A pet store has only cats and dogs. The ratio of the number of cats to the number of dogs in the pet store is 2:3 : 1\/4 of the cats and 1\/2 of the dogs wear collars. If there are 48 animals that wear collars, how many total animals are there in the pet store?<\/p>\n\n\n\n
The Correct Answer and Explanation is:<\/strong><\/p>\n\n\n\n Let’s break down the problem step-by-step:<\/p>\n\n\n\n Let the number of cats be CC and the number of dogs be DD.<\/p>\n\n\n\n Given the ratio of cats to dogs: CD=23\u2005\u200a\u27f9\u2005\u200aC=23D\\frac{C}{D} = \\frac{2}{3} \\implies C = \\frac{2}{3} D<\/p>\n\n\n\n The total number of animals wearing collars is given as 48: 14C+12D=48\\frac{1}{4} C + \\frac{1}{2} D = 48<\/p>\n\n\n\n 14\u00d723D+12D=48\\frac{1}{4} \\times \\frac{2}{3} D + \\frac{1}{2} D = 48<\/p>\n\n\n\n Simplify: 212D+12D=48\\frac{2}{12} D + \\frac{1}{2} D = 48 16D+12D=48\\frac{1}{6} D + \\frac{1}{2} D = 48<\/p>\n\n\n\n Convert 12\\frac{1}{2} to sixths: 16D+36D=48\\frac{1}{6} D + \\frac{3}{6} D = 48 46D=48\\frac{4}{6} D = 48<\/p>\n\n\n\n Simplify 46\\frac{4}{6} to 23\\frac{2}{3}: 23D=48\\frac{2}{3} D = 48<\/p>\n\n\n\n D=48\u00d732=72D = 48 \\times \\frac{3}{2} = 72<\/p>\n\n\n\n C=23\u00d772=48C = \\frac{2}{3} \\times 72 = 48<\/p>\n\n\n\n Total=C+D=48+72=120\\text{Total} = C + D = 48 + 72 = 120<\/p>\n\n\n\n The pet store has 120 animals<\/strong> in total.<\/p>\n\n\n\n This problem involves a ratio and fractional parts applied to a group of animals. First, understanding the ratio of cats to dogs, 2:3, tells us that for every 2 cats, there are 3 dogs. We represent this relationship with variables CC (cats) and DD (dogs), expressing cats as a fraction of dogs: C=23DC = \\frac{2}{3}D.<\/p>\n\n\n\n Next, the problem mentions that a certain fraction of these animals wear collars: 1\/4 of cats and 1\/2 of dogs. We use this information to write an equation relating the number of animals wearing collars to the total number of animals: 14C+12D=48\\frac{1}{4}C + \\frac{1}{2}D = 48.<\/p>\n\n\n\n By substituting the ratio expression for cats (C=23D)(C = \\frac{2}{3}D) into the collar equation, we reduce the problem to an equation in terms of DD only. Simplifying this equation involves converting fractions to a common denominator and combining terms.<\/p>\n\n\n\n Solving the simplified equation gives the exact number of dogs D=72D = 72. Then, using the ratio again, we find the number of cats C=48C = 48. Adding these together provides the total number of animals in the store, which is 120.<\/p>\n\n\n\n This approach ensures the problem is tackled methodically\u2014using algebra and fraction manipulation\u2014resulting in a clear and accurate solution.<\/p>\n","protected":false},"excerpt":{"rendered":" A pet store has only cats and dogs. The ratio of the number of cats to the number of dogs in the pet store is 2:3 : 1\/4 of the cats and 1\/2 of the dogs wear collars. If there are 48 animals that wear collars, how many total animals are there in the pet […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-217274","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217274","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=217274"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217274\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=217274"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=217274"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=217274"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\n\n\nProblem Recap:<\/h3>\n\n\n\n
\n
\n\n\n\nStep 1: Define variables<\/h3>\n\n\n\n
\n\n\n\nStep 2: Calculate animals wearing collars<\/h3>\n\n\n\n
\n
\n\n\n\nStep 3: Substitute C=23DC = \\frac{2}{3} D into the collar equation<\/h3>\n\n\n\n
\n\n\n\nStep 4: Solve for DD<\/h3>\n\n\n\n
\n\n\n\nStep 5: Find CC<\/h3>\n\n\n\n
\n\n\n\nStep 6: Find total animals<\/h3>\n\n\n\n
\n\n\n\nAnswer:<\/strong><\/h3>\n\n\n\n
\n\n\n\nDetailed Explanation:<\/h3>\n\n\n\n