{"id":217176,"date":"2025-05-21T06:45:32","date_gmt":"2025-05-21T06:45:32","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=217176"},"modified":"2025-05-21T06:45:35","modified_gmt":"2025-05-21T06:45:35","slug":"emilys-piggy-bank-contains-twice-as-many-dimes-as-nickels-it-contains-two-more-quarters-than-nickels","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/21\/emilys-piggy-bank-contains-twice-as-many-dimes-as-nickels-it-contains-two-more-quarters-than-nickels\/","title":{"rendered":"Emily&#8217;s piggy bank contains twice as many dimes as nickels. It contains two more quarters than nickels."},"content":{"rendered":"\n<p>Emily&#8217;s piggy bank contains twice as many dimes as nickels. It contains two more quarters than nickels. Emily calculates that if she had as many quarters as she has nickels, as many dimes as she has quarters, and as many nickels as she has dimes, then she would have $1.50 less than she actually has. How many nickels, how many dimes, and how many quarters are actually in Emily&#8217;s piggy bank? Emilyhas nickels, dimes, and quarters in her piggy bank. Clear all<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>Let\u2019s define the number of coins using a variable and then solve step-by-step:<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Define the variables<\/h3>\n\n\n\n<p>Let:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>nn = number of <strong>nickels<\/strong><\/li>\n\n\n\n<li>d=2nd = 2n = number of <strong>dimes<\/strong> (twice as many dimes as nickels)<\/li>\n\n\n\n<li>q=n+2q = n + 2 = number of <strong>quarters<\/strong> (two more quarters than nickels)<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Calculate the actual value of the coins<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Value of nickels: 0.05n0.05n<\/li>\n\n\n\n<li>Value of dimes: 0.10\u22c52n=0.20n0.10 \\cdot 2n = 0.20n<\/li>\n\n\n\n<li>Value of quarters: 0.25(n+2)0.25(n + 2)<\/li>\n<\/ul>\n\n\n\n<p>So the <strong>actual value<\/strong> is: 0.05n+0.20n+0.25(n+2)0.05n + 0.20n + 0.25(n + 2)<\/p>\n\n\n\n<p>Simplify: 0.05n+0.20n+0.25n+0.50=0.50n+0.500.05n + 0.20n + 0.25n + 0.50 = 0.50n + 0.50<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Value if coins were swapped<\/h3>\n\n\n\n<p>According to the problem:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Quarters become nickels<\/strong> \u2192 value: 0.05(n+2)0.05(n + 2)<\/li>\n\n\n\n<li><strong>Dimes become quarters<\/strong> \u2192 value: 0.25\u22c52n=0.50n0.25 \\cdot 2n = 0.50n<\/li>\n\n\n\n<li><strong>Nickels become dimes<\/strong> \u2192 value: 0.10n0.10n<\/li>\n<\/ul>\n\n\n\n<p>Total <strong>new value<\/strong>: 0.05(n+2)+0.50n+0.10n0.05(n + 2) + 0.50n + 0.10n<\/p>\n\n\n\n<p>Simplify: 0.05n+0.10+0.50n+0.10n=0.65n+0.100.05n + 0.10 + 0.50n + 0.10n = 0.65n + 0.10<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Use the condition from the problem<\/h3>\n\n\n\n<p>Emily says this new total is $1.50 <strong>less<\/strong> than the actual amount. 0.50n+0.50=0.65n+0.10+1.500.50n + 0.50 = 0.65n + 0.10 + 1.50<\/p>\n\n\n\n<p>Simplify the right-hand side: 0.50n+0.50=0.65n+1.600.50n + 0.50 = 0.65n + 1.60<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 5: Solve for nn<\/h3>\n\n\n\n<p>Subtract 0.50n0.50n from both sides: 0.50=0.15n+1.600.50 = 0.15n + 1.60<\/p>\n\n\n\n<p>Subtract 1.60 from both sides: \u22121.10=0.15n-1.10 = 0.15n<\/p>\n\n\n\n<p>Solve for nn: n=\u22121.100.15=\u22127.33n = \\frac{-1.10}{0.15} = -7.33<\/p>\n\n\n\n<p><strong>This gives a negative value<\/strong>, which is <strong>not possible<\/strong>.<\/p>\n\n\n\n<p>Let\u2019s check the equation setup.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Error Check:<\/h3>\n\n\n\n<p>Emily says: <em>if she had as many <strong>quarters as nickels<\/strong>, as many <strong>dimes as quarters<\/strong>, and as many <strong>nickels as dimes<\/strong><\/em>.<\/p>\n\n\n\n<p>So she <strong>swaps<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>quarters \u2192 nickels: qq nickels<\/li>\n\n\n\n<li>dimes \u2192 quarters: dd quarters<\/li>\n\n\n\n<li>nickels \u2192 dimes: nn dimes<\/li>\n<\/ul>\n\n\n\n<p><strong>New value<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Nickels (was quarters): 0.05q0.05q<\/li>\n\n\n\n<li>Quarters (was dimes): 0.25d0.25d<\/li>\n\n\n\n<li>Dimes (was nickels): 0.10n0.10n<\/li>\n<\/ul>\n\n\n\n<p>So: New&nbsp;value=0.05q+0.25d+0.10n\\text{New value} = 0.05q + 0.25d + 0.10n<\/p>\n\n\n\n<p>Original value: 0.05n+0.10d+0.25q0.05n + 0.10d + 0.25q<\/p>\n\n\n\n<p>Now set up: 0.05n+0.10d+0.25q=0.05q+0.25d+0.10n+1.500.05n + 0.10d + 0.25q = 0.05q + 0.25d + 0.10n + 1.50<\/p>\n\n\n\n<p>Substitute d=2nd = 2n, q=n+2q = n + 2<\/p>\n\n\n\n<p>Left side: 0.05n+0.10(2n)+0.25(n+2)=0.05n+0.20n+0.25n+0.50=0.50n+0.500.05n + 0.10(2n) + 0.25(n + 2) = 0.05n + 0.20n + 0.25n + 0.50 = 0.50n + 0.50<\/p>\n\n\n\n<p>Right side: 0.05(n+2)+0.25(2n)+0.10n+1.50=0.05n+0.10+0.50n+0.10n+1.50=0.65n+1.600.05(n + 2) + 0.25(2n) + 0.10n + 1.50 = 0.05n + 0.10 + 0.50n + 0.10n + 1.50 = 0.65n + 1.60<\/p>\n\n\n\n<p>Now set: 0.50n+0.50=0.65n+1.600.50n + 0.50 = 0.65n + 1.60<\/p>\n\n\n\n<p>Same result: solve \u22121.10=0.15n\u21d2n=\u22127.33-1.10 = 0.15n \\Rightarrow n = -7.33<\/p>\n\n\n\n<p>Still wrong \u2014 let\u2019s try <strong>reinterpreting<\/strong> the swap:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>If she had:<\/p>\n<\/blockquote>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>as many quarters as nickels<\/strong> \u2192 quarters = nn<\/li>\n\n\n\n<li><strong>as many dimes as quarters<\/strong> \u2192 dimes = qq<\/li>\n\n\n\n<li><strong>as many nickels as dimes<\/strong> \u2192 nickels = dd<\/li>\n<\/ul>\n\n\n\n<p>Now, we recalculate the <strong>new value<\/strong>:<\/p>\n\n\n\n<p>New:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Nickels (now = d): 0.05d0.05d<\/li>\n\n\n\n<li>Dimes (now = q): 0.10q0.10q<\/li>\n\n\n\n<li>Quarters (now = n): 0.25n0.25n<\/li>\n<\/ul>\n\n\n\n<p>Total new value: 0.05d+0.10q+0.25n0.05d + 0.10q + 0.25n<\/p>\n\n\n\n<p>Original: 0.05n+0.10d+0.25q0.05n + 0.10d + 0.25q<\/p>\n\n\n\n<p>Now use: 0.05n+0.10d+0.25q=0.05d+0.10q+0.25n+1.500.05n + 0.10d + 0.25q = 0.05d + 0.10q + 0.25n + 1.50<\/p>\n\n\n\n<p>Use d=2nd = 2n, q=n+2q = n + 2:<\/p>\n\n\n\n<p>Left: 0.05n+0.10(2n)+0.25(n+2)=0.05n+0.20n+0.25n+0.50=0.50n+0.500.05n + 0.10(2n) + 0.25(n + 2) = 0.05n + 0.20n + 0.25n + 0.50 = 0.50n + 0.50<\/p>\n\n\n\n<p>Right: 0.05(2n)+0.10(n+2)+0.25n+1.50=0.10n+0.10n+0.20+0.25n+1.50=0.45n+1.700.05(2n) + 0.10(n + 2) + 0.25n + 1.50 = 0.10n + 0.10n + 0.20 + 0.25n + 1.50 = 0.45n + 1.70<\/p>\n\n\n\n<p>Now solve: 0.50n+0.50=0.45n+1.700.50n + 0.50 = 0.45n + 1.70<\/p>\n\n\n\n<p>Subtract: 0.05n=1.20\u21d2n=1.200.05=240.05n = 1.20 \\Rightarrow n = \\frac{1.20}{0.05} = 24<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Nickels<\/strong> n=24n = 24<\/li>\n\n\n\n<li><strong>Dimes<\/strong> d=2n=48d = 2n = 48<\/li>\n\n\n\n<li><strong>Quarters<\/strong> q=n+2=26q = n + 2 = 26<\/li>\n<\/ul>\n\n\n\n<p>\u2705 <strong>Correct and verified.<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Emily&#8217;s piggy bank contains twice as many dimes as nickels. It contains two more quarters than nickels. Emily calculates that if she had as many quarters as she has nickels, as many dimes as she has quarters, and as many nickels as she has dimes, then she would have $1.50 less than she actually has. [&hellip;]<\/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-217176","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217176","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=217176"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217176\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=217176"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=217176"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=217176"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}