{"id":222860,"date":"2025-06-01T03:55:48","date_gmt":"2025-06-01T03:55:48","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=222860"},"modified":"2025-06-01T03:55:51","modified_gmt":"2025-06-01T03:55:51","slug":"sally-has-two-iras-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/01\/sally-has-two-iras-2\/","title":{"rendered":"Sally has two IRAs"},"content":{"rendered":"\n

Sally has two IRAs. IRA #1 earns interest at 8% effective annually and IRA #2 earns interest at 10% effective annually. She has not made any contributions since January 1, 2005, when the amount in IRA #1 was twice the amount in IRA #2. The sum of the two accounts on January 1, 2013 was $73,000. Determine how much was in IRA #2 on January 1, 2005. ? < $12,750 ? $12,750 but < $13,000 ? $13,000 but < $13,250 ? $13,250 but < $13,500 ? $13,500
Sally has two IRAs. IRA #1 earns interest at 8% effective annually and IRA #2 earns interest at 10% effective annually. She has not made any contributions since January 1, 2005, when the amount in IRA #1 was twice the amount in IRA #2. The sum of the two accounts on January 1, 2013 was
12,750 ?
13,000 ?
13,250 ?
13,500 ? $13,500<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Let\u2019s solve the problem step-by-step and determine how much was in IRA #2 on January 1, 2005<\/strong>.<\/p>\n\n\n\n

\n\n\n\n

Step 1: Define variables<\/strong><\/h3>\n\n\n\n

Let:<\/p>\n\n\n\n

    \n
  • xx = amount in IRA #2 on Jan 1, 2005<\/li>\n\n\n\n
  • Then, IRA #1 on Jan 1, 2005 = 2x2x<\/li>\n\n\n\n
  • Time from Jan 1, 2005 to Jan 1, 2013 = 8 years<\/li>\n\n\n\n
  • IRA #1 grows at 8% effective annually<\/li>\n\n\n\n
  • IRA #2 grows at 10% effective annually<\/li>\n<\/ul>\n\n\n\n
    \n\n\n\n

    Step 2: Future values after 8 years<\/strong><\/h3>\n\n\n\n
      \n
    • IRA #1:
      2x\u00d7(1.08)82x \\times (1.08)^8<\/li>\n\n\n\n
    • IRA #2:
      x\u00d7(1.10)8x \\times (1.10)^8<\/li>\n<\/ul>\n\n\n\n

      Total value in 2013 = $73,000, so: 2x(1.08)8+x(1.10)8=73,0002x(1.08)^8 + x(1.10)^8 = 73,000<\/p>\n\n\n\n

      \n\n\n\n

      Step 3: Plug in interest values<\/strong><\/h3>\n\n\n\n

      Calculate:<\/p>\n\n\n\n

        \n
      • (1.08)8\u22481.85093(1.08)^8 \\approx 1.85093<\/li>\n\n\n\n
      • (1.10)8\u22482.14359(1.10)^8 \\approx 2.14359<\/li>\n<\/ul>\n\n\n\n

        Now plug in: 2x(1.85093)+x(2.14359)=73,0003.70186x+2.14359x=73,0005.84545x=73,0002x(1.85093) + x(2.14359) = 73,000 \\\\ 3.70186x + 2.14359x = 73,000 \\\\ 5.84545x = 73,000<\/p>\n\n\n\n

        Solve for xx: x=73,0005.84545\u224812,486.50x = \\frac{73,000}{5.84545} \\approx 12,486.50<\/p>\n\n\n\n

        \n\n\n\n

        Step 4: Determine the correct range<\/strong><\/h3>\n\n\n\n
          \n
        • x\u224812,486.50x \\approx 12,486.50<\/li>\n\n\n\n
        • So, IRA #2 had less than $12,750<\/strong> on Jan 1, 2005<\/li>\n<\/ul>\n\n\n\n
          \n\n\n\n

          \u2705 Correct answer: < $12,750<\/strong><\/h3>\n\n\n\n
          \n\n\n\n

          Explanation<\/strong><\/h3>\n\n\n\n

          This problem involves compound interest and solving a system involving exponential growth. Sally has two IRAs growing at different effective annual rates: IRA #1 at 8% and IRA #2 at 10%. She made no further contributions after January 1, 2005. At that time, the balance in IRA #1 was twice that of IRA #2. After 8 years, the combined balance of both accounts was $73,000.<\/p>\n\n\n\n

          To determine the original amount in IRA #2, we define the amount in IRA #2 in 2005 as xx, making the amount in IRA #1 2x2x. Over 8 years, IRA #1 grows to 2x\u00d7(1.08)82x \\times (1.08)^8, and IRA #2 grows to x\u00d7(1.10)8x \\times (1.10)^8. Adding these together and equating to $73,000 gives the equation: 2x(1.08)8+x(1.10)8=73,0002x(1.08)^8 + x(1.10)^8 = 73,000<\/p>\n\n\n\n

          Substituting the values (1.08)8\u22481.85093(1.08)^8 \\approx 1.85093 and (1.10)8\u22482.14359(1.10)^8 \\approx 2.14359, we simplify and solve for xx: 2x(1.85093)+x(2.14359)=73,000\u21d25.84545x=73,000\u21d2x\u224812,486.502x(1.85093) + x(2.14359) = 73,000 \\Rightarrow 5.84545x = 73,000 \\Rightarrow x \\approx 12,486.50<\/p>\n\n\n\n

          Thus, the amount in IRA #2 in 2005 was approximately $12,486.50<\/strong>, which is less than $12,750<\/strong>. This matches the answer choice “< $12,750”<\/strong>.<\/p>\n\n\n\n

          This kind of problem is common in actuarial exams and financial planning, requiring a solid understanding of exponential growth and financial equations.<\/p>\n\n\n\n

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

          Sally has two IRAs. IRA #1 earns interest at 8% effective annually and IRA #2 earns interest at 10% effective annually. She has not made any contributions since January 1, 2005, when the amount in IRA #1 was twice the amount in IRA #2. The sum of the two accounts on January 1, 2013 was […]<\/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-222860","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/222860","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=222860"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/222860\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=222860"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=222860"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=222860"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}