{"id":37091,"date":"2025-06-25T12:51:36","date_gmt":"2025-06-25T12:51:36","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=37091"},"modified":"2025-06-25T12:51:42","modified_gmt":"2025-06-25T12:51:42","slug":"a-jar-of-coins-contains-nickels-dimes-and-quarters","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/a-jar-of-coins-contains-nickels-dimes-and-quarters\/","title":{"rendered":"A jar of coins contains nickels, dimes, and quarters"},"content":{"rendered":"\n<p>A jar of coins contains nickels, dimes, and quarters. The total number of coins is 12 and the total value is $2.50. How many of each coin are there?<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To find the number of each type of coin in the jar, we define:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Let <strong>n<\/strong> be the number of nickels<\/li>\n\n\n\n<li>Let <strong>d<\/strong> be the number of dimes<\/li>\n\n\n\n<li>Let <strong>q<\/strong> be the number of quarters<\/li>\n<\/ul>\n\n\n\n<p>We are given two pieces of information:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>The total number of coins is 12:<br><strong>n + d + q = 12<\/strong><\/li>\n\n\n\n<li>The total value is $2.50, which is 250 cents:<br><strong>5n + 10d + 25q = 250<\/strong><\/li>\n<\/ol>\n\n\n\n<p>To solve this, we can simplify the second equation by dividing every term by 5:<br><strong>n + 2d + 5q = 50<\/strong><\/p>\n\n\n\n<p>Now we have a system of two equations:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>n + d + q = 12<\/li>\n\n\n\n<li>n + 2d + 5q = 50<\/li>\n<\/ol>\n\n\n\n<p>We subtract the first equation from the second:<br>(n + 2d + 5q) &#8211; (n + d + q) = 50 &#8211; 12<br>n cancels out:<br><strong>(2d &#8211; d) + (5q &#8211; q) = 38<\/strong><br><strong>d + 4q = 38<\/strong><\/p>\n\n\n\n<p>Now solve for d:<br><strong>d = 38 &#8211; 4q<\/strong><\/p>\n\n\n\n<p>Now substitute back into the first equation:<br>n + d + q = 12<br>n + (38 &#8211; 4q) + q = 12<br>n = 12 &#8211; (38 &#8211; 4q + q)<br>n = 12 &#8211; (38 &#8211; 3q)<br>n = -26 + 3q<\/p>\n\n\n\n<p>Now test small integer values for q (since q must be a whole number):<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If q = 4:<br>d = 38 &#8211; 4(4) = 22<br>n = -26 + 3(4) = -14 (not valid)<\/li>\n\n\n\n<li>If q = 5:<br>d = 38 &#8211; 20 = 18<br>n = -26 + 15 = -11 (not valid)<\/li>\n\n\n\n<li>If q = 6:<br>d = 38 &#8211; 24 = 14<br>n = -26 + 18 = -8 (not valid)<\/li>\n\n\n\n<li>If q = 7:<br>d = 38 &#8211; 28 = 10<br>n = -26 + 21 = -5 (not valid)<\/li>\n\n\n\n<li>If q = 8:<br>d = 38 &#8211; 32 = 6<br>n = -26 + 24 = -2 (still invalid)<\/li>\n\n\n\n<li>If q = 9:<br>d = 38 &#8211; 36 = 2<br>n = -26 + 27 = 1 \u2192 valid<\/li>\n<\/ul>\n\n\n\n<p>Now check if n + d + q = 1 + 2 + 9 = 12 \u2714<br>Check value:<br>5(1) + 10(2) + 25(9) = 5 + 20 + 225 = 250 cents \u2714<\/p>\n\n\n\n<p><strong>Answer:<\/strong><br>Nickels: 1<br>Dimes: 2<br>Quarters: 9<\/p>\n\n\n\n<p>This is the only combination of non-negative integers that satisfies both conditions: a total of 12 coins and a value of $2.50.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A jar of coins contains nickels, dimes, and quarters. The total number of coins is 12 and the total value is $2.50. How many of each coin are there? The correct answer and explanation is: To find the number of each type of coin in the jar, we define: We are given two pieces of [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-37091","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/37091","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=37091"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/37091\/revisions"}],"predecessor-version":[{"id":37095,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/37091\/revisions\/37095"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=37091"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=37091"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=37091"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}