 \\text{ km\/h}<\/script><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1.2 Tipos Especiais de Raz\u00e3o<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">a) Escala<\/h4>\n\n\n\n<p class=\"\">Raz\u00e3o entre a medida no desenho e a medida real.<\/p>\n\n\n\n<p class=\"\"><strong>Exemplo:<\/strong> Uma planta baixa est\u00e1 na escala 1:100. Isso significa que 1 cm no desenho corresponde a 100 cm (1 metro) na realidade.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">b) Velocidade M\u00e9dia<\/h4>\n\n\n\n<p class=\"\">Raz\u00e3o entre dist\u00e2ncia percorrida e tempo gasto.<\/p>\n\n\n\n<p class=\"\"><span class=\"MathJax_Preview\" style=\"color: inherit;\"><\/span><span class=\"mjx-chtml MJXc-display\" style=\"text-align: center;\"><span id=\"MathJax-Element-5-Frame\" class=\"mjx-chtml MathJax_CHTML\" tabindex=\"0\" data-mathml=\"<math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;<msub&gt;<mi&gt;v<\/mi&gt;<mi&gt;m<\/mi&gt;<\/msub&gt;<mo&gt;=<\/mo&gt;<mfrac&gt;<mrow&gt;<mi mathvariant=&quot;normal&quot;&gt;&#x0394;<\/mi&gt;<mi&gt;S<\/mi&gt;<\/mrow&gt;<mrow&gt;<mi mathvariant=&quot;normal&quot;&gt;&#x0394;<\/mi&gt;<mi&gt;t<\/mi&gt;<\/mrow&gt;<\/mfrac&gt;<\/math&gt;\" role=\"presentation\" style=\"font-size: 113%; text-align: center; position: relative;\"><span id=\"MJXc-Node-48\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-49\" class=\"mjx-mrow\"><span id=\"MJXc-Node-50\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-51\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.199em; padding-bottom: 0.297em;\">v<\/span><\/span><\/span><span class=\"mjx-sub\" style=\"font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;\"><span id=\"MJXc-Node-52\" class=\"mjx-mi\" style=\"\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.199em; padding-bottom: 0.297em;\">m<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-53\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.101em; padding-bottom: 0.297em;\">=<\/span><\/span><span id=\"MJXc-Node-54\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\" style=\"width: 1.678em; padding: 0px 0.12em;\"><span class=\"mjx-numerator\" style=\"width: 1.678em; top: -1.418em;\"><span id=\"MJXc-Node-55\" class=\"mjx-mrow\"><span id=\"MJXc-Node-56\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.445em; padding-bottom: 0.347em;\">\u0394<\/span><\/span><span id=\"MJXc-Node-57\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.494em; padding-bottom: 0.297em; padding-right: 0.032em;\">S<\/span><\/span><\/span><\/span><span class=\"mjx-denominator\" style=\"width: 1.678em; bottom: -0.741em;\"><span id=\"MJXc-Node-58\" class=\"mjx-mrow\"><span id=\"MJXc-Node-59\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.445em; padding-bottom: 0.347em;\">\u0394<\/span><\/span><span id=\"MJXc-Node-60\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.396em; padding-bottom: 0.297em;\">t<\/span><\/span><\/span><\/span><span class=\"mjx-line\" style=\"border-bottom: 1.3px solid; top: -0.281em; width: 1.678em;\"><\/span><\/span><span class=\"mjx-vsize\" style=\"height: 2.158em; vertical-align: -0.741em;\"><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML MJX_Assistive_MathML_Block\" role=\"presentation\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><msub><mi>v<\/mi><mi>m<\/mi><\/msub><mo>=<\/mo><mfrac><mrow><mi mathvariant=\"normal\">\u0394<\/mi><mi>S<\/mi><\/mrow><mrow><mi mathvariant=\"normal\">\u0394<\/mi><mi>t<\/mi><\/mrow><\/mfrac><\/math><\/span><\/span><\/span><script>v_m = \\frac{\\Delta S}{\\Delta t}<\/script><\/p>\n\n\n\n<h4 class=\"wp-block-heading\">c) Densidade Demogr\u00e1fica<\/h4>\n\n\n\n<p class=\"\">Raz\u00e3o entre popula\u00e7\u00e3o e \u00e1rea territorial.<\/p>\n\n\n\n<p class=\"\">$$d = \\frac{\\text{popula\u00e7\u00e3o}}{\\text{\u00e1rea}}$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1.3 O que \u00e9 Propor\u00e7\u00e3o?<\/h3>\n\n\n\n<p class=\"\"><strong>Propor\u00e7\u00e3o<\/strong> \u00e9 a igualdade entre duas raz\u00f5es.<\/p>\n\n\n\n<p class=\"\"><strong>F\u00f3rmula:<\/strong> <span class=\"MathJax_Preview\" style=\"color: inherit;\"><\/span><span class=\"mjx-chtml MJXc-display\" style=\"text-align: center;\"><span id=\"MathJax-Element-7-Frame\" class=\"mjx-chtml MathJax_CHTML\" tabindex=\"0\" data-mathml=\"<math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;<mfrac&gt;<mi&gt;a<\/mi&gt;<mi&gt;b<\/mi&gt;<\/mfrac&gt;<mo&gt;=<\/mo&gt;<mfrac&gt;<mi&gt;c<\/mi&gt;<mi&gt;d<\/mi&gt;<\/mfrac&gt;<mtext&gt;&#xA0;ou&#xA0;<\/mtext&gt;<mi&gt;a<\/mi&gt;<mo&gt;:<\/mo&gt;<mi&gt;b<\/mi&gt;<mo&gt;=<\/mo&gt;<mi&gt;c<\/mi&gt;<mo&gt;:<\/mo&gt;<mi&gt;d<\/mi&gt;<\/math&gt;\" role=\"presentation\" style=\"font-size: 113%; text-align: center; position: relative;\"><span id=\"MJXc-Node-68\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-69\" class=\"mjx-mrow\"><span id=\"MJXc-Node-70\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\" style=\"width: 0.729em; padding: 0px 0.12em;\"><span class=\"mjx-numerator\" style=\"width: 0.729em; top: -1.143em;\"><span id=\"MJXc-Node-71\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.199em; padding-bottom: 0.297em;\">a<\/span><\/span><\/span><span class=\"mjx-denominator\" style=\"width: 0.729em; bottom: -0.722em;\"><span id=\"MJXc-Node-72\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.494em; padding-bottom: 0.297em;\">b<\/span><\/span><\/span><span class=\"mjx-line\" style=\"border-bottom: 1.3px solid; top: -0.281em; width: 0.729em;\"><\/span><\/span><span class=\"mjx-vsize\" style=\"height: 1.864em; vertical-align: -0.722em;\"><\/span><\/span><span id=\"MJXc-Node-73\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.101em; padding-bottom: 0.297em;\">=<\/span><\/span><span id=\"MJXc-Node-74\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\" style=\"width: 0.723em; padding: 0px 0.12em;\"><span class=\"mjx-numerator\" style=\"width: 0.723em; top: -1.144em;\"><span id=\"MJXc-Node-75\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.199em; padding-bottom: 0.297em;\">c<\/span><\/span><\/span><span class=\"mjx-denominator\" style=\"width: 0.723em; bottom: -0.721em;\"><span id=\"MJXc-Node-76\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.494em; padding-bottom: 0.297em; padding-right: 0.003em;\">d<\/span><\/span><\/span><span class=\"mjx-line\" style=\"border-bottom: 1.3px solid; top: -0.281em; width: 0.723em;\"><\/span><\/span><span class=\"mjx-vsize\" style=\"height: 1.864em; vertical-align: -0.721em;\"><\/span><\/span><span id=\"MJXc-Node-77\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.15em; padding-bottom: 0.347em;\">\u00a0ou\u00a0<\/span><\/span><span id=\"MJXc-Node-78\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.199em; padding-bottom: 0.297em;\">a<\/span><\/span><span id=\"MJXc-Node-79\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.15em; padding-bottom: 0.347em;\">:<\/span><\/span><span id=\"MJXc-Node-80\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.494em; padding-bottom: 0.297em;\">b<\/span><\/span><span id=\"MJXc-Node-81\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.101em; padding-bottom: 0.297em;\">=<\/span><\/span><span id=\"MJXc-Node-82\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.199em; padding-bottom: 0.297em;\">c<\/span><\/span><span id=\"MJXc-Node-83\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.15em; padding-bottom: 0.347em;\">:<\/span><\/span><span id=\"MJXc-Node-84\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.494em; padding-bottom: 0.297em; padding-right: 0.003em;\">d<\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML MJX_Assistive_MathML_Block\" role=\"presentation\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mfrac><mi>a<\/mi><mi>b<\/mi><\/mfrac><mo>=<\/mo><mfrac><mi>c<\/mi><mi>d<\/mi><\/mfrac><mtext>\u00a0ou\u00a0<\/mtext><mi>a<\/mi><mo>:<\/mo><mi>b<\/mi><mo>=<\/mo><mi>c<\/mi><mo>:<\/mo><mi>d<\/mi><\/math><\/span><\/span><\/span><script>\\frac{a}{b} = \\frac{c}{d} \\text{ ou } a:b = c:d<\/script><\/p>\n\n\n\n<p class=\"\">L\u00ea-se: &#8220;a est\u00e1 para b, assim como c est\u00e1 para d&#8221;<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Propriedade Fundamental das Propor\u00e7\u00f5es<\/h4>\n\n\n\n<p class=\"\">$$a \\times d = b \\times c$$<\/p>\n\n\n\n<p class=\"\">Os <strong>extremos<\/strong> (a e d) multiplicados s\u00e3o iguais aos <strong>meios<\/strong> (b e c) multiplicados.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Exemplo:<\/h4>\n\n\n\n<p class=\"\">Verificar se os n\u00fameros 2, 3, 4 e 6 formam uma propor\u00e7\u00e3o:<\/p>\n\n\n\n<p class=\"\"><span class=\"MathJax_Preview\" style=\"color: inherit;\"><\/span><span class=\"mjx-chtml MJXc-display\" style=\"text-align: center;\"><span id=\"MathJax-Element-9-Frame\" class=\"mjx-chtml MathJax_CHTML\" tabindex=\"0\" data-mathml=\"<math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;<mfrac&gt;<mn&gt;2<\/mn&gt;<mn&gt;3<\/mn&gt;<\/mfrac&gt;<mo&gt;=<\/mo&gt;<mfrac&gt;<mn&gt;4<\/mn&gt;<mn&gt;6<\/mn&gt;<\/mfrac&gt;<\/math&gt;\" role=\"presentation\" style=\"font-size: 113%; text-align: center; position: relative;\"><span id=\"MJXc-Node-94\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-95\" class=\"mjx-mrow\"><span id=\"MJXc-Node-96\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\" style=\"width: 0.7em; padding: 0px 0.12em;\"><span class=\"mjx-numerator\" style=\"width: 0.7em; top: -1.368em;\"><span id=\"MJXc-Node-97\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">2<\/span><\/span><\/span><span class=\"mjx-denominator\" style=\"width: 0.7em; bottom: -0.733em;\"><span id=\"MJXc-Node-98\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">3<\/span><\/span><\/span><span class=\"mjx-line\" style=\"border-bottom: 1.3px solid; top: -0.281em; width: 0.7em;\"><\/span><\/span><span class=\"mjx-vsize\" style=\"height: 2.1em; vertical-align: -0.733em;\"><\/span><\/span><span id=\"MJXc-Node-99\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.101em; padding-bottom: 0.297em;\">=<\/span><\/span><span id=\"MJXc-Node-100\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\" style=\"width: 0.7em; padding: 0px 0.12em;\"><span class=\"mjx-numerator\" style=\"width: 0.7em; top: -1.379em;\"><span id=\"MJXc-Node-101\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">4<\/span><\/span><\/span><span class=\"mjx-denominator\" style=\"width: 0.7em; bottom: -0.733em;\"><span id=\"MJXc-Node-102\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">6<\/span><\/span><\/span><span class=\"mjx-line\" style=\"border-bottom: 1.3px solid; top: -0.281em; width: 0.7em;\"><\/span><\/span><span class=\"mjx-vsize\" style=\"height: 2.111em; vertical-align: -0.733em;\"><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML MJX_Assistive_MathML_Block\" role=\"presentation\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mfrac><mn>2<\/mn><mn>3<\/mn><\/mfrac><mo>=<\/mo><mfrac><mn>4<\/mn><mn>6<\/mn><\/mfrac><\/math><\/span><\/span><\/span><script>\\frac{2}{3} = \\frac{4}{6}<\/script><\/p>\n\n\n\n<p class=\"\">Verificando: 2 x 6 = 12  e 3 x 4 = 12$ \u2713<\/p>\n\n\n\n<p class=\"\">A propor\u00e7\u00e3o \u00e9 verdadeira!<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1.4 Propriedades das Propor\u00e7\u00f5es<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">Propriedade 1: Troca dos Meios<\/h4>\n\n\n\n<p class=\"\"><span class=\"MathJax_Preview\" style=\"color: inherit;\"><\/span><span class=\"mjx-chtml MJXc-display\" style=\"text-align: center;\"><span id=\"MathJax-Element-10-Frame\" class=\"mjx-chtml MathJax_CHTML\" tabindex=\"0\" data-mathml=\"<math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;<mfrac&gt;<mi&gt;a<\/mi&gt;<mi&gt;b<\/mi&gt;<\/mfrac&gt;<mo&gt;=<\/mo&gt;<mfrac&gt;<mi&gt;c<\/mi&gt;<mi&gt;d<\/mi&gt;<\/mfrac&gt;<mo stretchy=&quot;false&quot;&gt;&#x21D2;<\/mo&gt;<mfrac&gt;<mi&gt;a<\/mi&gt;<mi&gt;c<\/mi&gt;<\/mfrac&gt;<mo&gt;=<\/mo&gt;<mfrac&gt;<mi&gt;b<\/mi&gt;<mi&gt;d<\/mi&gt;<\/mfrac&gt;<\/math&gt;\" role=\"presentation\" style=\"font-size: 113%; text-align: center; position: relative;\"><span id=\"MJXc-Node-103\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-104\" class=\"mjx-mrow\"><span id=\"MJXc-Node-105\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\" style=\"width: 0.729em; padding: 0px 0.12em;\"><span class=\"mjx-numerator\" style=\"width: 0.729em; top: -1.143em;\"><span id=\"MJXc-Node-106\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.199em; padding-bottom: 0.297em;\">a<\/span><\/span><\/span><span class=\"mjx-denominator\" style=\"width: 0.729em; bottom: -0.722em;\"><span id=\"MJXc-Node-107\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.494em; padding-bottom: 0.297em;\">b<\/span><\/span><\/span><span class=\"mjx-line\" style=\"border-bottom: 1.3px solid; top: -0.281em; width: 0.729em;\"><\/span><\/span><span class=\"mjx-vsize\" style=\"height: 1.864em; vertical-align: -0.722em;\"><\/span><\/span><span id=\"MJXc-Node-108\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.101em; padding-bottom: 0.297em;\">=<\/span><\/span><span id=\"MJXc-Node-109\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\" style=\"width: 0.723em; padding: 0px 0.12em;\"><span class=\"mjx-numerator\" style=\"width: 0.723em; top: -1.144em;\"><span id=\"MJXc-Node-110\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.199em; padding-bottom: 0.297em;\">c<\/span><\/span><\/span><span class=\"mjx-denominator\" style=\"width: 0.723em; bottom: -0.721em;\"><span id=\"MJXc-Node-111\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.494em; padding-bottom: 0.297em; padding-right: 0.003em;\">d<\/span><\/span><\/span><span class=\"mjx-line\" style=\"border-bottom: 1.3px solid; top: -0.281em; width: 0.723em;\"><\/span><\/span><span class=\"mjx-vsize\" style=\"height: 1.864em; vertical-align: -0.721em;\"><\/span><\/span><span id=\"MJXc-Node-112\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.248em; padding-bottom: 0.347em;\">\u21d2<\/span><\/span><span id=\"MJXc-Node-113\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\" style=\"width: 0.729em; padding: 0px 0.12em;\"><span class=\"mjx-numerator\" style=\"width: 0.729em; top: -1.143em;\"><span id=\"MJXc-Node-114\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.199em; padding-bottom: 0.297em;\">a<\/span><\/span><\/span><span class=\"mjx-denominator\" style=\"width: 0.729em; bottom: -0.722em;\"><span id=\"MJXc-Node-115\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.199em; padding-bottom: 0.297em;\">c<\/span><\/span><\/span><span class=\"mjx-line\" style=\"border-bottom: 1.3px solid; top: -0.281em; width: 0.729em;\"><\/span><\/span><span class=\"mjx-vsize\" style=\"height: 1.864em; vertical-align: -0.722em;\"><\/span><\/span><span id=\"MJXc-Node-116\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.101em; padding-bottom: 0.297em;\">=<\/span><\/span><span id=\"MJXc-Node-117\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\" style=\"width: 0.723em; padding: 0px 0.12em;\"><span class=\"mjx-numerator\" style=\"width: 0.723em; top: -1.396em;\"><span id=\"MJXc-Node-118\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.494em; padding-bottom: 0.297em;\">b<\/span><\/span><\/span><span class=\"mjx-denominator\" style=\"width: 0.723em; bottom: -0.721em;\"><span id=\"MJXc-Node-119\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.494em; padding-bottom: 0.297em; padding-right: 0.003em;\">d<\/span><\/span><\/span><span class=\"mjx-line\" style=\"border-bottom: 1.3px solid; top: -0.281em; width: 0.723em;\"><\/span><\/span><span class=\"mjx-vsize\" style=\"height: 2.116em; vertical-align: -0.721em;\"><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML MJX_Assistive_MathML_Block\" role=\"presentation\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mfrac><mi>a<\/mi><mi>b<\/mi><\/mfrac><mo>=<\/mo><mfrac><mi>c<\/mi><mi>d<\/mi><\/mfrac><mo stretchy=\"false\">\u21d2<\/mo><mfrac><mi>a<\/mi><mi>c<\/mi><\/mfrac><mo>=<\/mo><mfrac><mi>b<\/mi><mi>d<\/mi><\/mfrac><\/math><\/span><\/span><\/span><script>\\frac{a}{b} = \\frac{c}{d} \\Rightarrow \\frac{a}{c} = \\frac{b}{d}<\/script><\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Propriedade 2: Troca dos Extremos<\/h4>\n\n\n\n<p class=\"\">$$\\frac{a}{b} = \\frac{c}{d} \\Rightarrow \\frac{d}{b} = \\frac{c}{a}$$<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Propriedade 3: Soma dos Antecedentes e Consequentes<\/h4>\n\n\n\n<p class=\"\">$$\\frac{a}{b} = \\frac{c}{d} = \\frac{a+c}{b+d}$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1.5 Divis\u00e3o em Partes Proporcionais<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">Divis\u00e3o Diretamente Proporcional<\/h4>\n\n\n\n<p class=\"\"><strong>Exemplo Pr\u00e1tico:<\/strong> Dividir R$ 600,00 entre tr\u00eas pessoas (A, B e C) de forma diretamente proporcional a 2, 3 e 5.<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o:<\/strong><\/p>\n\n\n\n<p class=\"\">Soma dos valores: $2 + 3 + 5 = 10$<\/p>\n\n\n\n<p class=\"\">Constante de proporcionalidade: <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$k = \\frac{600}{10} = 60$$<\/div>\n\n\n\n<p class=\"\">C\u00e1lculo das partes:<\/p>\n\n\n\n<p class=\"\">A recebe: <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$2 \\times 60 = 120$$ <\/div>\n\n\n\n<p class=\"\">B recebe: <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$3 \\times 60 = 180$$<\/div>\n\n\n\n<p class=\"\">C recebe: $5 \\times 60 = 300$ reais<\/p>\n\n\n\n<p class=\"\"><strong>Verifica\u00e7\u00e3o:<\/strong> <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$120 + 180 + 300 = 600$$<\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Divis\u00e3o Inversamente Proporcional<\/h4>\n\n\n\n<p class=\"\"><strong>Exemplo Pr\u00e1tico:<\/strong> Dividir R$ 1.200,00 entre tr\u00eas funcion\u00e1rios inversamente proporcional ao n\u00famero de faltas: 2, 3 e 4 faltas.<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o:<\/strong><\/p>\n\n\n\n<p class=\"\">Inverter os n\u00fameros: <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$\\frac{1}{2}, \\frac{1}{3}, \\frac{1}{4}$$<\/div>\n\n\n\n<p class=\"\">MMC(2, 3, 4) = 12<\/p>\n\n\n\n<p class=\"\">Transformar em propor\u00e7\u00e3o direta: 6, 4, 3 (dividindo 12 por cada n\u00famero)<\/p>\n\n\n\n<p class=\"\">Soma: 6 + 4 + 3 = 13<\/p>\n\n\n\n<p class=\"\">Constante: <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$k = \\frac{1200}{13} \\approx 92,31$$<\/div>\n\n\n\n<p class=\"\">C\u00e1lculo:<\/p>\n\n\n\n<p class=\"\">1\u00ba funcion\u00e1rio: <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$6 \\times 92,31 = 553,84$$<\/div>\n\n\n\n<p class=\"\">2\u00ba funcion\u00e1rio: <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$4 \\times 92,31 = 369,24$$<\/div>\n\n\n\n<p class=\"\">3\u00ba funcion\u00e1rio: <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$3 \\times 92,31 = 276,92$$<\/div>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h2 class=\"wp-block-heading\">2. REGRA DE TR\u00caS SIMPLES<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">2.1 Conceito<\/h3>\n\n\n\n<p class=\"\">A <strong>regra de tr\u00eas simples<\/strong> \u00e9 um m\u00e9todo matem\u00e1tico usado para resolver problemas que envolvem <strong>duas grandezas diretamente ou inversamente proporcionais<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2.2 Grandezas Diretamente Proporcionais<\/h3>\n\n\n\n<p class=\"\">Duas grandezas s\u00e3o <strong>diretamente proporcionais<\/strong> quando:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\">Aumentando uma, a outra aumenta na mesma propor\u00e7\u00e3o<\/li>\n\n\n\n<li class=\"\">Diminuindo uma, a outra diminui na mesma propor\u00e7\u00e3o<\/li>\n<\/ul>\n\n\n\n<p class=\"\"><strong>Exemplo:<\/strong> Velocidade e dist\u00e2ncia (com tempo constante)<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2.3 Grandezas Inversamente Proporcionais<\/h3>\n\n\n\n<p class=\"\">Duas grandezas s\u00e3o <strong>inversamente proporcionais<\/strong> quando:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\">Aumentando uma, a outra diminui na mesma propor\u00e7\u00e3o<\/li>\n\n\n\n<li class=\"\">Diminuindo uma, a outra aumenta na mesma propor\u00e7\u00e3o<\/li>\n<\/ul>\n\n\n\n<p class=\"\"><strong>Exemplo:<\/strong> N\u00famero de oper\u00e1rios e tempo de trabalho<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2.4 Como Resolver Regra de Tr\u00eas Simples<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">Passo a Passo:<\/h4>\n\n\n\n<ol class=\"wp-block-list\">\n<li class=\"\"><strong>Identificar as grandezas<\/strong> envolvidas<\/li>\n\n\n\n<li class=\"\"><strong>Montar a tabela<\/strong> com as grandezas<\/li>\n\n\n\n<li class=\"\"><strong>Analisar a rela\u00e7\u00e3o<\/strong> (direta ou inversa)<\/li>\n\n\n\n<li class=\"\"><strong>Montar a propor\u00e7\u00e3o<\/strong><\/li>\n\n\n\n<li class=\"\"><strong>Resolver a equa\u00e7\u00e3o<\/strong><\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">2.5 Exemplos Resolvidos<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">Exemplo 1: Regra de Tr\u00eas Simples Direta<\/h4>\n\n\n\n<p class=\"\"><strong>Problema:<\/strong> Se 5 cadernos custam R$ 30,00, quanto custar\u00e3o 8 cadernos?<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o:<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Cadernos<\/th><th>Pre\u00e7o (R$)<\/th><\/tr><\/thead><tbody><tr><td>5<\/td><td>30<\/td><\/tr><tr><td>8<\/td><td>x<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"\"><strong>An\u00e1lise:<\/strong> Mais cadernos \u2192 mais caro (DIRETA)<\/p>\n\n\n\n<p class=\"\"><strong>Propor\u00e7\u00e3o:<\/strong> $$\\frac{5}{8} = \\frac{30}{x}$$<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o:<\/strong> $$5 \\times x = 8 \\times 30$$ $$5x = 240$$ $$x = \\frac{240}{5} = 48$$<\/p>\n\n\n\n<p class=\"\"><strong>Resposta:<\/strong> 8 cadernos custar\u00e3o R$ 48,00.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Exemplo 2: Regra de Tr\u00eas Simples Inversa<\/h4>\n\n\n\n<p class=\"\"><strong>Problema:<\/strong> 10 oper\u00e1rios constroem um muro em 6 dias. Em quantos dias 15 oper\u00e1rios construir\u00e3o o mesmo muro?<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o:<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Oper\u00e1rios<\/th><th>Dias<\/th><\/tr><\/thead><tbody><tr><td>10<\/td><td>6<\/td><\/tr><tr><td>15<\/td><td>x<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"\"><strong>An\u00e1lise:<\/strong> Mais oper\u00e1rios \u2192 menos dias (INVERSA)<\/p>\n\n\n\n<p class=\"\"><strong>Propor\u00e7\u00e3o (invertemos um lado):<\/strong> <span class=\"MathJax_Preview\" style=\"color: inherit;\"><\/span><span class=\"mjx-chtml MJXc-display\" style=\"text-align: center;\"><span id=\"MathJax-Element-17-Frame\" class=\"mjx-chtml MathJax_CHTML\" tabindex=\"0\" data-mathml=\"<math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;<mfrac&gt;<mn&gt;10<\/mn&gt;<mn&gt;15<\/mn&gt;<\/mfrac&gt;<mo&gt;=<\/mo&gt;<mfrac&gt;<mi&gt;x<\/mi&gt;<mn&gt;6<\/mn&gt;<\/mfrac&gt;<\/math&gt;\" role=\"presentation\" style=\"font-size: 113%; text-align: center; position: relative;\"><span id=\"MJXc-Node-189\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-190\" class=\"mjx-mrow\"><span id=\"MJXc-Node-191\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\" style=\"width: 1.2em; padding: 0px 0.12em;\"><span class=\"mjx-numerator\" style=\"width: 1.2em; top: -1.368em;\"><span id=\"MJXc-Node-192\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">10<\/span><\/span><\/span><span class=\"mjx-denominator\" style=\"width: 1.2em; bottom: -0.733em;\"><span id=\"MJXc-Node-193\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">15<\/span><\/span><\/span><span class=\"mjx-line\" style=\"border-bottom: 1.3px solid; top: -0.281em; width: 1.2em;\"><\/span><\/span><span class=\"mjx-vsize\" style=\"height: 2.1em; vertical-align: -0.733em;\"><\/span><\/span><span id=\"MJXc-Node-194\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.101em; padding-bottom: 0.297em;\">=<\/span><\/span><span id=\"MJXc-Node-195\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\" style=\"width: 0.772em; padding: 0px 0.12em;\"><span class=\"mjx-numerator\" style=\"width: 0.772em; top: -1.144em;\"><span id=\"MJXc-Node-196\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.199em; padding-bottom: 0.297em;\">x<\/span><\/span><\/span><span class=\"mjx-denominator\" style=\"width: 0.772em; bottom: -0.733em;\"><span id=\"MJXc-Node-197\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">6<\/span><\/span><\/span><span class=\"mjx-line\" style=\"border-bottom: 1.3px solid; top: -0.281em; width: 0.772em;\"><\/span><\/span><span class=\"mjx-vsize\" style=\"height: 1.876em; vertical-align: -0.733em;\"><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML MJX_Assistive_MathML_Block\" role=\"presentation\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mfrac><mn>10<\/mn><mn>15<\/mn><\/mfrac><mo>=<\/mo><mfrac><mi>x<\/mi><mn>6<\/mn><\/mfrac><\/math><\/span><\/span><\/span><script>\\frac{10}{15} = \\frac{x}{6}<\/script><\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o:<\/strong> <span class=\"MathJax_Preview\" style=\"color: inherit;\"><\/span><span class=\"mjx-chtml MJXc-display\" style=\"text-align: center;\"><span id=\"MathJax-Element-18-Frame\" class=\"mjx-chtml MathJax_CHTML\" tabindex=\"0\" data-mathml=\"<math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;<mn&gt;15<\/mn&gt;<mo&gt;&#x00D7;<\/mo&gt;<mi&gt;x<\/mi&gt;<mo&gt;=<\/mo&gt;<mn&gt;10<\/mn&gt;<mo&gt;&#x00D7;<\/mo&gt;<mn&gt;6<\/mn&gt;<\/math&gt;\" role=\"presentation\" style=\"font-size: 113%; text-align: center; position: relative;\"><span id=\"MJXc-Node-198\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-199\" class=\"mjx-mrow\"><span id=\"MJXc-Node-200\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">15<\/span><\/span><span id=\"MJXc-Node-201\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.199em; padding-bottom: 0.347em;\">\u00d7<\/span><\/span><span id=\"MJXc-Node-202\" class=\"mjx-mi MJXc-space2\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.199em; padding-bottom: 0.297em;\">x<\/span><\/span><span id=\"MJXc-Node-203\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.101em; padding-bottom: 0.297em;\">=<\/span><\/span><span id=\"MJXc-Node-204\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">10<\/span><\/span><span id=\"MJXc-Node-205\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.199em; padding-bottom: 0.347em;\">\u00d7<\/span><\/span><span id=\"MJXc-Node-206\" class=\"mjx-mn MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">6<\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML MJX_Assistive_MathML_Block\" role=\"presentation\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mn>15<\/mn><mo>\u00d7<\/mo><mi>x<\/mi><mo>=<\/mo><mn>10<\/mn><mo>\u00d7<\/mo><mn>6<\/mn><\/math><\/span><\/span><\/span><script>15 \\times x = 10 \\times 6<\/script> <span class=\"MathJax_Preview\" style=\"color: inherit;\"><\/span><span class=\"mjx-chtml MJXc-display\" style=\"text-align: center;\"><span id=\"MathJax-Element-19-Frame\" class=\"mjx-chtml MathJax_CHTML\" tabindex=\"0\" data-mathml=\"<math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;<mn&gt;15<\/mn&gt;<mi&gt;x<\/mi&gt;<mo&gt;=<\/mo&gt;<mn&gt;60<\/mn&gt;<\/math&gt;\" role=\"presentation\" style=\"font-size: 113%; text-align: center; position: relative;\"><span id=\"MJXc-Node-207\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-208\" class=\"mjx-mrow\"><span id=\"MJXc-Node-209\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">15<\/span><\/span><span id=\"MJXc-Node-210\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.199em; padding-bottom: 0.297em;\">x<\/span><\/span><span id=\"MJXc-Node-211\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.101em; padding-bottom: 0.297em;\">=<\/span><\/span><span id=\"MJXc-Node-212\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">60<\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML MJX_Assistive_MathML_Block\" role=\"presentation\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mn>15<\/mn><mi>x<\/mi><mo>=<\/mo><mn>60<\/mn><\/math><\/span><\/span><\/span><script>15x = 60<\/script> <span class=\"MathJax_Preview\" style=\"color: inherit;\"><\/span><span class=\"mjx-chtml MJXc-display\" style=\"text-align: center;\"><span id=\"MathJax-Element-20-Frame\" class=\"mjx-chtml MathJax_CHTML\" tabindex=\"0\" data-mathml=\"<math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;<mi&gt;x<\/mi&gt;<mo&gt;=<\/mo&gt;<mfrac&gt;<mn&gt;60<\/mn&gt;<mn&gt;15<\/mn&gt;<\/mfrac&gt;<mo&gt;=<\/mo&gt;<mn&gt;4<\/mn&gt;<\/math&gt;\" role=\"presentation\" style=\"font-size: 113%; text-align: center; position: relative;\"><span id=\"MJXc-Node-213\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-214\" class=\"mjx-mrow\"><span id=\"MJXc-Node-215\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.199em; padding-bottom: 0.297em;\">x<\/span><\/span><span id=\"MJXc-Node-216\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.101em; padding-bottom: 0.297em;\">=<\/span><\/span><span id=\"MJXc-Node-217\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\" style=\"width: 1.2em; padding: 0px 0.12em;\"><span class=\"mjx-numerator\" style=\"width: 1.2em; top: -1.368em;\"><span id=\"MJXc-Node-218\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">60<\/span><\/span><\/span><span class=\"mjx-denominator\" style=\"width: 1.2em; bottom: -0.733em;\"><span id=\"MJXc-Node-219\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">15<\/span><\/span><\/span><span class=\"mjx-line\" style=\"border-bottom: 1.3px solid; top: -0.281em; width: 1.2em;\"><\/span><\/span><span class=\"mjx-vsize\" style=\"height: 2.1em; vertical-align: -0.733em;\"><\/span><\/span><span id=\"MJXc-Node-220\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.101em; padding-bottom: 0.297em;\">=<\/span><\/span><span id=\"MJXc-Node-221\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">4<\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML MJX_Assistive_MathML_Block\" role=\"presentation\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mi>x<\/mi><mo>=<\/mo><mfrac><mn>60<\/mn><mn>15<\/mn><\/mfrac><mo>=<\/mo><mn>4<\/mn><\/math><\/span><\/span><\/span><script>x = \\frac{60}{15} = 4<\/script><\/p>\n\n\n\n<p class=\"\"><strong>Resposta:<\/strong> 15 oper\u00e1rios construir\u00e3o em 4 dias.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h2 class=\"wp-block-heading\">3. REGRA DE TR\u00caS COMPOSTA<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">3.1 Conceito<\/h3>\n\n\n\n<p class=\"\">A <strong>regra de tr\u00eas composta<\/strong> \u00e9 utilizada quando o problema envolve <strong>tr\u00eas ou mais grandezas<\/strong>, sendo necess\u00e1rio analisar a rela\u00e7\u00e3o entre cada uma delas com a grandeza desconhecida.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3.2 Como Resolver Regra de Tr\u00eas Composta<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">Passo a Passo:<\/h4>\n\n\n\n<ol class=\"wp-block-list\">\n<li class=\"\"><strong>Identificar todas as grandezas<\/strong><\/li>\n\n\n\n<li class=\"\"><strong>Montar a tabela<\/strong> completa<\/li>\n\n\n\n<li class=\"\"><strong>Isolar a grandeza com a inc\u00f3gnita<\/strong><\/li>\n\n\n\n<li class=\"\"><strong>Comparar cada grandeza com a inc\u00f3gnita<\/strong> (direta ou inversa)<\/li>\n\n\n\n<li class=\"\"><strong>Montar a propor\u00e7\u00e3o<\/strong> (inverter as grandezas inversas)<\/li>\n\n\n\n<li class=\"\"><strong>Resolver a equa\u00e7\u00e3o<\/strong><\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">3.3 Exemplos Resolvidos<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">Exemplo 1: Regra de Tr\u00eas Composta<\/h4>\n\n\n\n<p class=\"\"><strong>Problema:<\/strong> 12 oper\u00e1rios trabalhando 8 horas por dia constroem 60 metros de muro em 5 dias. Quantos metros de muro ser\u00e3o constru\u00eddos por 18 oper\u00e1rios trabalhando 6 horas por dia durante 10 dias?<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o:<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Oper\u00e1rios<\/th><th>Horas\/dia<\/th><th>Dias<\/th><th>Metros<\/th><\/tr><\/thead><tbody><tr><td>12<\/td><td>8<\/td><td>5<\/td><td>60<\/td><\/tr><tr><td>18<\/td><td>6<\/td><td>10<\/td><td>x<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"\"><strong>An\u00e1lise de cada grandeza em rela\u00e7\u00e3o a METROS:<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li class=\"\"><strong>Oper\u00e1rios x Metros:<\/strong> DIRETA (mais oper\u00e1rios \u2192 mais metros)<\/li>\n\n\n\n<li class=\"\"><strong>Horas\/dia x Metros:<\/strong> DIRETA (mais horas \u2192 mais metros)<\/li>\n\n\n\n<li class=\"\"><strong>Dias x Metros:<\/strong> DIRETA (mais dias \u2192 mais metros)<\/li>\n<\/ol>\n\n\n\n<p class=\"\"><strong>Montando a propor\u00e7\u00e3o:<\/strong> $$\\frac{60}{x} = \\frac{12}{18} \\times \\frac{8}{6} \\times \\frac{5}{10}$$<\/p>\n\n\n\n<p class=\"\"><strong>Simplificando:<\/strong> $$\\frac{60}{x} = \\frac{12 \\times 8 \\times 5}{18 \\times 6 \\times 10}$$<\/p>\n\n\n\n<p class=\"\">$$\\frac{60}{x} = \\frac{480}{1080}$$<\/p>\n\n\n\n<p class=\"\">$$\\frac{60}{x} = \\frac{4}{9}$$<\/p>\n\n\n\n<p class=\"\"><strong>Resolvendo:<\/strong> $$4x = 60 \\times 9$$ $$4x = 540$$ $$x = 135$$<\/p>\n\n\n\n<p class=\"\"><strong>Resposta:<\/strong> Ser\u00e3o constru\u00eddos 135 metros de muro.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Exemplo 2: Regra de Tr\u00eas Composta com Grandeza Inversa<\/h4>\n\n\n\n<p class=\"\"><strong>Problema:<\/strong> 8 m\u00e1quinas, trabalhando 6 horas por dia, produzem 2.400 pe\u00e7as em 5 dias. Quantas m\u00e1quinas ser\u00e3o necess\u00e1rias para produzir 3.000 pe\u00e7as em 4 dias, trabalhando 5 horas por dia?<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o:<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>M\u00e1quinas<\/th><th>Horas\/dia<\/th><th>Dias<\/th><th>Pe\u00e7as<\/th><\/tr><\/thead><tbody><tr><td>8<\/td><td>6<\/td><td>5<\/td><td>2400<\/td><\/tr><tr><td>x<\/td><td>5<\/td><td>4<\/td><td>3000<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"\"><strong>An\u00e1lise de cada grandeza em rela\u00e7\u00e3o a M\u00c1QUINAS:<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li class=\"\"><strong>Horas\/dia x M\u00e1quinas:<\/strong> INVERSA (mais horas \u2192 menos m\u00e1quinas)<\/li>\n\n\n\n<li class=\"\"><strong>Dias x M\u00e1quinas:<\/strong> INVERSA (mais dias \u2192 menos m\u00e1quinas)<\/li>\n\n\n\n<li class=\"\"><strong>Pe\u00e7as x M\u00e1quinas:<\/strong> DIRETA (mais pe\u00e7as \u2192 mais m\u00e1quinas)<\/li>\n<\/ol>\n\n\n\n<p class=\"\"><strong>Montando a propor\u00e7\u00e3o (invertendo as inversas):<\/strong> $$\\frac{8}{x} = \\frac{5}{6} \\times \\frac{4}{5} \\times \\frac{2400}{3000}$$<\/p>\n\n\n\n<p class=\"\"><strong>Simplificando:<\/strong> $$\\frac{8}{x} = \\frac{5 \\times 4 \\times 2400}{6 \\times 5 \\times 3000}$$<\/p>\n\n\n\n<p class=\"\">$$\\frac{8}{x} = \\frac{48000}{90000} = \\frac{8}{15}$$<\/p>\n\n\n\n<p class=\"\"><strong>Resolvendo:<\/strong> $$8 \\times 15 = 8x$$ $$120 = 8x$$ $$x = 15$$<\/p>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h2 class=\"wp-block-heading\">QUEST\u00d5ES ESTILO CONCURSO P\u00daBLICO RESOLVIDAS<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">QUEST\u00c3O 1 &#8211; N\u00edvel: M\u00e9dio (Estilo CESPE)<\/h3>\n\n\n\n<p class=\"\"><strong>Enunciado:<\/strong> Uma equipe de 15 servidores p\u00fablicos foi designada para analisar 450 processos administrativos. Ap\u00f3s 6 dias de trabalho, 180 processos j\u00e1 haviam sido analisados. Mantendo o mesmo ritmo de trabalho, em quantos dias, no total, todos os processos estar\u00e3o analisados?<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o Passo a Passo:<\/strong><\/p>\n\n\n\n<p class=\"\"><strong>Passo 1:<\/strong> Identificar as grandezas<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\">Processos analisados<\/li>\n\n\n\n<li class=\"\">Dias de trabalho<\/li>\n<\/ul>\n\n\n\n<p class=\"\"><strong>Passo 2:<\/strong> Organizar os dados<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Processos<\/th><th>Dias<\/th><\/tr><\/thead><tbody><tr><td>180<\/td><td>6<\/td><\/tr><tr><td>450<\/td><td>x<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"\"><strong>Passo 3:<\/strong> Analisar a rela\u00e7\u00e3o Mais processos \u2192 mais dias (DIRETA)<\/p>\n\n\n\n<p class=\"\"><strong>Passo 4:<\/strong> Montar a propor\u00e7\u00e3o $$\\frac{180}{450} = \\frac{6}{x}$$<\/p>\n\n\n\n<p class=\"\"><strong>Passo 5:<\/strong> Resolver $$180 \\times x = 450 \\times 6$$ $$180x = 2700$$ $$x = \\frac{2700}{180} = 15$$<\/p>\n\n\n\n<p class=\"\"><strong>Resposta:<\/strong> Os 450 processos estar\u00e3o analisados em 15 dias no total.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">QUEST\u00c3O 2 &#8211; N\u00edvel: M\u00e9dio (Estilo FCC)<\/h3>\n\n\n\n<p class=\"\"><strong>Enunciado:<\/strong> Um terreno de formato retangular tem 15 metros de largura por 25 metros de comprimento. Em uma planta baixa, esse terreno foi representado com 5 cm de largura. O comprimento do terreno na planta, em cent\u00edmetros, \u00e9:<\/p>\n\n\n\n<p class=\"\">a) 6,25 b) 7,50 c) 8,33 d) 10,00 e) 12,50<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o Passo a Passo:<\/strong><\/p>\n\n\n\n<p class=\"\"><strong>Passo 1:<\/strong> Converter para a mesma unidade<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\">Largura real: 15 m = 1.500 cm<\/li>\n\n\n\n<li class=\"\">Comprimento real: 25 m = 2.500 cm<\/li>\n\n\n\n<li class=\"\">Largura na planta: 5 cm<\/li>\n<\/ul>\n\n\n\n<p class=\"\"><strong>Passo 2:<\/strong> Encontrar a escala usando a largura $$\\text{Escala} = \\frac{\\text{medida na planta}}{\\text{medida real}} = \\frac{5}{1500} = \\frac{1}{300}$$<\/p>\n\n\n\n<p class=\"\"><strong>Passo 3:<\/strong> Aplicar a mesma escala ao comprimento $$\\frac{x}{2500} = \\frac{1}{300}$$<\/p>\n\n\n\n<p class=\"\"><strong>Passo 4:<\/strong> Resolver $$300x = 2500$$ $$x = \\frac{2500}{300} = 8,33 \\text{ cm}$$<\/p>\n\n\n\n<p class=\"\"><strong>Resposta:<\/strong> Alternativa C &#8211; 8,33 cm<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">QUEST\u00c3O 3 &#8211; N\u00edvel: Dif\u00edcil (Estilo FGV)<\/h3>\n\n\n\n<p class=\"\"><strong>Enunciado:<\/strong> Uma gr\u00e1fica possui 4 impressoras que, trabalhando 6 horas por dia, imprimem 9.600 folhetos em 8 dias. Para imprimir 18.000 folhetos em 6 dias, trabalhando 8 horas por dia, quantas impressoras ser\u00e3o necess\u00e1rias?<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o Passo a Passo:<\/strong><\/p>\n\n\n\n<p class=\"\"><strong>Passo 1:<\/strong> Organizar em tabela<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Impressoras<\/th><th>Horas\/dia<\/th><th>Dias<\/th><th>Folhetos<\/th><\/tr><\/thead><tbody><tr><td>4<\/td><td>6<\/td><td>8<\/td><td>9600<\/td><\/tr><tr><td>x<\/td><td>8<\/td><td>6<\/td><td>18000<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"\"><strong>Passo 2:<\/strong> Analisar cada grandeza em rela\u00e7\u00e3o a IMPRESSORAS<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li class=\"\"><strong>Horas\/dia x Impressoras:<\/strong> INVERSA (mais horas \u2192 menos impressoras)<\/li>\n\n\n\n<li class=\"\"><strong>Dias x Impressoras:<\/strong> INVERSA (mais dias \u2192 menos impressoras)<\/li>\n\n\n\n<li class=\"\"><strong>Folhetos x Impressoras:<\/strong> DIRETA (mais folhetos \u2192 mais impressoras)<\/li>\n<\/ol>\n\n\n\n<p class=\"\"><strong>Passo 3:<\/strong> Montar a propor\u00e7\u00e3o (invertendo as inversas) $$\\frac{4}{x} = \\frac{8}{6} \\times \\frac{6}{8} \\times \\frac{9600}{18000}$$<\/p>\n\n\n\n<p class=\"\"><strong>Passo 4:<\/strong> Simplificar $$\\frac{4}{x} = \\frac{8 \\times 6 \\times 9600}{6 \\times 8 \\times 18000}$$<\/p>\n\n\n\n<p class=\"\">$$\\frac{4}{x} = \\frac{460800}{864000}$$<\/p>\n\n\n\n<p class=\"\">$$\\frac{4}{x} = \\frac{9600}{18000} = \\frac{8}{15}$$<\/p>\n\n\n\n<p class=\"\"><strong>Passo 5:<\/strong> Resolver $$4 \\times 15 = 8x$$ $$60 = 8x$$ $$x = 7,5$$<\/p>\n\n\n\n<p class=\"\"><strong>Passo 6:<\/strong> Interpretar Como n\u00e3o \u00e9 poss\u00edvel ter 7,5 impressoras, s\u00e3o necess\u00e1rias <strong>8 impressoras<\/strong>.<\/p>\n\n\n\n<p class=\"\"><strong>Resposta:<\/strong> 8 impressoras<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">QUEST\u00c3O 4 &#8211; N\u00edvel: M\u00e9dio (Estilo VUNESP)<\/h3>\n\n\n\n<p class=\"\"><strong>Enunciado:<\/strong> Tr\u00eas servidores (Ana, Bruno e Carlos) devem dividir uma gratifica\u00e7\u00e3o de R$ 4.800,00 de forma proporcional ao tempo de servi\u00e7o de cada um: Ana tem 5 anos, Bruno tem 7 anos e Carlos tem 12 anos. Quanto cada um receber\u00e1?<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o Passo a Passo:<\/strong><\/p>\n\n\n\n<p class=\"\"><strong>Passo 1:<\/strong> Identificar os valores proporcionais<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\">Ana: 5<\/li>\n\n\n\n<li class=\"\">Bruno: 7<\/li>\n\n\n\n<li class=\"\">Carlos: 12<\/li>\n<\/ul>\n\n\n\n<p class=\"\"><strong>Passo 2:<\/strong> Somar os valores $$5 + 7 + 12 = 24$$<\/p>\n\n\n\n<p class=\"\"><strong>Passo 3:<\/strong> Calcular a constante de proporcionalidade $$k = \\frac{4800}{24} = 200$$<\/p>\n\n\n\n<p class=\"\"><strong>Passo 4:<\/strong> Calcular cada parte<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\">Ana: $5 \\times 200 = 1.000$ reais<\/li>\n\n\n\n<li class=\"\">Bruno: $7 \\times 200 = 1.400$ reais<\/li>\n\n\n\n<li class=\"\">Carlos: $12 \\times 200 = 2.400$ reais<\/li>\n<\/ul>\n\n\n\n<p class=\"\"><strong>Passo 5:<\/strong> Verificar $$1000 + 1400 + 2400 = 4800$$ \u2713<\/p>\n\n\n\n<p class=\"\"><strong>Resposta:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\">Ana: R$ 1.000,00<\/li>\n\n\n\n<li class=\"\">Bruno: R$ 1.400,00<\/li>\n\n\n\n<li class=\"\">Carlos: R$ 2.400,00<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">QUEST\u00c3O 5 &#8211; N\u00edvel: Dif\u00edcil (Estilo CESPE &#8211; Certo ou Errado)<\/h3>\n\n\n\n<p class=\"\"><strong>Enunciado:<\/strong> Uma equipe de 20 digitadores, trabalhando 5 horas por dia, digitou 300 p\u00e1ginas em 12 dias. Se uma equipe de 15 digitadores trabalhasse 6 horas por dia durante 20 dias, digitaria mais de 450 p\u00e1ginas.<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o Passo a Passo:<\/strong><\/p>\n\n\n\n<p class=\"\"><strong>Passo 1:<\/strong> Organizar os dados<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Digitadores<\/th><th>Horas\/dia<\/th><th>Dias<\/th><th>P\u00e1ginas<\/th><\/tr><\/thead><tbody><tr><td>20<\/td><td>5<\/td><td>12<\/td><td>300<\/td><\/tr><tr><td>15<\/td><td>6<\/td><td>20<\/td><td>x<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"\"><strong>Passo 2:<\/strong> Analisar as rela\u00e7\u00f5es com P\u00c1GINAS<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\">Digitadores x P\u00e1ginas: DIRETA<\/li>\n\n\n\n<li class=\"\">Horas\/dia x P\u00e1ginas: DIRETA<\/li>\n\n\n\n<li class=\"\">Dias x P\u00e1ginas: DIRETA<\/li>\n<\/ul>\n\n\n\n<p class=\"\"><strong>Passo 3:<\/strong> Montar a propor\u00e7\u00e3o $$\\frac{300}{x} = \\frac{20}{15} \\times \\frac{5}{6} \\times \\frac{12}{20}$$<\/p>\n\n\n\n<p class=\"\"><strong>Passo 4:<\/strong> Calcular $$\\frac{300}{x} = \\frac{20 \\times 5 \\times 12}{15 \\times 6 \\times 20}$$<\/p>\n\n\n\n<p class=\"\">$$\\frac{300}{x} = \\frac{1200}{1800} = \\frac{2}{3}$$<\/p>\n\n\n\n<p class=\"\"><strong>Passo 5:<\/strong> Resolver $$2x = 300 \\times 3$$ $$2x = 900$$ $$x = 450$$<\/p>\n\n\n\n<p class=\"\"><strong>Passo 6:<\/strong> Analisar a afirma\u00e7\u00e3o A quest\u00e3o diz &#8220;mais de 450 p\u00e1ginas&#8221;, mas o resultado \u00e9 exatamente 450 p\u00e1ginas.<\/p>\n\n\n\n<p class=\"\"><strong>Resposta:<\/strong> ERRADO<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">QUEST\u00c3O 6 &#8211; N\u00edvel: M\u00e9dio (Estilo FCC)<\/h3>\n\n\n\n<p class=\"\"><strong>Enunciado:<\/strong> Uma torneira enche um tanque em 6 horas. Outra torneira enche o mesmo tanque em 3 horas. Se as duas torneiras forem abertas simultaneamente, em quanto tempo encher\u00e3o o tanque?<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o Passo a Passo:<\/strong><\/p>\n\n\n\n<p class=\"\"><strong>Passo 1:<\/strong> Calcular o rendimento de cada torneira<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p class=\"\">Torneira A: <\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$\\frac{1}{6}$$<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p class=\"\">do tanque por hora<\/p>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p class=\"\">Torneira B:<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$\\frac{1}{3}$$<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p class=\"\">do tanque por hora<\/p>\n<\/div>\n<\/div>\n\n\n\n<p class=\"\"><strong>Passo 2:<\/strong> Somar os rendimentos $$\\frac{1}{6} + \\frac{1}{3} = \\frac{1}{6} + \\frac{2}{6} = \\frac{3}{6} = \\frac{1}{2}$$<\/p>\n\n\n\n<p class=\"\"><strong>Interpreta\u00e7\u00e3o:<\/strong> Juntas, enchem 1\/2 do tanque por hora<\/p>\n\n\n\n<p class=\"\"><strong>Passo 3:<\/strong> Calcular o tempo Se em 1 hora enchem 1\/2, ent\u00e3o: <span class=\"MathJax_Preview\" style=\"color: inherit;\"><\/span><span class=\"mjx-chtml MJXc-display\" style=\"text-align: center;\"><span id=\"MathJax-Element-59-Frame\" class=\"mjx-chtml MathJax_CHTML\" tabindex=\"0\" data-mathml=\"<math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;<mtext&gt;Tempo<\/mtext&gt;<mo&gt;=<\/mo&gt;<mfrac&gt;<mn&gt;1<\/mn&gt;<mfrac&gt;<mn&gt;1<\/mn&gt;<mn&gt;2<\/mn&gt;<\/mfrac&gt;<\/mfrac&gt;<mo&gt;=<\/mo&gt;<mn&gt;2<\/mn&gt;<mtext&gt;&#xA0;horas<\/mtext&gt;<\/math&gt;\" role=\"presentation\" style=\"font-size: 113%; text-align: center; position: relative;\"><span id=\"MJXc-Node-637\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-638\" class=\"mjx-mrow\"><span id=\"MJXc-Node-639\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.543em;\">Tempo<\/span><\/span><span id=\"MJXc-Node-640\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.101em; padding-bottom: 0.297em;\">=<\/span><\/span><span id=\"MJXc-Node-641\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\" style=\"width: 0.935em; padding: 0px 0.12em;\"><span class=\"mjx-numerator\" style=\"width: 0.935em; top: -1.368em;\"><span id=\"MJXc-Node-642\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">1<\/span><\/span><\/span><span class=\"mjx-denominator\" style=\"width: 0.935em; bottom: -1.282em;\"><span id=\"MJXc-Node-643\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\" style=\"width: 0.495em; padding: 0px 0.12em;\"><span class=\"mjx-numerator\" style=\"font-size: 70.7%; width: 0.7em; top: -1.286em;\"><span id=\"MJXc-Node-644\" class=\"mjx-mn\" style=\"\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">1<\/span><\/span><\/span><span class=\"mjx-denominator\" style=\"font-size: 70.7%; width: 0.7em; bottom: -0.579em;\"><span id=\"MJXc-Node-645\" class=\"mjx-mn\" style=\"\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">2<\/span><\/span><\/span><span class=\"mjx-line\" style=\"border-bottom: 1.3px solid; top: -0.281em; width: 0.495em;\"><\/span><\/span><span class=\"mjx-vsize\" style=\"height: 1.318em; vertical-align: -0.409em;\"><\/span><\/span><\/span><span class=\"mjx-line\" style=\"border-bottom: 1.3px solid; top: -0.281em; width: 0.935em;\"><\/span><\/span><span class=\"mjx-vsize\" style=\"height: 2.65em; vertical-align: -1.282em;\"><\/span><\/span><span id=\"MJXc-Node-646\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.101em; padding-bottom: 0.297em;\">=<\/span><\/span><span id=\"MJXc-Node-647\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">2<\/span><\/span><span id=\"MJXc-Node-648\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.396em; padding-bottom: 0.347em;\">\u00a0horas<\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML MJX_Assistive_MathML_Block\" role=\"presentation\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtext>Tempo<\/mtext><mo>=<\/mo><mfrac><mn>1<\/mn><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/mfrac><mo>=<\/mo><mn>2<\/mn><mtext>\u00a0horas<\/mtext><\/math><\/span><\/span><\/span><script>\\text{Tempo} = \\frac{1}{\\frac{1}{2}} = 2 \\text{ horas}<\/script><\/p>\n\n\n\n<p class=\"\"><strong>Resposta:<\/strong> 2 horas<\/p>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h2 class=\"wp-block-heading\">DICAS PARA CONCURSOS<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 Dicas de Ouro<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li class=\"\"><strong>Sempre identifique a rela\u00e7\u00e3o:<\/strong> Pergunte-se: &#8220;Se aumentar uma grandeza, a outra aumenta ou diminui?&#8221;<\/li>\n\n\n\n<li class=\"\"><strong>Organize os dados em tabela:<\/strong> Visualizar ajuda a n\u00e3o errar<\/li>\n\n\n\n<li class=\"\"><strong>Aten\u00e7\u00e3o \u00e0s unidades:<\/strong> Converta tudo para a mesma unidade antes de calcular<\/li>\n\n\n\n<li class=\"\"><strong>Simplifique antes de resolver:<\/strong> Facilita os c\u00e1lculos e economiza tempo<\/li>\n\n\n\n<li class=\"\"><strong>Verifique a resposta:<\/strong> Veja se o resultado faz sentido no contexto<\/li>\n\n\n\n<li class=\"\"><strong>Treine inversamente proporcional:<\/strong> \u00c9 onde a maioria erra!<\/li>\n\n\n\n<li class=\"\"><strong>Cuidado com pegadinhas:<\/strong> &#8220;No total&#8221; vs &#8220;ainda faltam&#8221; &#8211; leia com aten\u00e7\u00e3o<\/li>\n\n\n\n<li class=\"\"><strong>Pratique problemas mistos:<\/strong> Concursos adoram misturar conceitos<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83c\udfaf Erros Comuns a Evitar<\/h3>\n\n\n\n<p class=\"\">\u274c Confundir rela\u00e7\u00e3o direta com inversa \u274c N\u00e3o inverter as grandezas inversas na regra de tr\u00eas composta \u274c Misturar unidades de medida \u274c Esquecer de considerar todas as grandezas \u274c N\u00e3o arredondar quando necess\u00e1rio (ex: n\u00famero de pessoas)<\/p>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p class=\"\">Dominar varia\u00e7\u00e3o de grandezas, raz\u00e3o, propor\u00e7\u00e3o e regra de tr\u00eas \u00e9 essencial para qualquer concurso p\u00fablico. Esses conceitos aparecem em praticamente todas as provas, seja de forma direta ou aplicados a problemas contextualizados.<\/p>\n\n\n\n<p class=\"\">A chave do sucesso \u00e9:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\"><strong>Entender os conceitos fundamentais<\/strong><\/li>\n\n\n\n<li class=\"\"><strong>Praticar muito<\/strong> com quest\u00f5es variadas<\/li>\n\n\n\n<li class=\"\"><strong>Desenvolver racioc\u00ednio l\u00f3gico<\/strong> para identificar rela\u00e7\u00f5es<\/li>\n\n\n\n<li class=\"\"><strong>Criar estrat\u00e9gias pessoais<\/strong> de resolu\u00e7\u00e3o<\/li>\n<\/ul>\n\n\n\n<p class=\"\">Lembre-se: a matem\u00e1tica de concursos n\u00e3o exige genialidade, mas sim m\u00e9todo, pr\u00e1tica e aten\u00e7\u00e3o aos detalhes. Com dedica\u00e7\u00e3o e treino, voc\u00ea dominar\u00e1 esse conte\u00fado e garantir\u00e1 pontos valiosos na sua prova!<\/p>\n\n\n\n<p class=\"\"><strong>Bons estudos e sucesso na sua jornada! \ud83c\udf93\ud83d\udcda<\/strong><\/p>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n","protected":false},"excerpt":{"rendered":"<p>A varia\u00e7\u00e3o de grandezas \u00e9 um dos t\u00f3picos mais recorrentes em provas de concursos p\u00fablicos, aparecendo em quest\u00f5es de racioc\u00ednio l\u00f3gico, matem\u00e1tica e at\u00e9 em situa\u00e7\u00f5es pr\u00e1ticas de c\u00e1lculos administrativos. Dominar raz\u00e3o, propor\u00e7\u00e3o e regra de tr\u00eas \u00e9 fundamental para resolver problemas do dia a dia e garantir pontos preciosos em sua prova. Neste artigo, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":688,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"nf_dc_page":"","footnotes":""},"categories":[107,381],"tags":[46,357],"class_list":["post-4733","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-matematica","category-razao-proporcao-e-regra-de-tres","tag-estudo-completo","tag-dicas"],"acf":[],"_links":{"self":[{"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/posts\/4733","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/comments?post=4733"}],"version-history":[{"count":2,"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/posts\/4733\/revisions"}],"predecessor-version":[{"id":4736,"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/posts\/4733\/revisions\/4736"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/media\/688"}],"wp:attachment":[{"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/media?parent=4733"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/categories?post=4733"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/tags?post=4733"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}