{"id":216595,"date":"2025-05-19T15:17:38","date_gmt":"2025-05-19T15:17:38","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=216595"},"modified":"2025-05-19T15:17:40","modified_gmt":"2025-05-19T15:17:40","slug":"216595","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/19\/216595\/","title":{"rendered":""},"content":{"rendered":"\n

To solve this problem, we will use a system of equations<\/strong> based on the principles of weighted averages<\/strong> and total amount<\/strong>.<\/p>\n\n\n\n

\n\n\n\n

Let\u2019s define the variables:<\/strong><\/h3>\n\n\n\n
    \n
  • Let $x$ be the number of pounds of 31 cents per pound<\/strong> candy.<\/li>\n\n\n\n
  • Let $y$ be the number of pounds of 46 cents per pound<\/strong> candy.<\/li>\n<\/ul>\n\n\n\n

    We are told that the total mixture is 50 pounds<\/strong>, so:<\/p>\n\n\n\n

    $$
    x + y = 50 \\quad \\text{(Equation 1)}
    $$<\/p>\n\n\n\n

    The mixture costs 40 cents per pound<\/strong>, so the total cost of 50 pounds at 40 cents per pound is:<\/p>\n\n\n\n

    $$
    50 \\times 0.40 = 20 \\text{ dollars}
    $$<\/p>\n\n\n\n

    The cost of $x$ pounds of 31\u00a2 candy is:<\/p>\n\n\n\n

    $$
    0.31x
    $$<\/p>\n\n\n\n

    The cost of $y$ pounds of 46\u00a2 candy is:<\/p>\n\n\n\n

    $$
    0.46y
    $$<\/p>\n\n\n\n

    So the total cost equation is:<\/p>\n\n\n\n

    $$
    0.31x + 0.46y = 20 \\quad \\text{(Equation 2)}
    $$<\/p>\n\n\n\n

    \n\n\n\n

    Now solve the system:<\/strong><\/h3>\n\n\n\n

    From Equation 1:<\/p>\n\n\n\n

    $$
    y = 50 – x
    $$<\/p>\n\n\n\n

    Substitute into Equation 2:<\/p>\n\n\n\n

    $$
    0.31x + 0.46(50 – x) = 20
    $$<\/p>\n\n\n\n

    Distribute:<\/p>\n\n\n\n

    $$
    0.31x + 23 – 0.46x = 20
    $$<\/p>\n\n\n\n

    Combine like terms:<\/p>\n\n\n\n

    $$
    -0.15x + 23 = 20
    $$<\/p>\n\n\n\n

    Subtract 23 from both sides:<\/p>\n\n\n\n

    $$
    -0.15x = -3
    $$<\/p>\n\n\n\n

    Divide by -0.15:<\/p>\n\n\n\n

    $$
    x = 20
    $$<\/p>\n\n\n\n

    Now plug back in:<\/p>\n\n\n\n

    $$
    y = 50 – x = 50 – 20 = 30
    $$<\/p>\n\n\n\n

    \n\n\n\n

    \u2705 Final Answer:<\/strong><\/h3>\n\n\n\n
      \n
    • 20 pounds<\/strong> of 31\u00a2 candy<\/li>\n\n\n\n
    • 30 pounds<\/strong> of 46\u00a2 candy<\/li>\n<\/ul>\n\n\n\n
      \n\n\n\n

      Explanation (300+ words):<\/strong><\/h3>\n\n\n\n

      This is a classic mixture problem<\/strong> that involves combining two products with different costs to obtain a specific weight and price. The key to solving such problems is setting up equations that reflect both the quantity<\/strong> and the value<\/strong>.<\/p>\n\n\n\n

      We first define variables to represent how much of each type of candy is used. The total weight constraint gives one equation: $x + y = 50$, representing that the sum of the two candies must equal the final 50-pound mixture.<\/p>\n\n\n\n

      Next, we calculate the total cost of the final mixture. Since 50 pounds of candy is sold at 40 cents per pound, the total value is \\$20. This leads us to the second equation: the cost of the 31\u00a2 candy plus the cost of the 46\u00a2 candy must total \\$20. Using algebra, we substitute one variable (y) from the first equation into the second, which reduces the system to a single variable.<\/p>\n\n\n\n

      After simplifying and solving, we find that 20 pounds of the cheaper candy and 30 pounds of the more expensive candy produce a blend that averages out to 40 cents per pound. This makes intuitive sense too: since 40 cents is closer to 46 than 31, there must be more<\/strong> of the more expensive candy.<\/p>\n\n\n\n

      This problem demonstrates how algebra can be used to solve real-world scenarios involving cost-efficiency and product blending.<\/p>\n\n\n\n

      <\/p>\n","protected":false},"excerpt":{"rendered":"

      To solve this problem, we will use a system of equations based on the principles of weighted averages and total amount. Let\u2019s define the variables: We are told that the total mixture is 50 pounds, so: $$x + y = 50 \\quad \\text{(Equation 1)}$$ The mixture costs 40 cents per pound, so the total cost […]<\/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-216595","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/216595","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=216595"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/216595\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=216595"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=216595"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=216595"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}