{"id":228740,"date":"2026-05-12T14:52:38","date_gmt":"2026-05-12T17:52:38","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=228740"},"modified":"2026-05-12T14:52:38","modified_gmt":"2026-05-12T17:52:38","slug":"problemao-proporcoes-de-reais","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problemao-proporcoes-de-reais\/","title":{"rendered":".Problem\u00e3o: Propor\u00e7\u00f5es de reais"},"content":{"rendered":"<div class=\"g1n\" style=\"line-height: 25px; padding-bottom: 15px;\">\n<p class=\"tit\" style=\"line-height: 25px; margin: 10px;\">Problema<br \/>\n<span style=\"font-size: 12px; color: #800000;\"> (Indicado a partir do 7\u00ba ano do E. F.)<\/span><\/p>\n<hr style=\"height: 1px; width: 100%; background-color: #800000; margin-top: 10px; margin-bottom: 20px; border: 0 none;\" \/>\n<p>Sejam [tex]a, b[\/tex] e [tex]c[\/tex] n\u00fameros reais n\u00e3o nulos. Sabendo que as propor\u00e7\u00f5es [tex]\\dfrac{a+b}{c}=\\dfrac{b+c}{a}=\\dfrac{a+c}{b}[\/tex] s\u00e3o v\u00e1lidas, calcule os poss\u00edveis valores num\u00e9ricos da express\u00e3o [tex]\\dfrac{a+b}{c}[\/tex].<\/p>\n<p style=\"font-size: 12px; text-align: right; line-height: 1px;\">Adaptado de <strong> IME<\/strong>.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"g1n\" style=\"line-height: 28px;\">\n<p class=\"tit\" style=\"line-height: 28px; margin: 10px;\">Solu\u00e7\u00e3o <\/p>\n<hr style=\"height: 1px; width: 100%; background-color: #800000; margin-top: 1px; margin-bottom: 20px; border: 0 none;\" \/>\n<p>Se [tex]\\dfrac{p}{q}=\\dfrac{m}{n}[\/tex] e [tex]q+n \\ne 0[\/tex], ent\u00e3o [tex]\\dfrac{p}{q}=\\dfrac{m}{n}=\\dfrac{p+m}{q+n}[\/tex].<\/p>\n<p>Desta forma, pelo enunciado:<\/p>\n<p><center>[tex]\\dfrac {a+b}{c}=\\dfrac{b+c}{a}=\\dfrac{a+c}{b}=\\dfrac{2a+2b+2c}{a+b+c}=2[\/tex],<\/center><br \/>\n se [tex]a+b+c \\ne 0[\/tex]. <\/p>\n<p>No caso em que [tex]a+b+c=0[\/tex], ou seja, [tex]c=-(a+b)[\/tex], temos que: <\/p>\n<p><center>[tex]\\dfrac{a+b}{c}=\\dfrac {a+b}{-(a+b)}=-1[\/tex].<\/center><\/p>\n<p>Portanto, a express\u00e3o do enunciado pode assumir dois valores:<\/p>\n<p><center>[tex]\\dfrac {a+b}{c}=2[\/tex] ou [tex]\\dfrac {a+b}{c}=-1[\/tex].<\/center><\/p>\n<hr style=\"height: 1px; width: 100%; background-color: #800000; margin-top: 10px; margin-bottom: 15px; border: 0 none;\" \/>\n<p class=\"g2\">Solu\u00e7\u00e3o elaborada pelos <strong>Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"g3\">\nParticipou da discuss\u00e3o o Clube <strong>Ordem Benewest<\/strong>.<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Problema (Indicado a partir do 7\u00ba ano do E. F.) Sejam [tex]a, b[\/tex] e [tex]c[\/tex] n\u00fameros reais n\u00e3o nulos. Sabendo que as propor\u00e7\u00f5es [tex]\\dfrac{a+b}{c}=\\dfrac{b+c}{a}=\\dfrac{a+c}{b}[\/tex] s\u00e3o v\u00e1lidas, calcule os poss\u00edveis valores num\u00e9ricos da express\u00e3o [tex]\\dfrac{a+b}{c}[\/tex]. Adaptado de IME. &nbsp; Solu\u00e7\u00e3o Se [tex]\\dfrac{p}{q}=\\dfrac{m}{n}[\/tex] e [tex]q+n \\ne 0[\/tex], ent\u00e3o [tex]\\dfrac{p}{q}=\\dfrac{m}{n}=\\dfrac{p+m}{q+n}[\/tex]. Desta forma, pelo enunciado: [tex]\\dfrac {a+b}{c}=\\dfrac{b+c}{a}=\\dfrac{a+c}{b}=\\dfrac{2a+2b+2c}{a+b+c}=2[\/tex], se [tex]a+b+c &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problemao-proporcoes-de-reais\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":668,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-228740","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/228740","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/users\/668"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=228740"}],"version-history":[{"count":6,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/228740\/revisions"}],"predecessor-version":[{"id":229490,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/228740\/revisions\/229490"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=228740"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}