{"id":9573,"date":"2017-05-03T15:45:52","date_gmt":"2017-05-03T15:45:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9573"},"modified":"2024-04-29T23:04:46","modified_gmt":"2024-04-29T23:04:46","slug":"summary-solving-equations-that-contain-fractions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/summary-solving-equations-that-contain-fractions\/","title":{"raw":"Summary: Solving Equations Containing Fractions","rendered":"Summary: Solving Equations Containing Fractions"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li><strong>Summary of Fraction Operations<\/strong>\r\n<ul id=\"eip-id1170322929638\">\r\n \t<li><strong>Fraction multiplication:<\/strong> Multiply the numerators and multiply the denominators.\r\n[latex]\\Large\\frac{a}{b}\\cdot\\Large\\frac{c}{d}=\\Large\\frac{ac}{bd}[\/latex]<\/li>\r\n \t<li><strong>Fraction division:<\/strong> Multiply the first fraction by the reciprocal of the second.\r\n[latex]\\Large\\frac{a}{b}+\\Large\\frac{c}{d}=\\Large\\frac{a}{b}\\cdot\\Large\\frac{d}{c}[\/latex]<\/li>\r\n \t<li><strong>Fraction addition:<\/strong> Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.\r\n[latex]\\Large\\frac{a}{c}+\\Large\\frac{b}{c}=\\Large\\frac{a+b}{c}[\/latex]<\/li>\r\n \t<li><strong>Fraction subtraction:<\/strong> Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.\r\n[latex]\\Large\\frac{a}{c}-\\Large\\frac{b}{c}=\\Large\\frac{a-b}{c}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Simplify complex fractions.<\/strong>\r\n<ol id=\"eip-id1170321558052\" class=\"stepwise\">\r\n \t<li>Simplify the numerator.<\/li>\r\n \t<li>Simplify the denominator.<\/li>\r\n \t<li>Divide the numerator by the denominator.<\/li>\r\n \t<li>Simplify if possible.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<ul id=\"eip-809\">\r\n \t<li><strong>Determine whether a number is a solution to an equation.<\/strong>\r\n<ol id=\"eip-id1170324088217\" class=\"stepwise\">\r\n \t<li>Substitute the number for the variable in the equation.<\/li>\r\n \t<li>Simplify the expressions on both sides of the equation.<\/li>\r\n \t<li>Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><strong>Addition, Subtraction, and Division Properties of Equality<\/strong>\r\n<ul id=\"eip-id1170326379675\">\r\n \t<li>For any numbers [latex]a[\/latex], [latex]b[\/latex]<em>,<\/em> and [latex]c[\/latex], if [latex]a=b[\/latex] , then [latex]a+c=b+c[\/latex] . Addition Property of Equality<\/li>\r\n \t<li>if [latex]a=b[\/latex] , then [latex]a-c=b-c[\/latex] . Subtraction Property of Equality<\/li>\r\n \t<li>if [latex]a=b[\/latex] , then [latex]\\Large\\frac{a}{c}=\\Large\\frac{b}{c}[\/latex] , [latex]c\\ne 0[\/latex] . Division Property of Equality<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>The Multiplication Property of Equality<\/strong>\r\n<ul id=\"eip-id1170322834828\">\r\n \t<li>For any numbers [latex]a[\/latex], [latex]b[\/latex]<em>,<\/em> and [latex]c[\/latex], [latex]a=b[\/latex] , then [latex]ac=bc[\/latex] .<\/li>\r\n \t<li>If you multiply both sides of an equation by the same quantity, you still have equality.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<ul>\r\n \t<li><strong>Translate a word sentence to an algebraic equation.<\/strong>\r\n<ol id=\"eip-id1170324011027\" class=\"stepwise\">\r\n \t<li>Locate the \"equals\" word(s). Translate to an equal sign.<\/li>\r\n \t<li>Translate the words to the left of the \"equals\" word(s) into an algebraic expression.<\/li>\r\n \t<li>Translate the words to the right of the \"equals\" word(s) into an algebraic expression.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li><strong>Summary of Fraction Operations<\/strong>\n<ul id=\"eip-id1170322929638\">\n<li><strong>Fraction multiplication:<\/strong> Multiply the numerators and multiply the denominators.<br \/>\n[latex]\\Large\\frac{a}{b}\\cdot\\Large\\frac{c}{d}=\\Large\\frac{ac}{bd}[\/latex]<\/li>\n<li><strong>Fraction division:<\/strong> Multiply the first fraction by the reciprocal of the second.<br \/>\n[latex]\\Large\\frac{a}{b}+\\Large\\frac{c}{d}=\\Large\\frac{a}{b}\\cdot\\Large\\frac{d}{c}[\/latex]<\/li>\n<li><strong>Fraction addition:<\/strong> Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.<br \/>\n[latex]\\Large\\frac{a}{c}+\\Large\\frac{b}{c}=\\Large\\frac{a+b}{c}[\/latex]<\/li>\n<li><strong>Fraction subtraction:<\/strong> Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.<br \/>\n[latex]\\Large\\frac{a}{c}-\\Large\\frac{b}{c}=\\Large\\frac{a-b}{c}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li><strong>Simplify complex fractions.<\/strong>\n<ol id=\"eip-id1170321558052\" class=\"stepwise\">\n<li>Simplify the numerator.<\/li>\n<li>Simplify the denominator.<\/li>\n<li>Divide the numerator by the denominator.<\/li>\n<li>Simplify if possible.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<ul id=\"eip-809\">\n<li><strong>Determine whether a number is a solution to an equation.<\/strong>\n<ol id=\"eip-id1170324088217\" class=\"stepwise\">\n<li>Substitute the number for the variable in the equation.<\/li>\n<li>Simplify the expressions on both sides of the equation.<\/li>\n<li>Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.<\/li>\n<\/ol>\n<\/li>\n<li><strong>Addition, Subtraction, and Division Properties of Equality<\/strong>\n<ul id=\"eip-id1170326379675\">\n<li>For any numbers [latex]a[\/latex], [latex]b[\/latex]<em>,<\/em> and [latex]c[\/latex], if [latex]a=b[\/latex] , then [latex]a+c=b+c[\/latex] . Addition Property of Equality<\/li>\n<li>if [latex]a=b[\/latex] , then [latex]a-c=b-c[\/latex] . Subtraction Property of Equality<\/li>\n<li>if [latex]a=b[\/latex] , then [latex]\\Large\\frac{a}{c}=\\Large\\frac{b}{c}[\/latex] , [latex]c\\ne 0[\/latex] . Division Property of Equality<\/li>\n<\/ul>\n<\/li>\n<li><strong>The Multiplication Property of Equality<\/strong>\n<ul id=\"eip-id1170322834828\">\n<li>For any numbers [latex]a[\/latex], [latex]b[\/latex]<em>,<\/em> and [latex]c[\/latex], [latex]a=b[\/latex] , then [latex]ac=bc[\/latex] .<\/li>\n<li>If you multiply both sides of an equation by the same quantity, you still have equality.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ul>\n<li><strong>Translate a word sentence to an algebraic equation.<\/strong>\n<ol id=\"eip-id1170324011027\" class=\"stepwise\">\n<li>Locate the &#8220;equals&#8221; word(s). Translate to an equal sign.<\/li>\n<li>Translate the words to the left of the &#8220;equals&#8221; word(s) into an algebraic expression.<\/li>\n<li>Translate the words to the right of the &#8220;equals&#8221; word(s) into an algebraic expression.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-9573\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":37,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"}]","CANDELA_OUTCOMES_GUID":"160d148a7a4048f78f1b5c95e96ec4b9","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-9573","chapter","type-chapter","status-publish","hentry"],"part":6633,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/9573","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":16,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/9573\/revisions"}],"predecessor-version":[{"id":18886,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/9573\/revisions\/18886"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/6633"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/9573\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/media?parent=9573"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=9573"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=9573"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/license?post=9573"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}